Necessary and sufficient conditions for global asymptotic stability of a class of bimodal systems with discontinuous vector fields

Necessary and sufficient conditions for global asymptotic stability of a class of bimodal systems... Abstract In this article, we derive the necessary and sufficient conditions for global asymptotic stability of a class of bimodal piecewise linear systems in $$\mathbb{R}^{3}$$, where first mode has complex eigenvalues and the second mode has only real eigenvalues. The vector field is allowed to be discontinuous on the switching plane. The effect of discontinuity is illustrated by two examples. 1. Introduction During the past 20 years, the research on switched systems has gained a great momentum. This is basically due to the fact that there are many practical control applications which contain both continuous time dynamical systems and switching elements. The literature contains a rich variety of mathematical models proposed to represent behaviour of hybrid control systems. One of the typical models is the switched system which comprises a finite number of subsystems and a switching rule that determines the active subsystem at each instant. Piecewise linear systems (PLS) comprise a subclass of switched systems where the subsystems are linear, time invariant and switching is autonomous (state dependent). The class of PLS is one of the fundamental classes of hybrid dynamical systems, because the continuous dynamics is linear in each mode and the discrete dynamics is the simplest one. Therefore, study on PLS is important as a first step to constitute hybrid control theory. Bimodal piecewise linear system (BPLS) comprises a subclass of PLS, where there are only two subsystems. In spite of recent progress in hybrid control theory, there still remain fundamental issues to be clarified. Stability is one of them. One of the approaches used for the solution of stability problems with arbitrary switching is based on the existence of a common quadratic Lyapunov function for individual subsystems. Along this line, the works of Mori et al. (1997), Shorten & Narendra (1999), Shorten et al. (2004) and Sun (2010) can be cited. If a switched system has more than two subsystems, then finding a common quadratic Lyapunov function becomes a difficult task. In such cases, another approach, which is based on a nontraditional Lyapunov functions is employed. For the details of this approach, the works of Branicky (1998), Michel & Hu (1999), Liberzon & Morse (1999) and the survey papers by Decarlo et al. (2000), Shorten et al. (2007) may be referred to. The approaches, which are based on Lyapunov functions, usually yield sufficient conditions for stability. Therefore, we need a new approach to get a less conservative stability condition or to derive a set of necessary and sufficient conditions for the stability. To this end, we investigate the stability problem for BPLS from a different perspective. One of the main issues in BPLS is well posedness, i.e. the existence and uniqueness of the solutions. This problem is addressed in detail by Imura & van der Schaft (2000) for BPLS in $$\mathbb{R}^{n}$$. Later, it is shown by Eldem & Şahan (2014) and Şahan & Eldem (2015) that well-posedness conditions, given by Imura & van der Schaft (2000), induce a joint structure for subsystem matrices of BPLS in $$\mathbb{\mathbb{R}}^{n }$$. PLS are investigated extensively by many authors in the context of stability. The necessary and sufficient conditions for global asymptotic stability (GAS) of BPLS (with continuous vector fields) in $$\mathbb{\mathbb{R}}^{2}$$ is given by Çamlibel et al. (2003). The same problem (with discontinuous vector fields) is investigated by Iwatani & Hara (2006) and the importance of well-posedness is demonstrated by an example (Example 13 in Iwatani & Hara, 2006). Stability of BPLS in $$\mathbb{\mathbb{R}}^{3}$$ has also attracted considerable attention in the literature. Carmona et al. (2005, 2006) have considered the stability of BPLS in $$\mathbb{\mathbb{R}}^{3}$$ with continuous vector fields. In these papers, BPLS is transformed to the surface of the unit sphere in $$\mathbb{\mathbb{R}}^{3}$$ centred at the origin. In this framework, the authors searched for periodic solutions which is equivalent to the search for invariant cones of the original BPLS. Separate necessary and sufficient conditions for GAS of BPLS in $$\mathbb{\mathbb{R}}^{n},$$ where $$n>2$$ is given by Iwatani & Hara (2006). It is shown by Eldem & Şahan (2014) that the GAS of BPLS in $$\mathbb{\mathbb{R}}^{3}$$, where both modes have complex eigenvalues, reduces to the GAS of BPLS in $$\mathbb{\mathbb{R}}^{2}$$, if BPLS in $$\mathbb{R}^{3}$$ has a certain structure. Additionally, Eldem & Öner (2015) give necessary and sufficient conditions for GAS of BPLS where one mode has complex eigenvalues and the second mode have only real eigenvalues with algebraic multiplicity equal to three and geometric multiplicity is equal to one. In this article, the GAS of a class of BPLS in $$\mathbb{R}^{3}$$ is investigated under autonomous switching where first mode has complex eigenvalues and the second mode has only real eigenvalues. For the second mode, we have four cases of algebraic and geometric multiplicity, but we investigate only three cases. Because, for the case where the eigenvalues are $$\mu _{1}=\mu _{2}=\mu _{3},$$ the necessary and sufficient stability conditions of GAS of BPLS is given by Eldem & Öner (2015). The cases to be studied in this article create certain difficulties for the proof of the main result. Therefore, we need an additional assumption to overcome these difficulties. The article is organized as follows. Well-posedness conditions for BPLS and the behaviour of trajectories will be given in Section II. The behaviour of the trajectories is investigated towards a classification of trajectories. Stability of the classified trajectories is given in Section III. In Section IV, the main theorem of this article and some examples, which illustrate the theoretical results, are presented. Finally, the article ends with the conclusions in Section V. In order to make the reading smoother, the proofs of some of the results are given in the Appendix. 2. Preliminaries Throughout this article, we study GAS of the class of BPLS with the following mathematical model: Σ0:x˙={A1xifcTx≥0A2xifcTx≤0  (2.1) where $$\mathbf{x,c}\in \mathbb{\mathbb{R}}^{3}$$ and $$\mathbf{A}_{1},\mathbf{A}_{2}$$ are real $$3\times 3$$ matrices. For such systems, $$\mathcal{H}:=\left\{ \mathbf{x}\text{ }|\text{ }\mathbf{c}^{T}\mathbf{x}=0\right\} $$ denotes the plane which divides $$\mathbb{\mathbb{R}}^{3}$$ into two open half-spaces, $$\mathcal{H}^{+}=\left\{ \mathbf{x}\text{ }|\text{ }\mathbf{c}^{T}\mathbf{x>}0\right\} $$ and $$\mathcal{H}^{-}=\left\{ \mathbf{x}\text{ }|\text{ }\mathbf{c}^{T}\mathbf{x}<0\right\}\!.$$ Since both equations hold on $$\mathcal{H},$$ we first resolve the existence and uniqueness of the solutions (well-posedness) starting on $$\mathcal{H}$$. 2.1. Well-posedness and geometry of initial conditions in $$\mathbb{R}^{3}$$ Well-posedness simply means the existence and uniqueness of solutions. The first results about well-posedness of BPLS in $$\mathbb{R}^{n}$$ are given by Imura & van der Schaft (2000). Then, the structural issues of well-posednes are further investigated in $$\mathbb{R}^{3}$$ and $$\mathbb{R}^{n}$$ by Eldem & Şahan (2014) and Şahan & Eldem (2015), respectively. Classical solutions (which are continuously differentiable curves) fail to describe the behaviour of dynamical systems with discontinuous vector fields. In order to remedy this situation different solution notions (such as Carathéodory, Filippov, Krasovskii, sample-and-hold, Euler, Hermes...etc.) are introduced in the literature. An extensive account of these solution notions can be found in Cortes (2008). Among these, Carathéodory and Filippov solutions are the ones which are used most frequently. As explained in Cortes (2008), Carathéodory solutions can be considered as a generalization of classical solutions where the solution does not satisfy the differential equation on a set of measure zero. On the other hand, Filippov solutions replace the differential equations with differential inclusions which are defined by set-valued maps. Thus, instead of focusing the value of the vector field at a specific point $$x$$, Filippov solutions focus on a set of directions in the neighbourhood of $$x$$. Definitions of these solution notions are as follows. Definition 2.1 An absolutely continuous function $$x\mathbf{(}t\mathbf{):\mathbb{R} \rightarrow \mathbb{R}}^{n}$$ is said to be 1. Carathéodory solution of $$\it{\Sigma}_{0}$$ for the initial condition $$x_{0}$$ if $$x\mathbf{(}0\mathbf{)=} x_{0}$$ and $$x\mathbf{(}t\mathbf{)}$$ satisfies (2.1) for almost all $$t\in \mathbf{{\mathbb{R}}}$$. 2. Forward Carathéodory solution for the initial state $$x_{0},$$ if it is a solution in the sense of Carathéodory, and for each $$t_{0}\geq 0,$$ there exists $$\epsilon >0$$ such that either x˙=A1xandcTx≥0,orx˙=A2xandcTx≤0  holds for all $$t\in \left[t_{0},t_{0}+\epsilon \right] $$. 3. Filippov solution if $$x(t)$$ satisfies $${\dot{x}}\left(t\right) \in F\left(x\left(t\right) \right) $$ for almost all $$t\geq 0$$ where set valued function $$F$$ is given by F(x)={{A1x}ifcTx>0conv({A1x,A2x})if cTx=0{A2x}ifcTx<0  and $$conv(S)$$ denotes the convex hull of the set $$S.$$ In view of intuitive explanations and the definitions above, it is clear that Carathéodory and Filippov solutions are not related in general. In other words, there are vector fields with Carathéodory solutions which are not Filippov solutions and converse is also true. However, there are also vector fields where Carathéodory and Filippov solutions coincide. These fundamental questions on the relation between Carathéodory and Filippov solutions are investigated in Spraker & Biles (1996). For instance, Theorems 1 and 2 of Spraker & Biles (1996), provide conditions for the equivalence of Carathéodory and Filippov solutions. A detailed analysis of Carathéodory and Filippov solutions in the context of BPLS with discontinuous vector fields is given in Thuan & Çamllibel (2014). In Theorem 3.1 of Thuan & Çamllibel (2014), it is shown that if the equations in statements 5 and 6 of Theorem 3.1 hold, then every Filippov solution of the system is right unique and every Filippov solution is also both a forward and backward Carathéodory solution. It can be easily shown that equations in statements 5 and 6 of Theorem 3.1 in Thuan & Çamllibel (2014), also hold in our setup. In fact Assumptions 2.1 and 2.2 given below imply that the equations in statements 5 and 6 of Theorem 3.1 in Thuan & Çamllibel (2014) hold for bimodal systems considered in our article. Furthermore, bimodal systems described in Corollary 3.5 of Thuan & Çamllibel (2014) are exactly the systems described by equation (2.1) with the structure of (2.2) given below. Therefore, under the assumptions which guarantee well-posedness, it follows that forward Carathéodory solutions of (2.1) with the structure of (2.2) are also Filippov solutions. Consequently, the results on GAS presented in this article are also valid for Filippov solutions. In order to establish well-posedness, we need some assumptions. Assumption 2.1 The pairs $$\left(\mathbf{c}^{T}\boldsymbol{,\ } \mathbf{A}_{1}\right)$$ and $$\left(\mathbf{c}^{T} \boldsymbol{,}\mathbf{A}_{2}\right) $$ are observable and only $$ \left(\mathbf{c}^{T}\boldsymbol{,}\mathbf{A}_{2}\right) $$ is in observable canonical form. In view of this assumption the parameters of BPLS can be expressed as follows. A1=[a11 a12 a13 a21a22a23a31a32a33], A2=[00μ1μ2μ310−μ1μ2−μ1μ3−μ2μ301μ1+μ2+μ3]and c=[001]. The observability of the pair $$\left(\mathbf{c}^{T}\boldsymbol{,}\mathbf{A}_{i}\right),$$$$i=1,2$$ implies that $$\dim \left(\ker \mathbf{c}^{T}\cap {\ker }(\mathbf{c}^{T}\mathbf{A}_{i})\right) {\small =1}$$. Thus, $$\mathcal{L}_{i}:={\ker }\mathbf{c}^{T}\cap {\ker }(\mathbf{c}^{T}\mathbf{A}_{i})$$ is a line passing through the origin which divides $$\mathcal{H}$$ into two open half planes $$\mathcal{P}_{i}^{+}$$ and $$\mathcal{P}_{i}^{-}$$. On one side of this line $$\mathbf{c}^{T}\mathbf{A}_{i}\mathbf{x}>0$$$$\left(\mathcal{P}_{i}^{+}\right) $$ and one the other side $$\mathbf{c}^{T}\mathbf{A}_{i} \mathbf{x}<0$$$$\left(\mathcal{P}_{i}^{-}\right) $$. Similarly, the origin $${ \ker }\mathbf{c}^{T}\cap {\ker }\boldsymbol{(}\mathbf{c}^{T}\mathbf{A}_{i})\cap {\ker }(\mathbf{c}^{T}\mathbf{A}_{i}^{2})$$ divides $$\mathcal{L}_{i} $$ into two open half lines $$\mathcal{L}_{i}^{+}$$ where $$\mathbf{c}^{T} \mathbf{A}_{i}^{2}\mathbf{x}>0$$ and $$\mathcal{L}_{i}^{-}$$ where $$\mathbf{c} ^{T}\mathbf{A}_{i}^{2}\mathbf{x}<0$$. In view of the above development, and along the lines used in Imura & van der Schaft (2000), the set of initial conditions in $$\mathbb{R}^{3}$$ can be classified as follows. Definition 2.2 Let $$\mathcal{S}_{i}$$$$\left(i=1,2\right) $$ denote the set of initial conditions in $$\mathbb{R}^{3}$$ such that for every $$x_{0}\in \mathcal{S}_{i}$$ there exist $$\epsilon >0$$ and a unique forward Carathéodory solution of $$\it{\Sigma}_{0}$$ and $$\dot{x}\mathbf{=}A_{i}x$$ for all $$t\in \left[t_{0},t_{0}+\epsilon \right] $$. In this case, we say that the solution of$$\it{\Sigma}_{0}$$smoothly continues in$$\mathcal{S}_{i}$$. In view of the above definitions and Lemma 1 given in Eldem & Şahan (2014) or items (1) and (2) of Theorem 3.1 given in Şahan & Eldem (2015), we have the following assumption. Assumption 2.2 $$\ker c^{T}\cap \ker \left(c^{T}\mathbf{{A}_{1}} \right) =\ker c^{T}\cap \ker \left(c^{T}\mathbf{{A}_{2}}\right) $$ (equivalently $$a_{31}=0$$) and $$a_{21}>0$$, $$a_{32}>0,$$ i.e., $$\it{\Sigma}_{0}$$ is well-posed. It is easy to see that for a well-posed BPLS, we have $$\mathcal{P}_{1}^{+}=\mathcal{P}_{2}^{+},$$$$\mathcal{P}_{1}^{-}=\mathcal{P}_{2}^{-},$$$$\mathcal{L}_{1}^{+}= \mathcal{L}_{2}^{+},$$ and $$\mathcal{L}_{1}^{-}=\mathcal{L}_{2}^{-}$$. Consequently, it follows that $$\mathcal{S}_{1}=\mathcal{H}^{+}\cup \mathcal{P }_{1}^{+}\cup $$$$\mathcal{L}_{1}^{+}$$ and $$\mathcal{S}_{2}=\mathcal{H} ^{-}\cup \mathcal{P}_{2}^{-}\cup $$$$\mathcal{L}_{2}^{-}$$. This implies that $$\mathcal{S}_{1}\cup \mathcal{S}_{2}=\mathbb{R}^{3}$$ and $$\mathcal{S}_{1}\cap \mathcal{S}_{2}=\left\{ 0\right\} $$ and this guarantees well-posedness as defined in Imura & van der Schaft (2000). Under these assumptions, bimodal system is well-posed which guarantees the following fact. Fact 1: The vector field of both modes has the same sign on any initial condition on $$\mathcal{H}$$. Furthermore, the components of $$\it{\Sigma}_{0}$$ can be written as follows. A1=[a11 a12 a13 a21a22a230a32a33], A2=[00μ1μ2μ310−μ1μ2−μ1μ3−μ2μ301μ1+μ2+μ3] and c=[001]. (2.2) The last assumption, which is not related to well-posedness is given below. Assumption 2.3 The eigenvalues of $$\mathbf{A}_{1}$$ are $$\{\lambda_{1},\sigma_{1}\pm j\omega_{1}\}$$ where $$\lambda_{1}$$, $$\sigma_{1}$$ and $$ \omega_{1}>0$$ are real numbers such that $$\sigma_{1}-\lambda_{1}>0$$. The eigenvalues of $$\mathbf{A}_{2}$$ are $$\{\mu_{1},\mu_{2},\mu_{3}\}$$ where $$ \mu_{1},\mu_{2},$$ and $$\mu_{3}$$ are real numbers such that $$\mu_{1}\leq \mu_{2}\leq \mu_{3}.$$ 2.2. Behaviour of the trajectories in the first mode: The properties of the trajectories in the first mode are investigated in detail in Eldem & Şahan (2014). We give a summary of these results below. Let the eigenvectors of $$\mathbf{{A}_{1}}$$ be $$\left\{ \mathbf{r}_{1},\text{ }\mathbf{x}_{1}\pm j \mathbf{y}_{1}\right\} $$ where $$j=\sqrt{-1}$$. Since the solutions of $$\it{\Sigma}_{0}$$ starting from $$\mathcal{S}_{1}$$ can be expressed by eigenvectors of $$\mathbf{{A}_{1},}$$$$\left\{ \mathbf{r}_{1}, \text{ }\mathbf{x}_{1}\pm j\mathbf{y}_{1}\right\} $$ are chosen uniquely as follows. r1=[(a11−σ1)2+ω12+a12a21a32a21λ1−a33a321],x1=[(a11−λ1)(a11−σ1)+a12a21a32a21σ1−a33a321],y1=[ω1(a11−λ1)a32a21ω1a320]. (2.3) Let $$\mathbf{z}_{1}(t)$$ denote the solutions of $$\it{\Sigma}_{0}$$ with initial conditions in $$\mathcal{S}_{1}$$. Then, the behaviour of such trajectories (except the trajectories with $$\mathbf{z}_{1}(0)=$$$$K_{1}\mathbf{r}_{1},$$ where $$K_{1}>0$$) in the first mode can be expressed as follows. z1(t)=K1eλ1t{α1r1+e(σ1−λ1)t[x1sin⁡(θ1+ω1t)+y1cos⁡(θ1+ω1t)]}, (2.4) where $$K_{1}>0$$ is a real constant. Let f1(t):=α1+e(σ1−λ1)tsin⁡(θ1+ω1t). (2.5) Note that for any initial condition in $$\mathcal{S}_{1}$$, we have $$\mathbf{c} ^{T}\mathbf{z}_{1}(0)\geq 0$$. Then, it follows that cTz1(t)=K1eλ1t{α1+e(σ1−λ1)tsin⁡(θ1+ω1t)},=K1eλ1t{f1(t)}, (2.6) and $$\mathbf{c}^{T}\mathbf{z}_{1}(0)=K_{1}f_{1}(0)=K_{1}\left(\alpha_{1}+\sin \theta_{1}\right) \geq 0$$. Further note that the sign of $$\mathbf{c}^{T}\mathbf{z}_{1}(t)$$ and $$f_{1}(t)$$ are the same. The trajectories of $$\it{\Sigma}_{0}$$ may or may not change mode as shown in Eldem & Şahan (2014). From the point of view of stability, it is important to distinguish the trajectories which do not change mode. Towards this end, we introduce the following definitions and Lemmas from Eldem & Şahan (2014). Definition 2.3 (Eldem & Şahan, 2014, Definition 1) Let $$\mathbf{z}_{i}(t)$$ be a solution of $$\it{\Sigma}_{0}$$ with initial condition in $$\mathcal{S}_{i}$$$$\left(i=1,2\right) $$. If there exists a finite $$\tau_{i}>0$$ such that $$\mathbf{c}^{T}\mathbf{z}_{i}\left(\tau_{i}\right) =0$$ and $$\mathbf{z}_{i}(t)$$ changes mode at $$t=\tau_{i}$$, then $$ \mathbf{z}_{i}(t)$$ is called a transitive trajectory. Otherwise, it is called a non-transitive trajectory. Remark 2.1 The classification given in the above definition can be obtained easily by using equations (2.4)-(2.6). Towards this end, note that for any initial condition on $$\mathcal{H}\cap \mathcal{S}_{1}$$, we have, $$\alpha_{1}=-\sin {\theta_{1}}$$ in equation (2.6). Therefore, any initial condition $$\mathbf{z}_{1}(0)$$ in $$\mathcal{H}\cap \mathcal{S}_{1}$$ can be written as follows. z1(0)=K1(x^1sin⁡θ1+y1cos⁡θ1), where x^1:=x1−r1 and K1>0. (2.7) In order to determine the domain of $$\sin \theta_{1}$$ for initial conditions in $$\mathcal{H}\cap \mathcal{S}_{1}$$, the time derivative of $$ \mathbf{c^{T}z}_{1}(t)$$ at $$t=0$$ can be calculated easily as follows. cTA1z1(0)=R1sin⁡(θ1+ϕ1) where R1:=(σ1−λ1)2+ω12, b1:=cot⁡(ϕ1):=σ1−λ1ω1. (2.8) It is easy to see that $$\mathbf{c^{T}A_{1}z}_{1}\left(0\right) =0$$ and $$ \mathbf{c^{T}A_{1}^{2}z}_{1}\left(0\right) >0$$ at $$\theta_{1}=-\phi_{1}$$ whereas $$\mathbf{c^{T}A_{1}z}_{1}\left(0\right) =0$$ and $$\mathbf{c^{T}A_{1}^{2}z}_{1}\left(0\right) <0$$ at $$\theta_{1}=\pi -\phi_{1}$$. Consequently, the set of initial conditions in $$\mathcal{H}\cap \mathcal{S}_{1}$$ can be characterized as in equation (2.7) where $$-\phi_{1}\leq \theta_{1}<\pi -\phi_{1}$$. Similarly, in view of Fact 1, the set of initial conditions in $$\mathcal{H}\cap \mathcal{S}_{2}$$ can be characterized as in equation (2.7) where $$\pi -\phi_{1}\leq \theta_{1}<2\pi -\phi_{1}$$. On the other hand, let $$\tau_{1}>0$$ be the first instant where $$\mathbf{c^{T}z}_{1}(\tau_{1})=0$$. Then, it follows that z1(τ1)=K1eσ1τ1(x^1sin⁡(θ1+ω1τ1)+y1cos⁡(θ1+ω1τ1)) (2.9) e(σ1−λ1)τ1sin⁡(θ1+ω1τ1)=sin⁡θ1. (2.10) Since $$\sigma_{1}-\lambda_{1}>0$$ by Assumption 2.3, the existence of $$\tau_{1}>0$$ such that $$\mathbf{c^{T}z}_{1}(\tau_{1})=0$$ is obvious. Note further that since $$e^{-\left(\sigma_{1}-\lambda_{1}\right) \tau_{1}}\sin \theta_{1}$$ is an exponentially decreasing function and $$\sin \left(\theta_{1}+\omega_{1}\tau_{1}\right) $$ is a sinusoidal function, equation (2.10) implies that their first intersection after the start can be written as follows. if−ϕ1≤θ1≤0,⇒π≤θ1+ω1τ1<π+ϕ1, (2.11)  if 0≤θ1<π−ϕ1,⇒π−ϕ1<θ1+ω1τ1≤π. (2.12) If we define $$F_{1}(\theta_{1}):=\theta_{1}+\omega_{1}\tau_{1}$$. Then, using implicit differentiation in equation (2.10), it can be easily shown that dF1dθ1=cot⁡θ1+b1cot⁡(θ1+ω1τ1)+b1≤0, (2.13) where the equality holds only at $$\theta_{1}=-\phi_{1}$$. The details of the development given above can be found in Lemma’s 3,4, and 5 in Eldem & Şahan (2014). Note that $$\tau_{1}$$ and $$\theta_{1}+\omega_{1}\tau_{1}$$ given above are functions of $$\theta_{1}$$. In order to simplify the notation, we use θ^1:=θ1+ω1τ1 (2.14) in the rest of the article. 2.3. Behaviour of the trajectories in the second mode: The properties of trajectories starting from $$\mathcal{S}_{2}$$ are investigated in detail in Öner (2014). We give a summary of these results below without proof. The proofs can be found in Öner (2014). Since the solutions which start from $$\mathcal{S}_{2}$$ can be expressed in terms of eigenvectors of $$A_2$$, the choice of eigenvectors must be done appropriately as given by the following Lemma. Lemma 2.1 (Lemma 4.1 in Öner (2014)) Let $$\{\mathbf{s}_{1}\boldsymbol{,}\;\mathbf{s}_{2}\boldsymbol{,}\; \mathbf{s}_{3}\}$$ denote the eigenvectors or generalized eigenvectors of $$ \mathbf{A}_{2}$$. Then, eigenvectors or generalized eigenvectors can be uniquely chosen such that s1=[−μ2μ3μ2+μ3−1],s2=[−μ1μ3μ1+μ3−1],s3=[−μ1μ2μ1+μ2−1] for μ1<μ2<μ3. (2.15) For the other cases we have if μ1=μ2<μ3 then s2=[μ3−10]T. if μ1<μ2=μ3 then s3=[μ1−10]T. Let $$\mathbf{z}_{2}(t)$$ be a solution of $$\it{\Sigma}_{0}$$ with initial condition in $$\mathcal{S}_{2}$$. Then, the behaviour of such trajectories in the second mode can be written as z2(t)={d1eμ1ts1+d2eμ2tq2(t)+d3eμ3tq3(t)}, where $$d_{1},d_{2},d_{3}$$ are real constants and $$\{\mathbf{s}_{1} \boldsymbol{,}\mathbf{q}_{2}\left(t\right) \boldsymbol{,}\mathbf{q}_{3}\left(t\right) \}$$ are vector functions in $$\mathbb{R}^{3}$$ given as follows. ifμ1<μ2<μ3, then q2(t):=s2,q3(t):=s3, ifμ1=μ2<μ3, then q2(t):=s1t+s2,q3(t):=s3,ifμ1<μ2=μ3, then q2(t):=s2,q3(t):=s2t+s3,  (2.16) Let $$K_{2}:=\sqrt{d_{2}^{2}+d_{3}^{2}},$$$$\sin \theta_{2}:=\frac{d_{2}}{K_{2}},$$$$\cos \theta_{2}:=\frac{d_{3}}{K_{2}},$$ and $$\beta_{1}=\frac{d_{1} }{K_{2}}$$. Then, $$\mathbf{z}_{2}\mathbf{(}t\mathbf{)}$$ and $$\mathbf{c}^{T} \mathbf{z}_{2}\mathbf{(}t\mathbf{)}$$ can be written as follows. z2(t)=K2eμ1t[β1s1+q2(t)e(μ2−μ1)tsin⁡θ2+q3(t)e(μ3−μ1)tcos⁡θ2]. (2.17) cTz2(t)=K2eμ1t[−β1+(cTq2(t))e(μ2−μ1)tsin⁡θ2+(cTq3(t))e(μ3−μ1)tcos⁡θ2]. (2.18) Let $$f_{2}(t)$$ be defined as f2(t):=[−β1+(cTq2(t))e(μ2−μ1)tsin⁡θ2+(cTq3(t))e(μ3−μ1)tcos⁡θ2]. (2.19) Since $$\mathbf{c}^{T}\mathbf{z}_{2}\mathbf{(}t\mathbf{)}=K_{2}e^{\mu_{1}t}f_{2}(t)$$, the sign of $$f_{2}(t)$$ and the sign of $$\mathbf{c}^{T}\mathbf{z}_{2}\mathbf{(}t\mathbf{)}$$ are the same. 2.4. Smooth continuation of trajectories and bases for $$\mathcal{H}$$ In view of equation (2.18), a trajectory $$\mathbf{z}_{2}(t)$$ with initial condition not on $$\mathcal{H}$$, smoothly continues into $$\mathcal{S}_{2}$$ if $$f_{2}(0)<0$$. For trajectories starting from $$\mathcal{H}$$, the following result holds. Lemma 2.2 Suppose that $$\mathbf{c}^{T}\mathbf{z}_{2}(0)=0.$$ Then, $$\mathbf{z}_{2}(t)\,$$ smoothly continues into $$\mathcal{S}_{2}$$ if and only if $$-\phi_{2}\leq \theta_{2}<\pi -\phi_{2}$$, where cot⁡ϕ2=μ2−μ1μ3−μ1ifμ3>μ2>μ1,cot⁡ϕ2=1μ3−μ1ifμ3>μ2=μ1,cot⁡ϕ2=μ3−μ1ifμ3=μ2>μ1. We have two different bases for $$\mathcal{H}$$, namely $$\left\{ \hat{\mathbf{x}}_{1},\mathbf{y}_{1}\right\} $$ and $$\left\{ \hat{\mathbf{s}}_{2}, \hat{\mathbf{s}}_{3}\right\}\!,$$ where $$\left\{ \hat{\mathbf{x}}_{1},\mathbf{y}_{1}\right\} $$ is as defined by equations (2.3),(2.7) and $$\left\{ \hat{\mathbf{s}}_{2},\hat{\mathbf{s}}_{3}\right\} $$ is as defined by the following equations. s^2=s2+(cTs2)s1 and s^3=s3+(cTs3)s1. For trajectories starting from $$\mathcal{H}\cap \mathcal{S}_{2}$$, we have $$f_{2}(0)=0$$. Thus, equations (2.16) and (2.18) imply that β1=(cTs2)sin⁡θ2+(cTs3)cos⁡θ2 (2.20) and z2(0)=K2(β1s1+s2sin⁡θ2+s3cos⁡θ2). (2.21) It can be easily seen by Lemma 2.1 and equation (2.16) that $$\mathbf{c}^{T}\boldsymbol{\hat{s}}_{2}=\mathbf{c}^{T}\boldsymbol{\hat{s}}_{3}=0$$ for each case of multiplicities of eigenvalues of $$\mathbf{A}_{2}$$ and the pair $$\left\{ \boldsymbol{\hat{s}}_{2},\boldsymbol{\hat{s}}_{3}\right\} $$ is a basis for $$\mathcal{H}$$. Therefore, any initial condition on $$\mathcal{H}$$ can be expressed as K1(x^1sin⁡θ1+y1cos⁡θ1)orK2(s^2sin⁡θ2+s^3cos⁡θ2), (2.22) where $$-\phi_{i}\leq \theta_{i}\leq 2\pi -\phi_{i}$$. Accordingly, $$\mathbf{v}_{1}(\theta_{1})$$ and $$\mathbf{v}_{2}(\theta_{2})$$ are defined as follows. v1(θ1)=x^1sin⁡θ1+y1cos⁡θ1andv2(θ2)=s^2sin⁡θ2+s^3cos⁡θ2. (2.23) Definition 2.4 In the rest of the article $$\mathbf{v}_{1}(\theta_{1})$$ and $$\mathbf{v}_{2}(\theta_{2})$$ (as defined by equation (2.23) will be called directions in$$\mathcal{H}$$. The directions for which $$\eta \mathbf{v}_{1}(\theta_{1})=\mathbf{v}_{2}(\theta_{2})$$ hold for some constant $$\eta >0$$, will be called equivalent directions and denoted as $$\mathbf{v}_{1}(\theta_{1})\simeq \mathbf{v}_{2}(\theta_{2})$$. 2.5. Characterization of transitive and nontransitive trajectories for the second mode Since $$\sigma_{1}-\lambda_{1}>0$$, all trajectories starting from $$\mathcal{H}\cap \mathcal{S}_{1}$$ are transitive. However, this is not true for all trajectories starting from $$\mathcal{H}\cap \mathcal{S}_{2}$$. Note further that, since non-transitive trajectories in each mode do note change mode, they decay to the origin if and only if the corresponding mode has eigenvalues with negative real parts. Thus for GAS, we need to consider only the trajectories starting from $$\mathcal{H}$$ without loss of any generality. The following result gives the classification of trajectories starting from $$\mathcal{H}\cap \mathcal{S}_{2}$$. Corollary 2.1 Suppose that $$\mathbf{z}_{2}(t)$$ is a trajectory starting from $$\mathcal{H}\cap \mathcal{S}_{2}$$. Then, the following hold. 1. $$\mathbf{z}_{2}(t)$$ is a non-transitive trajectory if and only if $$\ \mathbf{z}_{2}(0)=K_{2}\mathbf{v}_{2}(\theta_{2})$$ where $$-\phi_{2}\leq \theta_{2}\leq \frac{\pi }{2}$$ and $$K_{2}>0.$$ 2. $$\mathbf{z}_{2}(t)$$ is a transitive trajectory if and only if $$\ \mathbf{z}_{2}(0)=K_{2}\mathbf{v}_{2}(\theta_{2})$$ where $$\frac{\pi }{2} < \theta_{2}<\pi -\phi_{2}$$ and $$K_{2}>0.$$ Proof. The proof of the Corollary is given Corollary 4.1 in Öner (2014). □ Definition 2.5 Note that the interval $$-\phi_{2}\leq \theta_{2}\leq \frac{\pi }{2}$$ describes a closed convex cone in $$\mathcal{S}_{2}\,\mathcal{\cap \,H}$$ bounded by the half-lines $$\mathcal{L(}\hat{\mathbf{s}}_{2})$$ and $$\mathcal{L}^{-}$$. This cone will be called mode-invariant cone of $$\mathcal{S}_{2}\cap \mathcal{H}$$ and denoted as $$\mathcal{C}^{-}$$. Remark 2.2 Suppose that $$\mathbf{z}_{2}(t)$$ be a transitive trajectory starting from $$\mathcal{S}_{2}\cap \mathcal{H}$$, and $$\tau_{2}>0$$ such that $$\mathbf{c}^{T}\mathbf{z}_{2}(\tau_{2})=0$$. Then, simultaneously solving equations $$f_{2}(\tau_{2})=0$$ and $$f_{2}(0)=0$$ for each case of multiplicity of eigenvalues, $$\tau_{2}$$ can be expressed as an implicit function of $$\theta_{2}$$ as given by the following equations. −cot⁡θ2=(e(μ2−μ1)τ2−1)(e(μ3−μ1)τ2−1), if μ1<μ2<μ3,−cot⁡θ2=τ2e(μ3−μ1)τ2−1, if μ1=μ2<μ3,−cot⁡θ2=1−e(μ1−μ3)τ2τ2, if μ1<μ2=μ3. (2.24) The equations above also imply that $$\tau_{2}\rightarrow 0$$ as $$\theta_{2}\rightarrow \pi -\phi_{2}$$ and $$\tau_{2}\rightarrow \infty $$ as $$\theta_{2}\rightarrow \frac{\pi }{2}$$. This is expected since at $$\theta_{2}=\frac{\pi }{2}$$ the initial condition is on the line $$\mathcal{L(}\hat{\mathbf{s}}_{2})$$ and the trajectories starting from these lines are nontransitive ($$\tau_{2}=\infty $$). On the other hand, as $$\theta_{2}\rightarrow \pi -\phi_{2},$$ the initial condition is on the line $$\mathcal{L}^{+}$$ and smooth continuation into $$\mathcal{S}_{2}$$ is not possible ($$\tau_{2}=0$$). Lemma 2.3 Let $$\mathbf{z}_{2}(t)$$ be a transitive trajectory starting from $$\mathcal{S}_{2}\cap \mathcal{H}$$ with initial condition $$\mathbf{z}_{2}(0)=\hat{\mathbf{s}}_{2}\sin \theta_{2}+\hat{\mathbf{s}}_{3}\cos \theta_{2}$$. Then, at $$t=\tau_{2}\,$$ where the trajectory changes mode, the following hold. 1. z2(τ2)=eμ3τ2N2{s^2sin⁡θ^2+s^3cos⁡θ^2} (2.25) where N2:=e2(μ2−μ3)τ2sin2⁡θ2+cos2⁡θ2  and cot⁡θ^2=e(μ3−μ2)τ2cot⁡θ2,ifμ1≤μ2<μ3,N2:=e2(μ1−μ3)τ2sin2⁡θ2+cos2⁡θ2  and cot⁡θ^2=e(μ3−μ1)τ2cot⁡θ2ifμ1<μ2=μ3, (2.26) 2. $$\tau_{2}$$ and $$\hat{\theta}_{2}$$ are decreasing functions of $$\theta_{2}$$. Furthermore, transitive trajectories of each mode hit $$\mathcal{H}$$ and change mode in the open conic regions bounded by the lines $$\mathcal{L}^{+}$$ and $$\mathcal{L}(-\hat{\mathbf{s}}_{2})$$ for the second mode. The map $$\mathcal{F}:\theta_{2}\rightarrow \hat{\theta}_{2}$$ is defined as $$\mathcal{F}\left(\theta_{2}\right) =\hat{\theta}_{2}$$. Furthermore, the following hold. (a) $$\hat{\theta}_{2}\rightarrow \pi -\phi_{2}$$ as $$\theta_{i}\rightarrow \pi -\phi_{2}$$ (equivalently as $$\tau_{2}\rightarrow 0$$) for all possible algebraic degrees of the eigenvalues, (b) $$\hat{\theta}_{2}\rightarrow \frac{3\pi }{4}$$ as $$\theta_{2}\rightarrow \frac{\pi }{2},$$ (equivalently as $$\tau_{2}\rightarrow \infty $$) if $$\mu_{1}<\mu_{2}<\mu_{3},$$ (c) $$\hat{\theta}_{2}\rightarrow \pi $$ as $$\theta_{2}\rightarrow \frac{\pi }{2},$$ (equivalently as $$\tau_{2}\rightarrow \infty $$) if $$\mu_{1}=\mu_{2}<\mu_{3}$$ or if $$\mu_{1}<\mu_{2}=\mu_{3},$$ Proof. The proof is given Lemma 4.3 in Öner (2014). □ 2.6. Mode change When a trajectory changes mode, we have to switch from one basis to the other for describing how the trajectory evolves after mode change. Before investigating the change of basis, we give the following definition, which is first introduced in Eldem & Öner (2015). Definition 2.6 The constant defined as B3:=λ1−a11−μ3a21ω1(1+b12) will be called the coupling constant in the rest of the article. Lemma 2.4 Let $$\mathbf{z}_{1}(t)$$ be a trajectory starting from $$\mathcal{S}_{1}\cap \mathcal{H}$$ with initial condition $$\mathbf{z}_{1}(0)=\mathbf{v}_{1}\left(\theta_{1}\right) $$ where $$-\phi_{1}\leq \theta_{1}<\pi -\phi_{1}$$. Then, at $$t=\tau_{1}$$ the trajectory changes mode and we have 1. z1(τ1)=η12(θ^1)eσ1τ1{s^2sin⁡θ2+s^3cos⁡θ2}, where θ2=cot−1⁡{−b2(B3(cot⁡θ^1+b1)+1)(cot⁡θ^1+b1)B2+1} for μ1≤μ2<μ3, =cot−1⁡{−b2(B3(cot⁡θ^1+b1)+1)(cot⁡θ^1+b1)B1+1} for μ1<μ2=μ3 (2.27) and η12(θ^1)=−ω1(cos⁡θ^1+b1sin⁡θ^1)a32(μ3−μ1)(cos⁡θ2+b2sin⁡θ2) for μ1≤μ2<μ3=−ω1(cos⁡θ^1+b1sin⁡θ^1)a32(cos⁡θ2+b2sin⁡θ2) for μ1<μ2=μ3. Moreover, $$B_{1}:=\frac{\left(\lambda_{1}-a_{11}-\mu_{1}a_{21}\right) }{\omega_{1}\left(1+b_{1}^{2}\right) }, B_{2}:=\frac{\left(\lambda_{1}-a_{11}-\mu_{2}a_{21}\right) }{\omega_{1}\left(1+b_{1}^{2}\right) }$$, and $$b_{2}:=\cot \phi_{2}$$ as defined in Lemma 2.2 for eigenvalues with different multiplicities. 2. Let $$\mathbf{z}_{2}(t)$$ be a trajectory starting from $$\mathcal{S}_{2}\cap \mathcal{H}$$ with initial condition $$\mathbf{z}_{2}(0)=\eta_{12}\left(\hat{\theta}_{1}\right) e^{\sigma_{1}\tau_{1}} \mathbf{v}_{2}\left(\theta_{2}\right) $$. If $$\mathbf{z}_{2}(t)$$ is a transitive trajectory ($$\frac{\pi }{2}<\theta_{2}<\pi -\phi_{2}$$) (or equivalently outside of the cone $$\mathcal{C}^{-}$$), then z2(τ2)=η12(θ^1)eμ3τ2eσ1τ1η21(θ2)N2(x^1sin⁡θ11+y1cos⁡θ11), where θ11=cot−1⁡{−(cot⁡θ^2+b2)b2B3+B2cot⁡θ^2−b1} for μ1≤μ2<μ3,=cot−1⁡{−(cot⁡θ^2+b2)b2B3+B1cot⁡θ^2−b1}for μ1<μ2=μ3, (2.28) and η21(θ2)=−a32(μ3−μ1)(cos⁡θ^2+b2sin⁡θ^2)ω1(cos⁡θ11+b1sin⁡θ11) for μ1≤μ2<μ3,=−a32(ω1(cos⁡θ^2+b2sin⁡θ^2)(b12+1))ω12(1+b12)(cos⁡θ11+b1sin⁡θ11)for μ1<μ2=μ3. Furthermore, $$-\phi_{1}<\theta_{11}<{-}\psi_{1}$$ where $$\cot \psi_{1}=\frac{1}{B_1}+b_{1}$$ and $$-{\hat{s}}_{2}\simeq{v}_{1} {(-}\psi_{1})$$. Proof. The proof is given in Appendix. □ 2.7. One loop around $$\mathcal{H}$$ plane Let $$\mathbf{z}_{1}(t)$$ be a transitive trajectory with initial condition $$\mathbf{z}_{1}(0)=\mathbf{v}_{1}\left(\theta_{1}\right) $$ in $$\mathcal{S}_{1}\cap \mathcal{H}$$. Suppose that $$\mathbf{z}_{1}(\tau_{1})$$ is outside the cone $$\mathcal{C}^{-}.$$ Then, the movement of the trajectories on $$\mathcal{H}$$ is controlled by the following functions. F1(θ1)=θ^1, G1(θ^1)=θ2, F2(θ2)=θ^2, G2(θ^2)=θ11. Thus, we can define a function $$\mathcal{T}_{1}:\mathcal{S}_{1}\cap \mathcal{H\rightarrow S}_{1}\cap \mathcal{H}$$ as follows. T1(θ1)=G2(F2(G1(F1(θ1))))=θ11. (2.29) Since change of basis is continuous, it follows that all the maps described above are continuous. Therefore, $$\mathcal{T}_{1}(\theta_{1})=\theta_{11}$$ is continuous. Note that this map represents the following four steps in the dynamic behaviour of the trajectories. v1(θ1)→eσ1τ1v1(θ^1)→η12(θ^1)eσ1τ1v2(θ2)→η12(θ^1)eμ2τ2eσ1τ1N2v2(θ^2)→eμ2τ2eσ1τ1η21(θ2)η12(θ^1)N2v1(θ11). Thus, $$\mathcal{T}_{1}(\theta_{1})$$ can be interpreted as a Poincaré full map. Also note that $$\mathcal{T}_{1}(\theta_{1})$$ is defined only for trajectories which change mode at least two times. For trajectories which changes mode $$2k$$ times$$,$$ we define $$\mathcal{T}_{1}^{k} (\theta_{1})$$ as follows. T1k(θ1):=T1(T1(⋅⋅⋅T1(θ1)⋅⋅⋅))(k times) and T1k(θ1):=θ1k. For transitive trajectories starting from $$\mathcal{S}_{2}\cap \mathcal{H}$$, we can also define $$\mathcal{T}_{2}:\mathcal{S}_{2}\cap \mathcal{H\rightarrow S}_{2}\cap \mathcal{H}$$ as $$\mathcal{T}_{2}^{k}\left(\theta_{2}\right) :=\theta_{2k}$$ in a completely similar way. 3. Stability of fix directions In view of the development given in the previous section, some trajectories may change mode only finite number of times and enter a mode-invariant cone and some may change mode infinitely many times as $$t\rightarrow \infty.$$ The trajectories, which change mode only a finite number of times as $$t\rightarrow \infty$$, decay to the origin if and only if $$\mu_{3}<0.$$ Therefore, we investigate only GAS of the class of trajectories which change mode infinitely many times as $$t\rightarrow \infty.$$ Towards this end, we give the following definitions to be consistent with the terminology used in Eldem & Şahan (2014). Definition 3.1 Let $$\theta_{i}^{\ast }$$ be a fixed point of $$\mathcal{T}_{i}(\theta_{i})$$ or equivalently $$\mathcal{T}_{i}(\theta_{i}^{\ast })=\theta_{i}^{\ast }$$ for $$i=1,2$$. Then, $$\mathbf{v}_{i}(\theta_{i}^{\ast })$$ is called a fixed direction. A fixed direction $$\mathbf{v}_{i}(\theta_{i}^{\ast })$$ is said to be attractive in an interval$$ I_{i}$$containing$$\theta_{i}^{\ast }$$ if for any $$\theta_{i}$$ in $$I_{i}$$ and for every $$\varepsilon >0$$ there exists a positive integer $$k$$ such that $$\left\vert \mathcal{T}_{i}^{k}(\theta_{i})-\theta_{i}^{\ast }\right\vert <\varepsilon $$ for $$i=1,2$$. If $$I_{i}$$ consists of only one point $$\theta_{i}^{\ast }$$, then the fixed point is said to be repulsive. In view of this definition, let $$\mathbf{z}_{i}^{\ast }(t)$$ be a trajectory starting from a fixed direction $$\mathbf{v}_{i}(\theta_{i}^{\ast })$$. To be more specific, $$\mathbf{z}_{i}^{\ast }(0)=K_{i}\mathbf{v}_{i}(\theta_{1}^{\ast })$$ where $$K_{i}>0$$ is a real constant. Then, in view of Lemma 2.4 $$K_{i}\mathbf{v}_{i}(\theta_{i}^{\ast })$$ is mapped as follows, Kivi(θi∗)→γ∗(θi∗)Kivi(θi∗), where γ∗(θ1∗):=eμ2τ2∗eσ1τ1∗η21(θ2∗)η12(θ^1∗)N2 Definition 3.2 In the sequel, $${\it {\gamma}} ^{\ast }(\theta_{i}^{\ast })$$ will be called convergence rate as in Eldem & Şahan (2014). It is clear from the development above that if $$\mathbf{z}_{1}^{\ast }(t)$$ is a trajectory starting from a fixed direction $$\mathbf{v}_{1}(\theta_{1}^{\ast })$$, then there exists a unique fixed direction in $$\mathcal{S}_{2}\cap \mathcal{H}$$ such that z1∗(τ1∗)=η12(θ1∗+ω1τ1∗)eσ1τ1∗v2(θ2∗). In view of the definition of $$\mathcal{T}(\theta_{1}^{\ast }),$$$$\mathbf{z}_{1}^{\ast }(\tau_{1}^{\ast })$$ is necessarily in the transitive region of $$ \mathcal{S}_{2}\cap \mathcal{H}$$ or equivalently is outside the cone $$ \mathcal{C}^{-}$$. This means that fixed directions exist as a pair $$\left(\theta_{1}^{\ast },\theta_{2}^{\ast }\right) $$. Therefore, we use the following notation for the convergence rate γ∗(θ1∗)=γ∗(θ1∗,θ2∗). In order to complete the picture, we also define below the attractiveness of mode-invariant cone $$\mathcal{C}^{-}$$ of $$\mathcal{S}_{2}\cap \mathcal{H}$$ as defined in Definition 2.5. Definition 3.3 $$\mathcal{C}^{-}$$ is said to be attractive in an interval$${\boldsymbol{I}}_{2}$$containing$$\mathcal{C}^{-}$$ if for any $$\theta_{2}$$$$\in $$$$I_{2}$$ there exists a finite non-negative integer $$k$$ such that $$-\phi_{2}< \mathcal{T}_{2}^{k}(\theta_{2})\leq \frac{\pi }{2}$$ or $$\mathcal{T}_{2}^{k}(\theta_{2})\rightarrow \frac{\pi }{2}$$ as $$k\rightarrow \infty $$. Lemma 3.1 Let $$\mathbf{v}_{i}(\theta_{i}^{\ast })$$ be a fixed direction which is attractive in an interval $$I_{i}$$. Then, the following hold. 1. If $$\mathbf{z}_{i}^{\ast }(t)$$ is a trajectory starting from the fixed direction $$\mathbf{v}_{i}^{\ast }(\theta_{i}^{\ast }),$$ then $$\mathbf{z}_{i}^{\ast }(t)\rightarrow 0$$ as $$t\rightarrow \infty $$ if and only if $${\it {\gamma}} ^{\ast }(\theta_{1}^{\ast },\theta_{2}^{\ast })<1$$. 2. If $$\mathbf{z}_{i}(t)$$ is a trajectory starting from $$\mathcal{S}_{i}\cap \mathcal{H}$$ with initial condition $$\mathbf{z}_{i}(0)=K_{i}\mathbf{v}_{i}(\theta_{i})$$ where $$\theta_{i}$$$$\in $$$$I_{i}$$, then $$\mathbf{z}_{i}(t)\rightarrow 0$$ as $$t\rightarrow \infty $$ if and only if $${\it {\gamma}}^{\ast }(\theta_{1}^{\ast },\theta_{2}^{\ast })<1$$. 3. Suppose that $$\mathcal{C}^{-}$$ is attractive in an interval $$I_{2}$$ containing $$\mathcal{C}^{-}$$. Let $$\mathbf{z}_{2}(0)=$$$$\mathbf{v}_{2}\left(\theta_{2}\right) $$ where $$\theta_{2}$$$$\in $$$$I_{2}$$. Then $$z_{2}(t)\rightarrow 0$$ as $$t\rightarrow \infty $$ if and only if $$\mu_{3}<0$$. Proof. The proof of this Lemma can be done by following the similar lines as in the proof of Lemma 5 in Eldem & Öner (2015). To avoid duplication, the proof is omitted here. □ Corollary 3.1 A general expression for $${\it {\gamma}} ^{\ast }(\theta_{1}^{\ast })$$ is γ∗(θ1∗,θ2∗)=eμ2τ2eσ1τ1cos⁡θ^1∗+b1sin⁡θ^1∗cos⁡θ1∗+b1sin⁡θ1∗cot⁡θ^2∗+b2cot⁡θ2∗+b2, and if $$\theta_{1}^{\ast }\neq 0,$$ we have γ∗(θ1∗,θ2∗)=eμ2τ2eλ1τ11+B2(cot⁡θ^1∗+b1)1+B2(cot⁡θ1∗+b1). (3.1) Proof. Using the definition of convergence rate, we have $${\it {\gamma}} ^{\ast }(\theta_{1}^{\ast }):=e^{\mu_{2}\tau_{2}^{\ast }}e^{\sigma_{1}\tau_{1}^{\ast }}\eta_{21}(\theta_{2}^{\ast })\eta_{12}\left(\hat{\theta}_{1}^{\ast }\right) N_{2}.$$ Substituting for $$\eta_{21}(\theta_{2}^{\ast })$$, $$\eta_{12}\left(\hat{\theta}_{1}^{\ast }\right) $$ and $$N_{2}$$ given in Lemma 2.3 and Lemma 2.4, we obtain γ∗(θ1∗)=eμ2τ2eσ1τ1cos⁡θ^1∗+b1sin⁡θ^1∗cos⁡θ1∗+b1sin⁡θ1∗cot⁡θ^2∗+b2cot⁡θ2∗+b2. Note that $$\frac{\cot \hat{\theta}_{2}^{\ast }+b_{2}}{\cot \theta_{2}^{\ast }+b_{2}}=\frac{\cot \theta_{1}^{\ast }+b_{1}}{\cot \hat{\theta}_{1}^{\ast }+b}\frac{1+B_{2}\left(\cot \hat{\theta}_{1}^{\ast }+b_{1}\right) }{1+B_{2}\left(\cot \theta_{1}^{\ast }+b_{1}\right) }\!.$$ Therefore, if $$\theta_{1}^{\ast }\neq 0$$, then $$\sin \theta_{1}^{\ast }$$$$\neq 0$$ and we get γ∗(θ1∗)=eμ2τ2eλ1τ11+B2(cot⁡θ^1∗+b1)1+B2(cot⁡θ1∗+b1). □ Let the interval $$I$$ be in the domain of $$\mathcal{T}_1 (.)$$ and $$\theta_1 \in I$$. We show below that the derivative of $$\mathcal{T}_1 (\theta_1)$$ with respect to $$\theta_1$$ is positive. Lemma 3.2 The derivative of $$\theta_{11}$$ with respect to $$\theta_{1}$$ is positive in domain of $$\mathcal{T}_{1}$$ and dT1dθ1=(sin⁡(θ11+ψ1)sin⁡(θ^1+ψ1))(sin⁡(θ11+ψ2)sin⁡(θ^1+ψ2))(sin⁡(θ11+ψ3)sin⁡(θ^1+ψ3))(sin⁡(θ1+ϕ1)sin⁡(θ11+ϕ1)) ×sin⁡θ^1sin⁡θ1 for μ1<μ2<μ3=(sin⁡(θ11+ψ2)sin⁡(θ^1+ψ2))2(sin⁡(θ11+ψ3)sin⁡(θ^1+ψ3))(sin⁡(θ1+ϕ1)sin⁡(θ11+ϕ1))sin⁡θ^1sin⁡θ1 for μ1=μ2<μ3=(sin⁡(θ11+ψ2)sin⁡(θ^1+ψ2))2(sin⁡(θ11+ψ1)sin⁡(θ^1+ψ1))(sin⁡(θ1+ϕ1)sin⁡(θ11+ϕ1))sin⁡θ^1sin⁡θ1 for μ1<μ2=μ3 where $$\cot \psi_{1}=\frac{1}{B_{1}}+b_{1},$$$$\cot \psi_{2}=\frac{1}{B_{2}}+b_{1}$$ and $$\cot \psi_{3}=\frac{1}{B_{3}}+b_{1}$$. Proof. The proof of the Lemma will be given in Appendix. □ Using Lemma 2.3 and Lemma 2.4, it is easy to show that v1(−ψ1)≃v2(3π/4),v1(−ψ2)≃v2(π)=−s^3,v1(−ψ3)≃v2(3π/2)=−s^2. (3.2) More precisely, the pairs $$\{\mathbf{v}_{1}(-\psi_{1}),\mathbf{v}_{2}(3\pi /4)\}$$, $$\{\mathbf{v}_{1}(-\psi_{2}),\mathbf{v}_{2}(\pi)\}$$, and $$\{\mathbf{v}_{1}(-\psi_{3}),\mathbf{v}_{2}(3\pi /2)\}$$ are equivalent directions as defined in Definition 2.4. Since $$B_{3}<B_{2}<B_{1}$$, it also follows that $$\psi_{3}<\psi_{2}<\psi_{1}<\phi_{1}$$. Suppose that a trajectory starting from $$\mathcal{S}_{1}\cap \mathcal{H}$$ along the direction $$\mathbf{v}_{1}(\theta_{1})$$ where $$-\phi_{1}<\theta_{1}<0,$$ hits $$\mathcal{S}_{2}\cap \mathcal{H}$$ outside the cone $$\mathcal{C}^{-}$$ and continues in the second mode. Then, in view of Lemma 2.3.2.b, this trajectory hits back $$\mathcal{S}_{1}\cap \mathcal{H}$$ along the direction $$\mathbf{v}_{1}(\theta_{11})$$ where $$-\phi_{1}<\theta_{11}<-\psi_{1}$$. In view of this observation, we can assume without loss of any generality that, $$\mathcal{T}_{1}(\theta_{1}) $$ is a continuous and differentiable function which maps $$[-\phi_{1},-\psi_{1}]$$ into itself. Thus, the existence of a fixed point of $$\mathcal{T}_{1}(\theta_{1}) $$ is clear. We have to show that the fixed point is unique and attractive in the interval $$[-\phi_{1},-\psi_{1}]$$. Towards this end, let ρ(θ1):=T1(θ1)−θ1. (3.3) Note that $$\rho (-\phi_1)>0$$ and $$\rho (-\psi_1)<0$$. Now we can state and prove the following results. Lemma 3.3 If $$\phi_{1}<2\psi_{2}$$, then $$\frac{d\rho \left(\theta_{1}\right) }{d\theta_{1}}<0 $$ whenever $$\rho \left(\theta_{1} \right) =0$$. Proof. We give the proof only for the case $$\mu_{1}<\mu_{2}<\mu_{3}.$$ The proof for the other cases can be done completely in a similar manner. If $$\rho (\theta_1)=\mathcal{T}_{1}(\theta_{1})-\theta_1=\theta_{11}-\theta_1=0$$, then $$\theta_{11}=\theta_1$$ and this implies that dT1(θ1)dθ1∣θ1=θ11=(sin⁡(θ1+ψ1)sin⁡(θ^1+ψ1))(sin⁡(θ1+ψ2)sin⁡(θ^1+ψ2))(sin⁡(θ1+ψ3)sin⁡(θ^1+ψ3))sin⁡θ^1sin⁡θ1. Since ddx(sin⁡(θ1+x)sin⁡(θ^1+x))=sin⁡(θ^1−θ1)(sin⁡(θ^1+x))2<0, we get sin⁡θ1sin⁡θ^1>sin⁡(θ1+ψ3)sin⁡(θ^1+ψ3)>sin⁡(θ1+ψ2)sin⁡(θ^1+ψ2)>sin⁡(θ1+ψ1)sin⁡(θ^1+ψ1). (3.4) This implies that sin⁡(θ1+ψ3)sin⁡(θ^1+ψ3)sin⁡θ^1sin⁡θ1<1. Since $$\frac{\sin \left(\theta_{1}+\psi_{2}\right) }{\sin \left(\hat{ \theta}_{1}+\psi_{2}\right) }>\frac{\sin \left(\theta_{1}+\psi_{1}\right) }{\sin \left(\hat{\theta}_{1}+\psi_{1}\right) }$$, it is enough to show that $$\frac{\sin \left(\theta_{1}+\psi_{2}\right) }{\sin \left(\hat{ \theta}_{1}+\psi_{2}\right) }<1.$$ Note that in view of Remark 2.1, we have $$\hat{\theta}_{1}=\pi +y_{1} $$ where $$y_{1}<\phi_{1}$$. Furthermore, since $$-\phi_{1}<-\theta_{11}<-\psi_{1}$$, we get sin⁡(θ1+ψ2)sin⁡(θ^1+ψ2)=sin⁡(|θ1|−ψ2)sin⁡(y+ψ2). Since $$\phi_{1}<2\psi_{2}$$, it follows that ϕ1<2ψ2⇔ϕ1−ψ2<ψ2<y+ψ2. If $$y+\psi_{2}\leq \frac{\pi }{2},$$ the inequalities given above imply that $$\frac{\sin \left(\theta_{1}+\psi_{2}\right) }{\sin \left(\hat{\theta}_{1}+\psi_{2}\right) }<\frac{\sin \left(\phi_{1}-\psi_{2}\right) }{\sin \left(y+\psi_{2}\right) }<1$$. On the other hand, if $$y+\psi_{2}>\frac{\pi }{2}$$ and $$\phi_{1}-\psi_{2}> \pi -y-\psi_{2}$$, then $$\phi_{1}+y>\pi.$$ This is a contradiction because $$\phi_{1}+y<\pi.$$ Therefore, if $$y+\psi_{2}>\frac{\pi }{2}$$ we must have $$\phi_{1}-\psi_{2}<\pi -y-\psi_{2}$$. This is equivalent to the fact that sin⁡(θ1+ψ2)sin⁡(θ^1+ψ2)<sin⁡(ϕ1−ψ2)sin⁡(y+ψ2)=sin⁡(ϕ1−ψ2)sin⁡(π−y−ψ2)<1. Therefore, if $$\phi_{1}<2\psi_{2},$$ using the last inequality and inequality (3.4) we have the following relations: 1>sin⁡(θ1+ψ2)sin⁡(θ^1+ψ2)>sin⁡(θ1+ψ1)sin⁡(θ^1+ψ1). Consequently, we get $$\frac{d\mathcal{T}_{1}}{d\theta_{1}}<1$$ whenever $$\theta_{11}=\theta_1$$ or equivalently whenever $$\rho (\theta_1)=0$$. □ Lemma 3.4 There is a unique and attractive fix point $$\theta_{1}^{\ast }$$ of $$ \mathcal{T}_{1}$$ if $$\frac{d\rho \left(\theta_{1}\right) }{d\theta_{1}}<0 $$ whenever $$\rho \left(\theta_{1} \right) =0.$$ Proof. Suppose that $$\theta_{1}^{1}$$ is the first and $$\theta_{1}^{2}>\theta_{1}^{1}$$ is the second fixed point of $$\mathcal{T}_{1}$$ such that $$-\phi_1<\theta_{1}^{1}< \theta_{1}^{2}<-\psi_1$$. Since $$\rho (-\phi_1)>0$$, it follows that $$\rho (\theta_1)>0$$ in the interval $$[-\phi_1, \theta_{1}^{1})$$. Since $$\frac{d\rho \left(\theta_{1}\right) }{d\theta_{1}}<0$$ at $$\theta_1=\theta_{1}^{1}$$ it also follows that $$\rho (\theta_1)<0$$ in the interval $$(\theta_{1}^{1}, \theta_{1}^{1}+\epsilon]$$ for some $$\epsilon>0$$ where $$\theta_{1}^{1}+\epsilon<\theta_{1}^{2}$$. Since $$\theta_{1}^{2}$$ is the second fixed point after $$\theta_{1}^{1}$$, we have $$\rho (\theta_1)<0$$ in the interval $$(\theta_{1}^{1}, \theta_{1}^{2})$$. Note that $$\rho (\theta_{1}^{2})=0$$ and $$\frac{d\rho \left(\theta_{1}\right) }{d\theta_{1}}<0$$ at $$\theta_1=\theta_{1}^{2}$$. This implies that again $$\rho (\theta_1)<0$$ in the interval $$[\theta_{1}^{2}-\epsilon) \cup (\theta_{1}^{2},\theta_{1}^{2}+\epsilon]$$ for some $$\epsilon>0$$. Consequently, $$\rho (\theta_{1}^{2})$$ is a local maximum. Since $$\rho (\theta_1)$$ is a differentiable function, it must hold that $$\frac{d\rho \left(\theta_{1}\right) }{d\theta_{1}}=0$$ at $$\theta_1=\theta_{1}^{2}$$. This contradicts the hypothesis that $$\frac{d\rho \left(\theta_{1}\right) }{d\theta_{1}}<0 $$ whenever $$\rho \left(\theta_{1} \right) =0.$$ Therefore, there is no second fixed point of $$\mathcal{T}_1$$ or equivalently the fixed point $$\theta_{1}^{1}$$ is unique. Attractivity follows from the fact that $$\mathcal{T}_1 (\theta_1)>\theta_1$$ in the interval $$[-\phi_1, \theta_{1}^{1})$$ and $$\mathcal{T}_1 (\theta_1)>\theta_1$$ in the interval $$(\theta_{1}^{1}, -\psi_1]$$. □ In view of the results given above, we have the following assumption. Assumption 3.1 $$\phi_{1}<2\psi_{2}.$$ 4. Main result In this section, we consider three different geometric structures given as follows. 1. $$-\mathbf{y}_{1}$$ is outside the cone $$\mathcal{C}^{-}$$$$\left(B_{3}>0\right) $$. 2. $$\mathbf{y}_{1}$$ and $$\hat{\mathbf{s}}_{2}$$ are on the same line $$\left(B_{3}=0\right) $$. 3. $$-\mathbf{y}_{1}$$ is in the cone $$\mathcal{C}^{-}$$$$\left(B_{3}<0\right) $$. For the first two cases above we provide the necessary and sufficient conditions for GAS of bimodal systems being considered. Furthermore, we demonstrate the effect of the discontinuity of the vector field on GAS of bimodal systems by two examples. In these examples, we basically show that the eigenvectors of one of the subsystems can be changed without changing the eigenvalues of the subsystem and this change can make bimodal system GAS or unstable. This change is achieved by changing the coupling constant given in Definition 2.6. The case where $$B_3<0$$ deserves a separate investigation. Our comments for this case is given in Remark 4.1. Theorem 4.1 Given the bimodal system $$\it{\Sigma}_{0}$$ with Assumptions 2.1, 2.2, 2.3 and 3.1, the following hold. If $$B_{3}\geq 0$$, then the unique fixed direction $$\theta_1^{\ast}$$ in the interval $$[-\phi_1,-\psi_1]$$ is attractive in the interval $$[-\phi_1, \bar{\theta}_1)$$ where $$\mathcal{G}_1 (\mathcal{F}_1 (\bar{\theta}_1))=\frac{\pi}{2}$$, and mode-invariant cone $$\mathcal{C}^{-}$$ is repulsive. If $$B_{3}\geq 0$$, then the system is GAS if and only if real eigenvalues of both modes are negative and the convergence rate $${\it {\gamma}} ^{\ast }(\theta_{1}^{\ast },\theta_{2}^{\ast })<1$$. Proof. The proof is given only for the case $$\mu_{1}<\mu_{2}<\mu_{3}.$$ The proof for the other cases can be done completely in a similar manner. Consider a trajectory starting from $$\mathcal{S}_{1}\cap \mathcal{H}$$. Suppose that $$B_{3}>0$$, i.e., $$-\mathbf{y}_{1}$$ is outside the cone $$\mathcal{C}^{-}$$. Since $$\mathcal{F}_{1}(\theta_{1})$$ is a continuous function which maps $$[0,\pi -\phi_{1})$$, to $$(\pi -\phi_{1},\pi]$$, there exists a unique $$\bar{\theta}_{1}\in \lbrack 0,\pi -\phi_{1})$$ such that $$ \mathcal{G}_{1}\mathcal{(F}_{1}\mathcal{(\bar{\theta}}_{1}))=\frac{\pi }{2}$$. Therefore, if $$\mathbf{z}_{1}(0)=\mathbf{v}_{1}(\theta_{1})$$ where $$ \theta_{1}\,\geq \,\bar{\theta}_{1}$$, then $$z_{1}(t)$$ will enter the cone $$ \mathcal{C}^{-}$$ after one mode change and stay in $$\mathcal{S}_{2}$$ for all $$t\geq \tau_{1}$$. On the other hand, for any $$\theta_{1}\in \lbrack -\phi_{1},\,\bar{\theta}_{1})$$, it follows that $$\mathbf{z}_{1}(\hat{\theta}_{1})$$ is outside the cone $$\mathcal{C}^{-}.$$ Consequently, $$\mathcal{T}_{1}(\theta_{1})$$$$\in \lbrack -\phi_{1},-\psi_{1})$$ by Lemma 2.4.2. Furthermore, for any $$\theta_{1} \in $$$$[-\phi_{1},-\psi_{1}]$$ it also follows that $$\mathcal{T}_{1}(\theta_{1}) \in $$$$[-\phi_{1},-\psi_{1})$$. Also note that if $$\theta_{1}>-\psi_{1},$$ we get $$\mathcal{T}_{1}(\theta_{1})<\theta_{1}$$ and if $$\theta_{1}=-\phi_{1}$$ we have $$\mathcal{T}_{1}(\theta_{1})>\theta_{1}.$$ This implies that there exists $$\theta_{1}^{\ast }$$ such that $$\mathcal{T}_{1}(\theta_{1}^{\ast })=\theta_{1}^{\ast }.$$ Then, in view of Lemmas 3.3 and 3.4 it follows that $$\theta_{1}^{\ast }$$ is a unique fix point which is attractive in $$[-\phi,\bar{\theta_{1}})$$ and and mode-invariant cone $$\mathcal{C}^{-}$$ is repulsive. Consequently, Lemma 3.1 implies that bimodal system is GAS if and only if real eigenvalues of both modes are negative and the convergence rate $${\it {\gamma}} ^{\ast }(\theta_{1}^{\ast },\theta_{2}^{\ast })<1$$. On the other hand, if $$B_{3}=0$$ then $$-\mathbf{y}_{1}$$ and $$\hat{\mathbf{s}}_{2}$$ are equivalent directions. This means that $$\bar{\theta}_{1}=0$$. As in the previous case, if $$\mathbf{z}_{1}(0)=\mathbf{v}_{1}(\theta_{1})$$ where $$\theta_{1}\in \lbrack 0,\pi -\phi_{1})$$, then $$\mathbf{z}_{1}(t)$$ will enter the cone $$\mathcal{C}^{-}$$ after one mode change and stay in $$\mathcal{S}_{2}$$ for all $$t\geq \tau_{1}$$. If $$\mathbf{z}_{1}(0)=\mathbf{v}_{1}(\theta_{1})$$ where $$\theta_{1}\in \lbrack -\phi_{1},0), $$ then $$\hat{\theta}_{1}\in (\pi,\pi +\phi_{1})$$ by equation (2.11) and this implies that $$\mathbf{z}_{1}\left(\tau_{1}\right) $$ is outside $$\mathcal{C}^{-}$$. Continuing along the similar lines as in the previous case, it follows that there exists a unique fixed point $$\theta_{1}^{\ast }$$ of $$\mathcal{T}_{1}(\theta_{1})$$ which is attractive in $$[-\phi,\bar{\theta_{1}})$$ and mode-invariant cone $$\mathcal{C}^{-}$$ is repulsive. Therefore, we again conclude that bimodal system is GAS if and only if real eigenvalues of both modes are negative and $${\it {\gamma}} ^{\ast }(\theta_{1}^{\ast },\theta_{2}^{\ast })<1$$. □ Remark 4.1 1. Suppose that $$B_{3}<0$$ and $$\mathcal{G}_1 (\mathcal{F}_1 (-\phi_1)) \leq \frac{\pi}{2}$$. This means that $$\mathcal{G}_1 (\mathcal{F}_1 (\theta_1))$$ maps $$[-\phi_1, \pi - \phi_1)$$ into mode-invariant cone $$\mathcal{C}^{-}$$. Therefore, $$\mathcal{C}^{-}$$ is attractive in $$\mathcal{S}_2 \cap \mathcal{H}$$ and there is no fixed direction. 2. If $$\mathcal{G}_1 (\mathcal{F}_1 (-\phi_1)) > \frac{\pi}{2},$$ then there exists $$\bar{\theta}_{1} \in \lbrack -\phi_{1},0) $$ such that $$\mathcal{G}_{1} (\mathcal{(F}_{1} (\bar{\theta}_{1}))=\frac{\pi }{2}$$ and the following points need a separate investigation. (a) If $$-\psi_{1}>\bar{\theta}_{1},$$ then there may or may not be a fixed point. Furthermore, the fixed point may or may not be attractive. Therefore, a separate analysis is necessary for these issues. (b) If $$-\psi_{1} \leq \bar{\theta}_{1}$$ and $$B_2 >0$$, then Lemmas 3.3 and 3.4 hold and intuitively we expect that bimodal system is GAS if only if real eigenvalues of both modes are negative and $${\it {\gamma}} ^{\ast }(\theta_{1}^{\ast },\theta_{2}^{\ast })<1$$. However, this needs a separate proof. The following examples are included to demonstrate the effect of the coupling constant on GAS of bimodal systems. Example 4.1 Consider the following bimodal system where $${A}_{2}$$ is in observable canonical form as described in (2.2) with eigenvalues $$\left\{ -4,-3,-2\right\} $$ and the eigenvalues of $${A}_{1}$$ (given below) are $$\left\{ -1,\frac{3}{10}+4i,\frac{3}{10}-4i\right\}\!.$$ A1=[−28391070−293130227269198811449162412782−410441391[9pt]0107130−23365]. (4.1) Additionally $$\phi_{1}<2\psi_{2}\left(1.2566<2.2196\right).$$ This system is unstable with $${\it {\gamma}} ^{\ast }\left(\theta_{1}^{\ast },\theta_{2}^{\ast }\right) =2.9111$$ and the coupling constant is $$B_{3}=4.0073$$. Note that we can change the coupling constant while keeping the eigenvalues of both modes fixed with $$\mathbf{A}_{1}$$ as given below. A1=[6791287−2295273436407943−58121930528289128702455607449364079430−6441219305015911818890800267521−2170413282890]. (4.2) We again have $$\phi_{1}<2\psi_{2}$$$$\left(1.2566<1.3912\right)$$. This system is stable with $${\it {\gamma}} ^{\ast }\left(\theta_{1}^{\ast },\theta_{2}^{\ast }\right) =0.4259$$ and $$B_{3}=$$$$0.6486$$. The trajectories of the systems above are depicted in Figs 1 and 2 (In figures, the arrows show the direction of the trajectories). Fig. 1. View largeDownload slide Unstable trajectories. Fig. 1. View largeDownload slide Unstable trajectories. Fig. 2. View largeDownload slide Stable trajectories. Fig. 2. View largeDownload slide Stable trajectories. Example 4.2 Consider the following bimodal system. The eigenvalues of $$\mathbf{A}_{1}$$ are $$\left\{-\frac{1}{3},-\frac{1}{40}+\frac{1}{5}i,\right.$$$$\left.-\frac{1}{40}-\frac{1}{5}i\right\} $$. $$\mathbf{A}_{2}$$ is in observable canonical form as described in (2.2) with eigenvalues $$\left\{ -4,-3,-2\right\}$$. A1=[−5511560−196513073202726019719559233153660−557260020617310472−6551576]. Both modes of this system are stable, but bimodal system is unstable with $${\it {\gamma}} ^{\ast }\left(\theta_{1}^{\ast },\theta_{2}^{\ast }\right) =1.4011. $$ The coupling constant is $$B=$$$$3.0212$$ and $$\phi_{1}<2\psi_{2}$$$$\left(0.5754<1.0319\right)$$. We can change the coupling constant while keeping the eigenvalues of both modes fixed. Consider the bimodal system given below. A1=[−4811832−289240−603802022852441−3217327480−170734580022910802120], where the eigenvalues of $$\mathbf{A}_{1}$$ are again $$\left\{ -\frac{1}{3},- \frac{1}{40}+\frac{1}{5}i,-\frac{1}{40}-\frac{1}{5}i\right\}$$ and $$\mathbf{A}_{2}$$ is in observable canonical form as described in (2.2) with the same eigenvalues $$\left\{ -4,-3,-2\right\} $$. Furthermore, we also have $$ \phi_{1}<2\psi_{2}$$$$\left(0.5754<0.8660\right)$$. This system is stable with $${\it {\gamma}} ^{\ast }\left(\theta_{1}^{\ast },\theta_{2}^{\ast }\right) =0.4641$$ and with the coupling constant $$B= 1.0375$$. 5. Conclusions In this article, we gave the necessary and sufficient conditions for GAS of a class of BPLS in $$\mathbb{R}^{3}$$ with discontinuous vector fields. Unlike the article Eldem & Öner (2015), we take into account three cases ($$\mu_{1}=\mu_{2}<\mu_{3},$$$$\mu_{1}<\mu_{2}=\mu_{3}$$ and $$\mu_{1}<\mu_{2}<\mu_{3}.$$) for the second mode. It is clear that the classification of the trajectories as $$(i)$$ the trajectories which change mode finite number of times as $$t \rightarrow \infty $$ and $$(ii)$$ the trajectories which change mode infinite number of times as $$t \rightarrow \infty $$ plays a crucial role in GAS. In addition, it is demonstrated by the example above that the coupling constant and the discontinuity of the vector field will play a crucial role in such cases, too. For future work, the other classes of BPLS in $$\mathbb{R}^{3}$$ can be investigated within the same framework. Moreover, finding the necessary and sufficient conditions for GAS of BPLS in $$\mathbb{R}^{n}$$ is still an open and non-trivial problem and can be considered as a challenging future work. References Branicky M. S. ( 1998 ) Multiple Lyapunov functions and other analysis tools for switched and hybrid systems . IEEE Trans. Automat. Contr., 43 , 475 – 482 . Google Scholar CrossRef Search ADS Çamlibel M. K. 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In view of Remark 2.1 there exists $$\tau_{1}>0 $$ such that $$c^{T}z_{1}(\tau_{1})=0$$ and the trajectory changes mode. Using equation (2.9) we get z1(τ1)=eσ1τ1v1(θ^1), where $$\hat{\theta}_{1}$$ in $$(\pi -\phi_{1},\pi +\phi_{1})$$. Then, in view of Definition 2.4 there exists a unique $$\theta_{2}$$ in $$\left[-\phi_{2},2\pi -\phi_{2}\right] $$ and $$\eta_{12}(\hat{\theta}_{1})>0$$ such that $$\mathbf{v}_{1}(\hat{\theta}_{1})=\eta_{12}(\hat{\theta}_{1})\mathbf{v}_{2}(\theta_{2}).$$ Therefore, we get eσ1τ1v1(θ^1)=η12(θ^1)eσ1τ1(s^2sin⁡θ2+s^3cos⁡θ2) which implies that x^1sin⁡(θ^1)+y1cos⁡(θ^1)=η12(θ^1)(s^2sin⁡θ2+s^3cos⁡θ2) or equivalently Γ1[sin⁡θ^1cos⁡θ^1]=η12(θ^1)Γ2[sin⁡θ2cos⁡θ2]. The equation above can be written more explicitly as follows. Γ1[sin⁡θ^1cos⁡θ^1]=η12Γ2[sin⁡θ2cos⁡θ2]. Let $Q_{1}=\left[\begin{array}{@{}cc}1 & 0 \\b_{1} & 1\end{array}\right] $, $Q_{2}=\left[\begin{array}{@{}cc}1 & 0 \\ \frac{\mu_{2}-\mu_{1}}{\mu_{3}-\mu_{1}} & 1\end{array}\right] $ and $$Q_{3}=$$ $\left[\begin{array}{@{}cc}1 & \mu_{2} \\0 & 1\end{array}\right].$ Then, we get Q3Γ1Q1−1Q1[sin⁡θ^1cos⁡θ^1]=η12Q3Γ2Q2−1Q2[sin⁡θ2cos⁡θ2] and we obtain, η12[sin⁡θ2cos⁡θ2+b2sin⁡θ2]=Γ2−1Γ1[sin⁡θ^1cos⁡θ^1+b1sin⁡θ^1] where $$b_{2}=\cot \phi_{2}=\frac{\mu_{2}-\mu_{1}}{\mu_{3}-\mu_{1}}$$ and Γ2−1Γ1=ω1a32a21(μ2−μ1)(μ3−μ2)[−ω1(b12+1)−(λ1−a11−μ2a21)0−b2a21(μ3−μ2)]. More explicitly, we have η12[sin⁡θ2cos⁡θ2+b2sin⁡θ2]=−ω1a32(μ2−μ1)(B2−B3)[1B20b2(B2−B3)][sin⁡θ^1cos⁡θ^1+b1sin⁡θ^1]. Using the equations above, we get the following solutions. η12sin⁡θ2=−ω1(sin⁡θ^1+B2(cos⁡θ^1+b1sin⁡θ^1))a32(μ2−μ1)(B2−B3)η12(cos⁡θ2+b2sin⁡θ2)=−ω1b2(cos⁡θ^1+b1sin⁡θ^1)a32(μ2−μ1)θ2=cot−1⁡{(B2−B3)b2(cot⁡θ^1+b1)(cot⁡θ^1+b1)B2+1−b2}=cot−1⁡{−b2(B3(cot⁡θ^1+b1)+1)(cot⁡θ^1+b1)B2+1} 2. Let $$\mu_{1}<\mu_{2}<\mu_{3}$$ and $$\mathbf{z}_{2}(t)$$ be a trajectory starting from $$\mathcal{S}_{2}\cap \mathcal{H}$$ with initial condition $$\mathbf{z}_{2}(0)=\eta_{12}\left(\hat{\theta}_{1}\right) e^{\sigma_{1}\tau_{1}}\mathbf{v}_{2}\left(\theta_{2}\right) $$. If $$\mathbf{z}_{2}(t)$$ is a transitive trajectory ($$\frac{\pi }{2}<\theta_{2}<\pi -\phi_{2}$$) (or equivalently outside of the cone $$\mathcal{C}^{-}$$), then there exists a $$\tau_{2}>0$$ such that $$\mathbf{c^{T}}\mathbf{\ z}_{2}(\tau_{2})=0$$ and the trajectory changes mode at $$t=\tau_{2}$$. Note that $$\mathbf{\ z}_{2}(\tau_{2})$$ can be expressed as follows. z2(τ2)=η12(θ^1)eσ1τ1eμ2τ2N2{s^2sin⁡θ^2+s^3cos⁡θ^2}. Then, in view of Definition 2.4 there exists a unique $$\theta_{11}$$ in $$(-\phi_{1},-\psi_{1})$$ and $$\eta_{21}(\theta_{2})>0$$ such that $$ \mathbf{v}_{2}(\hat{\theta}_{2})=\eta_{21}(\hat{\theta}_{2})\mathbf{v}_{1}(\theta_{11}).$$ Therefore, we have z2(τ2)=η12(θ^1)eσ1τ1eμ2τ2N2{s^2sin⁡θ^2+s^3cos⁡θ^2}=η12(θ^1)eμ2τ2eσ1τ1η21(θ^2)N2(x^1sin⁡θ11+y1cos⁡θ11) where s^2sin⁡θ^2+s^3cos⁡θ^2:=η21(θ^2){x^1sin⁡θ11+y1cos⁡θ11}. This equation can also be written as follows. Γ2[sin⁡θ^2cos⁡θ^2]=η21Γ1[sin⁡θ11cos⁡θ11]. Using the same approach as in the previous case, let $Q_{1}=\left[\begin{array}{@{}cc} 1 & 0 \\ b_{1} & 1 \end{array} \right] $, $Q_{2}=\left[\begin{array}{@{}cc} 1 & 0 \\ \frac{\mu_{2}-\mu_{1}}{\mu_{3}-\mu_{1}} & 1 \end{array} \right] $ and $Q_{3}=\left[\begin{array}{@{}cc} a_{21} & \left(\lambda_{1}-a_{11}\right) \\ 0 & 1 \end{array} \right] $. Then, we obtain Q3Γ2Q2−1Q2[10b21][sin⁡θ^2cos⁡θ^2]=η21Q3Γ1Q1−1Q1. which yields η21[sin⁡θ11cos⁡θ11+b1sin⁡θ11]=Γ1−1Γ2[sin⁡θ^2cos⁡θ^2+b2sin⁡θ^2] where Γ1−1Γ2=(μ3−μ1)a32ω1[b2(B3−B2)B20−1]. More explicitly, we have η21[sin⁡θ11cos⁡θ11+b1sin⁡θ11]=(μ3−μ1)a32ω1[b2(B3−B2)B20−1][sin⁡θ^2cos⁡θ^2+b2sin⁡θ^2]. The equation above implies that η21sin⁡θ11=(μ3−μ1)a32ω1[b2B3sin⁡θ^2+B2cos⁡θ^2]η21(cos⁡θ11+b1sin⁡θ11)=−(μ3−μ1)a32ω1(cos⁡θ^2+b2sin⁡θ2)θ11=cot−1⁡{−(cot⁡θ^2+b2)b2B3+B2cot⁡θ^2−b1}. Furthermore, using Lemma 2.3.2.b we get $$\hat{\theta}_{2}\rightarrow \frac{3\pi }{4}$$ and $$\tau_{2} \rightarrow \infty$$ as $$\theta_{2}\rightarrow \frac{\pi }{2}$$. Substituting $$ \hat{\theta}_{2}\rightarrow \frac{3\pi }{4}$$, in equation (2.28) and using the equality $$b_{2}B_{3}-B_{2}=B_{1}\left(b_{2}-1\right),$$ we get $$ \cot \theta_{11}=-\frac{1}{B_{1}}-b_{1}= \cot(-\psi_{1})$$. This implies that $$-\phi_{1}<\theta_{11}<-\psi_{1}.$$ □ Proof of Lemma 3.2. We calculate the derivative of $$\theta_{11}$$ with respect to $$\theta_{1},$$ for all cases and we get following results: dθ11dθ1=dθ11dθ^2dθ^2dτ2dτ2dθ2dθ2dθ^1dθ^1dθ1 1. Firstly, we give the proof the case $$\mu_{1}<\mu_{2}<\mu_{3}$$. Using the equation (2.28),we calculate $$\frac{d\theta_{11}}{d\hat{\theta}_{2}},$$ dθ11dθ^2=b2sin2⁡θ11(B2−B3)sin2⁡θ^2(b2B3+B2cot⁡θ^2)2 and using equations (2.24) and (2.26), we get −cot⁡θ^2=e(μ3−μ2)τ2(e(μ2−μ1)τ2−1)(e(μ3−μ1)τ2−1). Calculating the derivative of above equation with respect to $$\tau_{2},$$ we get the following equation 1sin2⁡θ^2dθ^2dτ2=e(μ3−μ2)τ2(e(μ3−μ1)τ2−1){(μ3−μ1)e(μ2−μ1)τ2+(μ3−μ1)e(μ3−μ1)τ2cot⁡θ2−(μ3−μ2)}. Since −cot⁡θ2=(e(μ2−μ1)τ2−1)(e(μ3−μ1)τ2−1)⇒e(μ3−μ1)τ2cot⁡θ2+e(μ2−μ1)τ2=cot⁡θ2+1 we get dθ^2dτ2=sin2⁡θ^2e(μ3−μ2)τ2(μ3−μ1)(cot⁡θ2+b2)(e(μ3−μ1)τ2−1). Calculate the derivative of $$\cot \theta_{2}$$ with respect to $$\hat{\theta}_{1},$$ we get 1sin2⁡θ2dθ2dθ^1=−b2B3((cot⁡θ^1+b1)B2+1)sin2⁡θ^1+b2B2(B3(cot⁡θ^1+b1)+1)sin2⁡θ^1((cot⁡θ^1+b1)B2+1)2dθ2dθ^1=b2(B2−B3)sin2⁡θ2sin2⁡θ^1((cot⁡θ^1+b1)B2+1)2 Additionally, we calculate $$\frac{d\tau_{2}}{d\theta_{2}},$$ using the equation (2.24) 1sin2⁡θ2=dτ2dθ2[(μ2−μ1)e(μ2−μ1)τ2(e(μ3−μ1)τ2−1)−(μ3−μ1)e(μ3−μ1)τ2(e(μ2−μ1)τ2−1)](exp⁡((μ3−μ1)τ2)−1)21sin2⁡θ2=dτ2dθ2[(μ2−μ1)e(μ2−μ1)τ2+(μ3−μ1)e(μ3−μ1)τ2cot⁡θ2](exp⁡((μ3−μ1)τ2)−1)dτ2dθ2=(e(μ3−μ1)τ2−1)(μ3−μ1)sin2⁡θ2e(μ2−μ1)τ2(b2+cot⁡θ^2) and then dθ11dθ1=dθ11dθ^2dθ^2dτ2dτ2dθ2dθ2dθ^1dθ^1dθ1=(b2(B2−B3)b2B3+B2cot⁡θ^2)2(e(μ3−μ2)τ2(b2+cot⁡θ2)e(μ2−μ1)τ2(b2+cot⁡θ^2))(dF1dθ1sin2⁡θ11sin2⁡θ^1((cot⁡θ^1+b1)B2+1)2) Since $$-\left(\cot \theta_{11}+b_{1}\right) =\frac{\cot \hat{\theta}_{2}+b_{2}}{b_{2}B_{3}+B_{2}\cot \hat{\theta}_{2}}$$ and $$\cot \theta_{2}+b_{2}=\frac{\left(B_{2}-B_{3}\right) b_{2}\left(\cot \hat{\theta}_{1}+b_{1}\right) }{\left(\cot \hat{\theta}_{1}+b_{1}\right) B_{2}+1}$$ we get dθ11dθ1=dF1dθ1e(μ3−μ2)τ2sin2⁡θ11e(μ2−μ1)τ2sin2⁡θ^1(cot⁡θ11+b1cot⁡θ^2+b2)2(cot⁡θ2+b2cot⁡θ^2+b2)(cot⁡θ2+b2cot⁡θ^1+b1)2=dF1dθ1e(μ3−μ2)τ2sin2⁡θ11e(μ2−μ1)τ2sin2⁡θ^1(cot⁡θ11+b1cot⁡θ^1+b1)2(cot⁡θ2+b2cot⁡θ^2+b2)3 We know that $$e^{\left(\mu_{3}-\mu_{2}\right) \tau_{2}}=\frac{\cot \hat{ \theta}_{2}}{\cot \theta_{2}}=\left(\frac{\left(\cot \theta_{11}+b_{1}\right) B_{3}+1}{\left(\cot \theta_{11}+b_{1}\right) B_{2}+1} \right) \left(\frac{\left(\cot \hat{\theta}_{1}+b_{1}\right) B_{2}+1}{ \left(\cot \hat{\theta}_{1}+b_{1}\right) B_{3}+1}\right) $$, e(μ1−μ2)τ2=1+cot⁡θ^21+cot⁡θ2=((cot⁡θ11+b1)B1+1(cot⁡θ^1+b1)B1+1)((cot⁡θ^1+b1)B2+1(cot⁡θ11+b1)B2+1) and $$\frac{\cot \theta_{2}+b_{2}}{\cot \hat{\theta}_{2}+b_{2}}=\frac{\frac{ \left(B_{2}-B_{3}\right) b_{2}\left(\cot \hat{\theta}_{1}+b_{1}\right) }{ \left(\cot \hat{\theta}_{1}+b_{1}\right) B_{2}+1}}{\frac{\left(B_{2}-B_{3}\right) b_{2}\left(\cot \theta_{11}+b_{1}\right) }{\left(\cot \theta_{11}+b_{1}\right) B_{2}+1}}$$ then dθ11dθ1=sin2⁡θ11e(μ2−μ1)τ2sin2⁡θ^1(cot⁡θ1+b1)(cot⁡θ11+b1)((cot⁡θ11+b1)B3+1(cot⁡θ^1+b1)B3+1)((cot⁡θ11+b1)B2+1(cot⁡θ^1+b1)B2+1)2=e(μ1−μ2)τ2(sin⁡(θ11+ψ2)sin⁡(θ^1+ψ2))2(cot⁡θ1+b1)(cot⁡θ11+b1)((cot⁡θ11+b1)B3+1(cot⁡θ^1+b1)B3+1)=(sin⁡(θ11+ψ1)sin⁡(θ^1+ψ1))(sin⁡(θ11+ψ2)sin⁡(θ^1+ψ2))(sin⁡(θ1+ϕ1)sin⁡(θ11+ϕ1))(sin⁡(θ11+ψ3)sin⁡(θ^1+ψ3))sin⁡θ^1sin⁡θ1 where $$\cot \psi_{2}=\frac{1}{B_{2}}+b_{1}$$ and $$\cot \psi_{3}=\frac{1}{B_{3}}+b_{1}$$. 2. In view of the above development, we have dθ11dθ1=e(μ1−μ2)τ2(sin⁡(θ11+ψ2)sin⁡(θ^1+ψ2))2(cot⁡θ1+b1)(cot⁡θ11+b1)((cot⁡θ11+b1)B3+1(cot⁡θ^1+b1)B3+1). Using this equation and $$\mu_{1}=\mu_{2},$$ (and $$B_{1}=B_{2}$$), then we get dθ11dθ1=(sin⁡(θ11+ψ2)sin⁡(θ^1+ψ2))2(cot⁡θ1+b1)(cot⁡θ11+b1)((cot⁡θ11+b1)B3+1(cot⁡θ^1+b1)B3+1)=(sin⁡(θ11+ψ2)sin⁡(θ^1+ψ2))2(sin⁡(θ1+ϕ1)sin⁡(θ11+ϕ1))(sin⁡(θ11+ψ3)sin⁡(θ^1+ψ3))sin⁡θ^1sin⁡θ1. 3. We get following equation for the first case. dθ11dθ1=sin2⁡θ11e(μ2−μ1)τ2sin2⁡θ^1(cot⁡θ1+b1)(cot⁡θ11+b1)((cot⁡θ11+b1)B3+1(cot⁡θ^1+b1)B3+1)((cot⁡θ11+b1)B2+1(cot⁡θ^1+b1)B2+1)2 For $$\mu_{1}<\mu_{2}=\mu_{3}$$, we know that $$B_{3}=B_{2}$$, then we get dθ11dθ1=sin2⁡θ11e(μ3−μ1)τ2sin2⁡θ^1(cot⁡θ1+b1)(cot⁡θ11+b1)((cot⁡θ11+b1)B3+1(cot⁡θ^1+b1)B3+1)3 and for this case we know that e(μ3−μ1)τ2=cot⁡θ^2cot⁡θ2=((cot⁡θ11+b1)B3+1(cot⁡θ11+b1)B1+1)((cot⁡θ^1+b1)B1+1(cot⁡θ^1+b1)B3+1). Therefore dθ11dθ1=sin2⁡θ11sin2⁡θ^1(cot⁡θ1+b1cot⁡θ11+b1)((cot⁡θ11+b1)B3+1(cot⁡θ^1+b1)B3+1)2((cot⁡θ11+b1)B1+1(cot⁡θ^1+b1)B1+1)=(sin⁡(θ11+ψ3)sin⁡(θ^1+ψ3))2(cot⁡θ1+b1cot⁡θ11+b1)((cot⁡θ11+b1)B1+1(cot⁡θ^1+b1)B1+1)=(sin⁡(θ11+ψ3)sin⁡(θ^1+ψ3))2(sin⁡(θ1+ϕ1)sin⁡(θ11+ϕ1))(sin⁡(θ11+ψ1)sin⁡(θ^1+ψ1))sin⁡θ^1sin⁡θ1. □ © The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png IMA Journal of Mathematical Control and Information Oxford University Press

Necessary and sufficient conditions for global asymptotic stability of a class of bimodal systems with discontinuous vector fields

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Oxford University Press
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© The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
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0265-0754
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1471-6887
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10.1093/imamci/dnx018
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Abstract

Abstract In this article, we derive the necessary and sufficient conditions for global asymptotic stability of a class of bimodal piecewise linear systems in $$\mathbb{R}^{3}$$, where first mode has complex eigenvalues and the second mode has only real eigenvalues. The vector field is allowed to be discontinuous on the switching plane. The effect of discontinuity is illustrated by two examples. 1. Introduction During the past 20 years, the research on switched systems has gained a great momentum. This is basically due to the fact that there are many practical control applications which contain both continuous time dynamical systems and switching elements. The literature contains a rich variety of mathematical models proposed to represent behaviour of hybrid control systems. One of the typical models is the switched system which comprises a finite number of subsystems and a switching rule that determines the active subsystem at each instant. Piecewise linear systems (PLS) comprise a subclass of switched systems where the subsystems are linear, time invariant and switching is autonomous (state dependent). The class of PLS is one of the fundamental classes of hybrid dynamical systems, because the continuous dynamics is linear in each mode and the discrete dynamics is the simplest one. Therefore, study on PLS is important as a first step to constitute hybrid control theory. Bimodal piecewise linear system (BPLS) comprises a subclass of PLS, where there are only two subsystems. In spite of recent progress in hybrid control theory, there still remain fundamental issues to be clarified. Stability is one of them. One of the approaches used for the solution of stability problems with arbitrary switching is based on the existence of a common quadratic Lyapunov function for individual subsystems. Along this line, the works of Mori et al. (1997), Shorten & Narendra (1999), Shorten et al. (2004) and Sun (2010) can be cited. If a switched system has more than two subsystems, then finding a common quadratic Lyapunov function becomes a difficult task. In such cases, another approach, which is based on a nontraditional Lyapunov functions is employed. For the details of this approach, the works of Branicky (1998), Michel & Hu (1999), Liberzon & Morse (1999) and the survey papers by Decarlo et al. (2000), Shorten et al. (2007) may be referred to. The approaches, which are based on Lyapunov functions, usually yield sufficient conditions for stability. Therefore, we need a new approach to get a less conservative stability condition or to derive a set of necessary and sufficient conditions for the stability. To this end, we investigate the stability problem for BPLS from a different perspective. One of the main issues in BPLS is well posedness, i.e. the existence and uniqueness of the solutions. This problem is addressed in detail by Imura & van der Schaft (2000) for BPLS in $$\mathbb{R}^{n}$$. Later, it is shown by Eldem & Şahan (2014) and Şahan & Eldem (2015) that well-posedness conditions, given by Imura & van der Schaft (2000), induce a joint structure for subsystem matrices of BPLS in $$\mathbb{\mathbb{R}}^{n }$$. PLS are investigated extensively by many authors in the context of stability. The necessary and sufficient conditions for global asymptotic stability (GAS) of BPLS (with continuous vector fields) in $$\mathbb{\mathbb{R}}^{2}$$ is given by Çamlibel et al. (2003). The same problem (with discontinuous vector fields) is investigated by Iwatani & Hara (2006) and the importance of well-posedness is demonstrated by an example (Example 13 in Iwatani & Hara, 2006). Stability of BPLS in $$\mathbb{\mathbb{R}}^{3}$$ has also attracted considerable attention in the literature. Carmona et al. (2005, 2006) have considered the stability of BPLS in $$\mathbb{\mathbb{R}}^{3}$$ with continuous vector fields. In these papers, BPLS is transformed to the surface of the unit sphere in $$\mathbb{\mathbb{R}}^{3}$$ centred at the origin. In this framework, the authors searched for periodic solutions which is equivalent to the search for invariant cones of the original BPLS. Separate necessary and sufficient conditions for GAS of BPLS in $$\mathbb{\mathbb{R}}^{n},$$ where $$n>2$$ is given by Iwatani & Hara (2006). It is shown by Eldem & Şahan (2014) that the GAS of BPLS in $$\mathbb{\mathbb{R}}^{3}$$, where both modes have complex eigenvalues, reduces to the GAS of BPLS in $$\mathbb{\mathbb{R}}^{2}$$, if BPLS in $$\mathbb{R}^{3}$$ has a certain structure. Additionally, Eldem & Öner (2015) give necessary and sufficient conditions for GAS of BPLS where one mode has complex eigenvalues and the second mode have only real eigenvalues with algebraic multiplicity equal to three and geometric multiplicity is equal to one. In this article, the GAS of a class of BPLS in $$\mathbb{R}^{3}$$ is investigated under autonomous switching where first mode has complex eigenvalues and the second mode has only real eigenvalues. For the second mode, we have four cases of algebraic and geometric multiplicity, but we investigate only three cases. Because, for the case where the eigenvalues are $$\mu _{1}=\mu _{2}=\mu _{3},$$ the necessary and sufficient stability conditions of GAS of BPLS is given by Eldem & Öner (2015). The cases to be studied in this article create certain difficulties for the proof of the main result. Therefore, we need an additional assumption to overcome these difficulties. The article is organized as follows. Well-posedness conditions for BPLS and the behaviour of trajectories will be given in Section II. The behaviour of the trajectories is investigated towards a classification of trajectories. Stability of the classified trajectories is given in Section III. In Section IV, the main theorem of this article and some examples, which illustrate the theoretical results, are presented. Finally, the article ends with the conclusions in Section V. In order to make the reading smoother, the proofs of some of the results are given in the Appendix. 2. Preliminaries Throughout this article, we study GAS of the class of BPLS with the following mathematical model: Σ0:x˙={A1xifcTx≥0A2xifcTx≤0  (2.1) where $$\mathbf{x,c}\in \mathbb{\mathbb{R}}^{3}$$ and $$\mathbf{A}_{1},\mathbf{A}_{2}$$ are real $$3\times 3$$ matrices. For such systems, $$\mathcal{H}:=\left\{ \mathbf{x}\text{ }|\text{ }\mathbf{c}^{T}\mathbf{x}=0\right\} $$ denotes the plane which divides $$\mathbb{\mathbb{R}}^{3}$$ into two open half-spaces, $$\mathcal{H}^{+}=\left\{ \mathbf{x}\text{ }|\text{ }\mathbf{c}^{T}\mathbf{x>}0\right\} $$ and $$\mathcal{H}^{-}=\left\{ \mathbf{x}\text{ }|\text{ }\mathbf{c}^{T}\mathbf{x}<0\right\}\!.$$ Since both equations hold on $$\mathcal{H},$$ we first resolve the existence and uniqueness of the solutions (well-posedness) starting on $$\mathcal{H}$$. 2.1. Well-posedness and geometry of initial conditions in $$\mathbb{R}^{3}$$ Well-posedness simply means the existence and uniqueness of solutions. The first results about well-posedness of BPLS in $$\mathbb{R}^{n}$$ are given by Imura & van der Schaft (2000). Then, the structural issues of well-posednes are further investigated in $$\mathbb{R}^{3}$$ and $$\mathbb{R}^{n}$$ by Eldem & Şahan (2014) and Şahan & Eldem (2015), respectively. Classical solutions (which are continuously differentiable curves) fail to describe the behaviour of dynamical systems with discontinuous vector fields. In order to remedy this situation different solution notions (such as Carathéodory, Filippov, Krasovskii, sample-and-hold, Euler, Hermes...etc.) are introduced in the literature. An extensive account of these solution notions can be found in Cortes (2008). Among these, Carathéodory and Filippov solutions are the ones which are used most frequently. As explained in Cortes (2008), Carathéodory solutions can be considered as a generalization of classical solutions where the solution does not satisfy the differential equation on a set of measure zero. On the other hand, Filippov solutions replace the differential equations with differential inclusions which are defined by set-valued maps. Thus, instead of focusing the value of the vector field at a specific point $$x$$, Filippov solutions focus on a set of directions in the neighbourhood of $$x$$. Definitions of these solution notions are as follows. Definition 2.1 An absolutely continuous function $$x\mathbf{(}t\mathbf{):\mathbb{R} \rightarrow \mathbb{R}}^{n}$$ is said to be 1. Carathéodory solution of $$\it{\Sigma}_{0}$$ for the initial condition $$x_{0}$$ if $$x\mathbf{(}0\mathbf{)=} x_{0}$$ and $$x\mathbf{(}t\mathbf{)}$$ satisfies (2.1) for almost all $$t\in \mathbf{{\mathbb{R}}}$$. 2. Forward Carathéodory solution for the initial state $$x_{0},$$ if it is a solution in the sense of Carathéodory, and for each $$t_{0}\geq 0,$$ there exists $$\epsilon >0$$ such that either x˙=A1xandcTx≥0,orx˙=A2xandcTx≤0  holds for all $$t\in \left[t_{0},t_{0}+\epsilon \right] $$. 3. Filippov solution if $$x(t)$$ satisfies $${\dot{x}}\left(t\right) \in F\left(x\left(t\right) \right) $$ for almost all $$t\geq 0$$ where set valued function $$F$$ is given by F(x)={{A1x}ifcTx>0conv({A1x,A2x})if cTx=0{A2x}ifcTx<0  and $$conv(S)$$ denotes the convex hull of the set $$S.$$ In view of intuitive explanations and the definitions above, it is clear that Carathéodory and Filippov solutions are not related in general. In other words, there are vector fields with Carathéodory solutions which are not Filippov solutions and converse is also true. However, there are also vector fields where Carathéodory and Filippov solutions coincide. These fundamental questions on the relation between Carathéodory and Filippov solutions are investigated in Spraker & Biles (1996). For instance, Theorems 1 and 2 of Spraker & Biles (1996), provide conditions for the equivalence of Carathéodory and Filippov solutions. A detailed analysis of Carathéodory and Filippov solutions in the context of BPLS with discontinuous vector fields is given in Thuan & Çamllibel (2014). In Theorem 3.1 of Thuan & Çamllibel (2014), it is shown that if the equations in statements 5 and 6 of Theorem 3.1 hold, then every Filippov solution of the system is right unique and every Filippov solution is also both a forward and backward Carathéodory solution. It can be easily shown that equations in statements 5 and 6 of Theorem 3.1 in Thuan & Çamllibel (2014), also hold in our setup. In fact Assumptions 2.1 and 2.2 given below imply that the equations in statements 5 and 6 of Theorem 3.1 in Thuan & Çamllibel (2014) hold for bimodal systems considered in our article. Furthermore, bimodal systems described in Corollary 3.5 of Thuan & Çamllibel (2014) are exactly the systems described by equation (2.1) with the structure of (2.2) given below. Therefore, under the assumptions which guarantee well-posedness, it follows that forward Carathéodory solutions of (2.1) with the structure of (2.2) are also Filippov solutions. Consequently, the results on GAS presented in this article are also valid for Filippov solutions. In order to establish well-posedness, we need some assumptions. Assumption 2.1 The pairs $$\left(\mathbf{c}^{T}\boldsymbol{,\ } \mathbf{A}_{1}\right)$$ and $$\left(\mathbf{c}^{T} \boldsymbol{,}\mathbf{A}_{2}\right) $$ are observable and only $$ \left(\mathbf{c}^{T}\boldsymbol{,}\mathbf{A}_{2}\right) $$ is in observable canonical form. In view of this assumption the parameters of BPLS can be expressed as follows. A1=[a11 a12 a13 a21a22a23a31a32a33], A2=[00μ1μ2μ310−μ1μ2−μ1μ3−μ2μ301μ1+μ2+μ3]and c=[001]. The observability of the pair $$\left(\mathbf{c}^{T}\boldsymbol{,}\mathbf{A}_{i}\right),$$$$i=1,2$$ implies that $$\dim \left(\ker \mathbf{c}^{T}\cap {\ker }(\mathbf{c}^{T}\mathbf{A}_{i})\right) {\small =1}$$. Thus, $$\mathcal{L}_{i}:={\ker }\mathbf{c}^{T}\cap {\ker }(\mathbf{c}^{T}\mathbf{A}_{i})$$ is a line passing through the origin which divides $$\mathcal{H}$$ into two open half planes $$\mathcal{P}_{i}^{+}$$ and $$\mathcal{P}_{i}^{-}$$. On one side of this line $$\mathbf{c}^{T}\mathbf{A}_{i}\mathbf{x}>0$$$$\left(\mathcal{P}_{i}^{+}\right) $$ and one the other side $$\mathbf{c}^{T}\mathbf{A}_{i} \mathbf{x}<0$$$$\left(\mathcal{P}_{i}^{-}\right) $$. Similarly, the origin $${ \ker }\mathbf{c}^{T}\cap {\ker }\boldsymbol{(}\mathbf{c}^{T}\mathbf{A}_{i})\cap {\ker }(\mathbf{c}^{T}\mathbf{A}_{i}^{2})$$ divides $$\mathcal{L}_{i} $$ into two open half lines $$\mathcal{L}_{i}^{+}$$ where $$\mathbf{c}^{T} \mathbf{A}_{i}^{2}\mathbf{x}>0$$ and $$\mathcal{L}_{i}^{-}$$ where $$\mathbf{c} ^{T}\mathbf{A}_{i}^{2}\mathbf{x}<0$$. In view of the above development, and along the lines used in Imura & van der Schaft (2000), the set of initial conditions in $$\mathbb{R}^{3}$$ can be classified as follows. Definition 2.2 Let $$\mathcal{S}_{i}$$$$\left(i=1,2\right) $$ denote the set of initial conditions in $$\mathbb{R}^{3}$$ such that for every $$x_{0}\in \mathcal{S}_{i}$$ there exist $$\epsilon >0$$ and a unique forward Carathéodory solution of $$\it{\Sigma}_{0}$$ and $$\dot{x}\mathbf{=}A_{i}x$$ for all $$t\in \left[t_{0},t_{0}+\epsilon \right] $$. In this case, we say that the solution of$$\it{\Sigma}_{0}$$smoothly continues in$$\mathcal{S}_{i}$$. In view of the above definitions and Lemma 1 given in Eldem & Şahan (2014) or items (1) and (2) of Theorem 3.1 given in Şahan & Eldem (2015), we have the following assumption. Assumption 2.2 $$\ker c^{T}\cap \ker \left(c^{T}\mathbf{{A}_{1}} \right) =\ker c^{T}\cap \ker \left(c^{T}\mathbf{{A}_{2}}\right) $$ (equivalently $$a_{31}=0$$) and $$a_{21}>0$$, $$a_{32}>0,$$ i.e., $$\it{\Sigma}_{0}$$ is well-posed. It is easy to see that for a well-posed BPLS, we have $$\mathcal{P}_{1}^{+}=\mathcal{P}_{2}^{+},$$$$\mathcal{P}_{1}^{-}=\mathcal{P}_{2}^{-},$$$$\mathcal{L}_{1}^{+}= \mathcal{L}_{2}^{+},$$ and $$\mathcal{L}_{1}^{-}=\mathcal{L}_{2}^{-}$$. Consequently, it follows that $$\mathcal{S}_{1}=\mathcal{H}^{+}\cup \mathcal{P }_{1}^{+}\cup $$$$\mathcal{L}_{1}^{+}$$ and $$\mathcal{S}_{2}=\mathcal{H} ^{-}\cup \mathcal{P}_{2}^{-}\cup $$$$\mathcal{L}_{2}^{-}$$. This implies that $$\mathcal{S}_{1}\cup \mathcal{S}_{2}=\mathbb{R}^{3}$$ and $$\mathcal{S}_{1}\cap \mathcal{S}_{2}=\left\{ 0\right\} $$ and this guarantees well-posedness as defined in Imura & van der Schaft (2000). Under these assumptions, bimodal system is well-posed which guarantees the following fact. Fact 1: The vector field of both modes has the same sign on any initial condition on $$\mathcal{H}$$. Furthermore, the components of $$\it{\Sigma}_{0}$$ can be written as follows. A1=[a11 a12 a13 a21a22a230a32a33], A2=[00μ1μ2μ310−μ1μ2−μ1μ3−μ2μ301μ1+μ2+μ3] and c=[001]. (2.2) The last assumption, which is not related to well-posedness is given below. Assumption 2.3 The eigenvalues of $$\mathbf{A}_{1}$$ are $$\{\lambda_{1},\sigma_{1}\pm j\omega_{1}\}$$ where $$\lambda_{1}$$, $$\sigma_{1}$$ and $$ \omega_{1}>0$$ are real numbers such that $$\sigma_{1}-\lambda_{1}>0$$. The eigenvalues of $$\mathbf{A}_{2}$$ are $$\{\mu_{1},\mu_{2},\mu_{3}\}$$ where $$ \mu_{1},\mu_{2},$$ and $$\mu_{3}$$ are real numbers such that $$\mu_{1}\leq \mu_{2}\leq \mu_{3}.$$ 2.2. Behaviour of the trajectories in the first mode: The properties of the trajectories in the first mode are investigated in detail in Eldem & Şahan (2014). We give a summary of these results below. Let the eigenvectors of $$\mathbf{{A}_{1}}$$ be $$\left\{ \mathbf{r}_{1},\text{ }\mathbf{x}_{1}\pm j \mathbf{y}_{1}\right\} $$ where $$j=\sqrt{-1}$$. Since the solutions of $$\it{\Sigma}_{0}$$ starting from $$\mathcal{S}_{1}$$ can be expressed by eigenvectors of $$\mathbf{{A}_{1},}$$$$\left\{ \mathbf{r}_{1}, \text{ }\mathbf{x}_{1}\pm j\mathbf{y}_{1}\right\} $$ are chosen uniquely as follows. r1=[(a11−σ1)2+ω12+a12a21a32a21λ1−a33a321],x1=[(a11−λ1)(a11−σ1)+a12a21a32a21σ1−a33a321],y1=[ω1(a11−λ1)a32a21ω1a320]. (2.3) Let $$\mathbf{z}_{1}(t)$$ denote the solutions of $$\it{\Sigma}_{0}$$ with initial conditions in $$\mathcal{S}_{1}$$. Then, the behaviour of such trajectories (except the trajectories with $$\mathbf{z}_{1}(0)=$$$$K_{1}\mathbf{r}_{1},$$ where $$K_{1}>0$$) in the first mode can be expressed as follows. z1(t)=K1eλ1t{α1r1+e(σ1−λ1)t[x1sin⁡(θ1+ω1t)+y1cos⁡(θ1+ω1t)]}, (2.4) where $$K_{1}>0$$ is a real constant. Let f1(t):=α1+e(σ1−λ1)tsin⁡(θ1+ω1t). (2.5) Note that for any initial condition in $$\mathcal{S}_{1}$$, we have $$\mathbf{c} ^{T}\mathbf{z}_{1}(0)\geq 0$$. Then, it follows that cTz1(t)=K1eλ1t{α1+e(σ1−λ1)tsin⁡(θ1+ω1t)},=K1eλ1t{f1(t)}, (2.6) and $$\mathbf{c}^{T}\mathbf{z}_{1}(0)=K_{1}f_{1}(0)=K_{1}\left(\alpha_{1}+\sin \theta_{1}\right) \geq 0$$. Further note that the sign of $$\mathbf{c}^{T}\mathbf{z}_{1}(t)$$ and $$f_{1}(t)$$ are the same. The trajectories of $$\it{\Sigma}_{0}$$ may or may not change mode as shown in Eldem & Şahan (2014). From the point of view of stability, it is important to distinguish the trajectories which do not change mode. Towards this end, we introduce the following definitions and Lemmas from Eldem & Şahan (2014). Definition 2.3 (Eldem & Şahan, 2014, Definition 1) Let $$\mathbf{z}_{i}(t)$$ be a solution of $$\it{\Sigma}_{0}$$ with initial condition in $$\mathcal{S}_{i}$$$$\left(i=1,2\right) $$. If there exists a finite $$\tau_{i}>0$$ such that $$\mathbf{c}^{T}\mathbf{z}_{i}\left(\tau_{i}\right) =0$$ and $$\mathbf{z}_{i}(t)$$ changes mode at $$t=\tau_{i}$$, then $$ \mathbf{z}_{i}(t)$$ is called a transitive trajectory. Otherwise, it is called a non-transitive trajectory. Remark 2.1 The classification given in the above definition can be obtained easily by using equations (2.4)-(2.6). Towards this end, note that for any initial condition on $$\mathcal{H}\cap \mathcal{S}_{1}$$, we have, $$\alpha_{1}=-\sin {\theta_{1}}$$ in equation (2.6). Therefore, any initial condition $$\mathbf{z}_{1}(0)$$ in $$\mathcal{H}\cap \mathcal{S}_{1}$$ can be written as follows. z1(0)=K1(x^1sin⁡θ1+y1cos⁡θ1), where x^1:=x1−r1 and K1>0. (2.7) In order to determine the domain of $$\sin \theta_{1}$$ for initial conditions in $$\mathcal{H}\cap \mathcal{S}_{1}$$, the time derivative of $$ \mathbf{c^{T}z}_{1}(t)$$ at $$t=0$$ can be calculated easily as follows. cTA1z1(0)=R1sin⁡(θ1+ϕ1) where R1:=(σ1−λ1)2+ω12, b1:=cot⁡(ϕ1):=σ1−λ1ω1. (2.8) It is easy to see that $$\mathbf{c^{T}A_{1}z}_{1}\left(0\right) =0$$ and $$ \mathbf{c^{T}A_{1}^{2}z}_{1}\left(0\right) >0$$ at $$\theta_{1}=-\phi_{1}$$ whereas $$\mathbf{c^{T}A_{1}z}_{1}\left(0\right) =0$$ and $$\mathbf{c^{T}A_{1}^{2}z}_{1}\left(0\right) <0$$ at $$\theta_{1}=\pi -\phi_{1}$$. Consequently, the set of initial conditions in $$\mathcal{H}\cap \mathcal{S}_{1}$$ can be characterized as in equation (2.7) where $$-\phi_{1}\leq \theta_{1}<\pi -\phi_{1}$$. Similarly, in view of Fact 1, the set of initial conditions in $$\mathcal{H}\cap \mathcal{S}_{2}$$ can be characterized as in equation (2.7) where $$\pi -\phi_{1}\leq \theta_{1}<2\pi -\phi_{1}$$. On the other hand, let $$\tau_{1}>0$$ be the first instant where $$\mathbf{c^{T}z}_{1}(\tau_{1})=0$$. Then, it follows that z1(τ1)=K1eσ1τ1(x^1sin⁡(θ1+ω1τ1)+y1cos⁡(θ1+ω1τ1)) (2.9) e(σ1−λ1)τ1sin⁡(θ1+ω1τ1)=sin⁡θ1. (2.10) Since $$\sigma_{1}-\lambda_{1}>0$$ by Assumption 2.3, the existence of $$\tau_{1}>0$$ such that $$\mathbf{c^{T}z}_{1}(\tau_{1})=0$$ is obvious. Note further that since $$e^{-\left(\sigma_{1}-\lambda_{1}\right) \tau_{1}}\sin \theta_{1}$$ is an exponentially decreasing function and $$\sin \left(\theta_{1}+\omega_{1}\tau_{1}\right) $$ is a sinusoidal function, equation (2.10) implies that their first intersection after the start can be written as follows. if−ϕ1≤θ1≤0,⇒π≤θ1+ω1τ1<π+ϕ1, (2.11)  if 0≤θ1<π−ϕ1,⇒π−ϕ1<θ1+ω1τ1≤π. (2.12) If we define $$F_{1}(\theta_{1}):=\theta_{1}+\omega_{1}\tau_{1}$$. Then, using implicit differentiation in equation (2.10), it can be easily shown that dF1dθ1=cot⁡θ1+b1cot⁡(θ1+ω1τ1)+b1≤0, (2.13) where the equality holds only at $$\theta_{1}=-\phi_{1}$$. The details of the development given above can be found in Lemma’s 3,4, and 5 in Eldem & Şahan (2014). Note that $$\tau_{1}$$ and $$\theta_{1}+\omega_{1}\tau_{1}$$ given above are functions of $$\theta_{1}$$. In order to simplify the notation, we use θ^1:=θ1+ω1τ1 (2.14) in the rest of the article. 2.3. Behaviour of the trajectories in the second mode: The properties of trajectories starting from $$\mathcal{S}_{2}$$ are investigated in detail in Öner (2014). We give a summary of these results below without proof. The proofs can be found in Öner (2014). Since the solutions which start from $$\mathcal{S}_{2}$$ can be expressed in terms of eigenvectors of $$A_2$$, the choice of eigenvectors must be done appropriately as given by the following Lemma. Lemma 2.1 (Lemma 4.1 in Öner (2014)) Let $$\{\mathbf{s}_{1}\boldsymbol{,}\;\mathbf{s}_{2}\boldsymbol{,}\; \mathbf{s}_{3}\}$$ denote the eigenvectors or generalized eigenvectors of $$ \mathbf{A}_{2}$$. Then, eigenvectors or generalized eigenvectors can be uniquely chosen such that s1=[−μ2μ3μ2+μ3−1],s2=[−μ1μ3μ1+μ3−1],s3=[−μ1μ2μ1+μ2−1] for μ1<μ2<μ3. (2.15) For the other cases we have if μ1=μ2<μ3 then s2=[μ3−10]T. if μ1<μ2=μ3 then s3=[μ1−10]T. Let $$\mathbf{z}_{2}(t)$$ be a solution of $$\it{\Sigma}_{0}$$ with initial condition in $$\mathcal{S}_{2}$$. Then, the behaviour of such trajectories in the second mode can be written as z2(t)={d1eμ1ts1+d2eμ2tq2(t)+d3eμ3tq3(t)}, where $$d_{1},d_{2},d_{3}$$ are real constants and $$\{\mathbf{s}_{1} \boldsymbol{,}\mathbf{q}_{2}\left(t\right) \boldsymbol{,}\mathbf{q}_{3}\left(t\right) \}$$ are vector functions in $$\mathbb{R}^{3}$$ given as follows. ifμ1<μ2<μ3, then q2(t):=s2,q3(t):=s3, ifμ1=μ2<μ3, then q2(t):=s1t+s2,q3(t):=s3,ifμ1<μ2=μ3, then q2(t):=s2,q3(t):=s2t+s3,  (2.16) Let $$K_{2}:=\sqrt{d_{2}^{2}+d_{3}^{2}},$$$$\sin \theta_{2}:=\frac{d_{2}}{K_{2}},$$$$\cos \theta_{2}:=\frac{d_{3}}{K_{2}},$$ and $$\beta_{1}=\frac{d_{1} }{K_{2}}$$. Then, $$\mathbf{z}_{2}\mathbf{(}t\mathbf{)}$$ and $$\mathbf{c}^{T} \mathbf{z}_{2}\mathbf{(}t\mathbf{)}$$ can be written as follows. z2(t)=K2eμ1t[β1s1+q2(t)e(μ2−μ1)tsin⁡θ2+q3(t)e(μ3−μ1)tcos⁡θ2]. (2.17) cTz2(t)=K2eμ1t[−β1+(cTq2(t))e(μ2−μ1)tsin⁡θ2+(cTq3(t))e(μ3−μ1)tcos⁡θ2]. (2.18) Let $$f_{2}(t)$$ be defined as f2(t):=[−β1+(cTq2(t))e(μ2−μ1)tsin⁡θ2+(cTq3(t))e(μ3−μ1)tcos⁡θ2]. (2.19) Since $$\mathbf{c}^{T}\mathbf{z}_{2}\mathbf{(}t\mathbf{)}=K_{2}e^{\mu_{1}t}f_{2}(t)$$, the sign of $$f_{2}(t)$$ and the sign of $$\mathbf{c}^{T}\mathbf{z}_{2}\mathbf{(}t\mathbf{)}$$ are the same. 2.4. Smooth continuation of trajectories and bases for $$\mathcal{H}$$ In view of equation (2.18), a trajectory $$\mathbf{z}_{2}(t)$$ with initial condition not on $$\mathcal{H}$$, smoothly continues into $$\mathcal{S}_{2}$$ if $$f_{2}(0)<0$$. For trajectories starting from $$\mathcal{H}$$, the following result holds. Lemma 2.2 Suppose that $$\mathbf{c}^{T}\mathbf{z}_{2}(0)=0.$$ Then, $$\mathbf{z}_{2}(t)\,$$ smoothly continues into $$\mathcal{S}_{2}$$ if and only if $$-\phi_{2}\leq \theta_{2}<\pi -\phi_{2}$$, where cot⁡ϕ2=μ2−μ1μ3−μ1ifμ3>μ2>μ1,cot⁡ϕ2=1μ3−μ1ifμ3>μ2=μ1,cot⁡ϕ2=μ3−μ1ifμ3=μ2>μ1. We have two different bases for $$\mathcal{H}$$, namely $$\left\{ \hat{\mathbf{x}}_{1},\mathbf{y}_{1}\right\} $$ and $$\left\{ \hat{\mathbf{s}}_{2}, \hat{\mathbf{s}}_{3}\right\}\!,$$ where $$\left\{ \hat{\mathbf{x}}_{1},\mathbf{y}_{1}\right\} $$ is as defined by equations (2.3),(2.7) and $$\left\{ \hat{\mathbf{s}}_{2},\hat{\mathbf{s}}_{3}\right\} $$ is as defined by the following equations. s^2=s2+(cTs2)s1 and s^3=s3+(cTs3)s1. For trajectories starting from $$\mathcal{H}\cap \mathcal{S}_{2}$$, we have $$f_{2}(0)=0$$. Thus, equations (2.16) and (2.18) imply that β1=(cTs2)sin⁡θ2+(cTs3)cos⁡θ2 (2.20) and z2(0)=K2(β1s1+s2sin⁡θ2+s3cos⁡θ2). (2.21) It can be easily seen by Lemma 2.1 and equation (2.16) that $$\mathbf{c}^{T}\boldsymbol{\hat{s}}_{2}=\mathbf{c}^{T}\boldsymbol{\hat{s}}_{3}=0$$ for each case of multiplicities of eigenvalues of $$\mathbf{A}_{2}$$ and the pair $$\left\{ \boldsymbol{\hat{s}}_{2},\boldsymbol{\hat{s}}_{3}\right\} $$ is a basis for $$\mathcal{H}$$. Therefore, any initial condition on $$\mathcal{H}$$ can be expressed as K1(x^1sin⁡θ1+y1cos⁡θ1)orK2(s^2sin⁡θ2+s^3cos⁡θ2), (2.22) where $$-\phi_{i}\leq \theta_{i}\leq 2\pi -\phi_{i}$$. Accordingly, $$\mathbf{v}_{1}(\theta_{1})$$ and $$\mathbf{v}_{2}(\theta_{2})$$ are defined as follows. v1(θ1)=x^1sin⁡θ1+y1cos⁡θ1andv2(θ2)=s^2sin⁡θ2+s^3cos⁡θ2. (2.23) Definition 2.4 In the rest of the article $$\mathbf{v}_{1}(\theta_{1})$$ and $$\mathbf{v}_{2}(\theta_{2})$$ (as defined by equation (2.23) will be called directions in$$\mathcal{H}$$. The directions for which $$\eta \mathbf{v}_{1}(\theta_{1})=\mathbf{v}_{2}(\theta_{2})$$ hold for some constant $$\eta >0$$, will be called equivalent directions and denoted as $$\mathbf{v}_{1}(\theta_{1})\simeq \mathbf{v}_{2}(\theta_{2})$$. 2.5. Characterization of transitive and nontransitive trajectories for the second mode Since $$\sigma_{1}-\lambda_{1}>0$$, all trajectories starting from $$\mathcal{H}\cap \mathcal{S}_{1}$$ are transitive. However, this is not true for all trajectories starting from $$\mathcal{H}\cap \mathcal{S}_{2}$$. Note further that, since non-transitive trajectories in each mode do note change mode, they decay to the origin if and only if the corresponding mode has eigenvalues with negative real parts. Thus for GAS, we need to consider only the trajectories starting from $$\mathcal{H}$$ without loss of any generality. The following result gives the classification of trajectories starting from $$\mathcal{H}\cap \mathcal{S}_{2}$$. Corollary 2.1 Suppose that $$\mathbf{z}_{2}(t)$$ is a trajectory starting from $$\mathcal{H}\cap \mathcal{S}_{2}$$. Then, the following hold. 1. $$\mathbf{z}_{2}(t)$$ is a non-transitive trajectory if and only if $$\ \mathbf{z}_{2}(0)=K_{2}\mathbf{v}_{2}(\theta_{2})$$ where $$-\phi_{2}\leq \theta_{2}\leq \frac{\pi }{2}$$ and $$K_{2}>0.$$ 2. $$\mathbf{z}_{2}(t)$$ is a transitive trajectory if and only if $$\ \mathbf{z}_{2}(0)=K_{2}\mathbf{v}_{2}(\theta_{2})$$ where $$\frac{\pi }{2} < \theta_{2}<\pi -\phi_{2}$$ and $$K_{2}>0.$$ Proof. The proof of the Corollary is given Corollary 4.1 in Öner (2014). □ Definition 2.5 Note that the interval $$-\phi_{2}\leq \theta_{2}\leq \frac{\pi }{2}$$ describes a closed convex cone in $$\mathcal{S}_{2}\,\mathcal{\cap \,H}$$ bounded by the half-lines $$\mathcal{L(}\hat{\mathbf{s}}_{2})$$ and $$\mathcal{L}^{-}$$. This cone will be called mode-invariant cone of $$\mathcal{S}_{2}\cap \mathcal{H}$$ and denoted as $$\mathcal{C}^{-}$$. Remark 2.2 Suppose that $$\mathbf{z}_{2}(t)$$ be a transitive trajectory starting from $$\mathcal{S}_{2}\cap \mathcal{H}$$, and $$\tau_{2}>0$$ such that $$\mathbf{c}^{T}\mathbf{z}_{2}(\tau_{2})=0$$. Then, simultaneously solving equations $$f_{2}(\tau_{2})=0$$ and $$f_{2}(0)=0$$ for each case of multiplicity of eigenvalues, $$\tau_{2}$$ can be expressed as an implicit function of $$\theta_{2}$$ as given by the following equations. −cot⁡θ2=(e(μ2−μ1)τ2−1)(e(μ3−μ1)τ2−1), if μ1<μ2<μ3,−cot⁡θ2=τ2e(μ3−μ1)τ2−1, if μ1=μ2<μ3,−cot⁡θ2=1−e(μ1−μ3)τ2τ2, if μ1<μ2=μ3. (2.24) The equations above also imply that $$\tau_{2}\rightarrow 0$$ as $$\theta_{2}\rightarrow \pi -\phi_{2}$$ and $$\tau_{2}\rightarrow \infty $$ as $$\theta_{2}\rightarrow \frac{\pi }{2}$$. This is expected since at $$\theta_{2}=\frac{\pi }{2}$$ the initial condition is on the line $$\mathcal{L(}\hat{\mathbf{s}}_{2})$$ and the trajectories starting from these lines are nontransitive ($$\tau_{2}=\infty $$). On the other hand, as $$\theta_{2}\rightarrow \pi -\phi_{2},$$ the initial condition is on the line $$\mathcal{L}^{+}$$ and smooth continuation into $$\mathcal{S}_{2}$$ is not possible ($$\tau_{2}=0$$). Lemma 2.3 Let $$\mathbf{z}_{2}(t)$$ be a transitive trajectory starting from $$\mathcal{S}_{2}\cap \mathcal{H}$$ with initial condition $$\mathbf{z}_{2}(0)=\hat{\mathbf{s}}_{2}\sin \theta_{2}+\hat{\mathbf{s}}_{3}\cos \theta_{2}$$. Then, at $$t=\tau_{2}\,$$ where the trajectory changes mode, the following hold. 1. z2(τ2)=eμ3τ2N2{s^2sin⁡θ^2+s^3cos⁡θ^2} (2.25) where N2:=e2(μ2−μ3)τ2sin2⁡θ2+cos2⁡θ2  and cot⁡θ^2=e(μ3−μ2)τ2cot⁡θ2,ifμ1≤μ2<μ3,N2:=e2(μ1−μ3)τ2sin2⁡θ2+cos2⁡θ2  and cot⁡θ^2=e(μ3−μ1)τ2cot⁡θ2ifμ1<μ2=μ3, (2.26) 2. $$\tau_{2}$$ and $$\hat{\theta}_{2}$$ are decreasing functions of $$\theta_{2}$$. Furthermore, transitive trajectories of each mode hit $$\mathcal{H}$$ and change mode in the open conic regions bounded by the lines $$\mathcal{L}^{+}$$ and $$\mathcal{L}(-\hat{\mathbf{s}}_{2})$$ for the second mode. The map $$\mathcal{F}:\theta_{2}\rightarrow \hat{\theta}_{2}$$ is defined as $$\mathcal{F}\left(\theta_{2}\right) =\hat{\theta}_{2}$$. Furthermore, the following hold. (a) $$\hat{\theta}_{2}\rightarrow \pi -\phi_{2}$$ as $$\theta_{i}\rightarrow \pi -\phi_{2}$$ (equivalently as $$\tau_{2}\rightarrow 0$$) for all possible algebraic degrees of the eigenvalues, (b) $$\hat{\theta}_{2}\rightarrow \frac{3\pi }{4}$$ as $$\theta_{2}\rightarrow \frac{\pi }{2},$$ (equivalently as $$\tau_{2}\rightarrow \infty $$) if $$\mu_{1}<\mu_{2}<\mu_{3},$$ (c) $$\hat{\theta}_{2}\rightarrow \pi $$ as $$\theta_{2}\rightarrow \frac{\pi }{2},$$ (equivalently as $$\tau_{2}\rightarrow \infty $$) if $$\mu_{1}=\mu_{2}<\mu_{3}$$ or if $$\mu_{1}<\mu_{2}=\mu_{3},$$ Proof. The proof is given Lemma 4.3 in Öner (2014). □ 2.6. Mode change When a trajectory changes mode, we have to switch from one basis to the other for describing how the trajectory evolves after mode change. Before investigating the change of basis, we give the following definition, which is first introduced in Eldem & Öner (2015). Definition 2.6 The constant defined as B3:=λ1−a11−μ3a21ω1(1+b12) will be called the coupling constant in the rest of the article. Lemma 2.4 Let $$\mathbf{z}_{1}(t)$$ be a trajectory starting from $$\mathcal{S}_{1}\cap \mathcal{H}$$ with initial condition $$\mathbf{z}_{1}(0)=\mathbf{v}_{1}\left(\theta_{1}\right) $$ where $$-\phi_{1}\leq \theta_{1}<\pi -\phi_{1}$$. Then, at $$t=\tau_{1}$$ the trajectory changes mode and we have 1. z1(τ1)=η12(θ^1)eσ1τ1{s^2sin⁡θ2+s^3cos⁡θ2}, where θ2=cot−1⁡{−b2(B3(cot⁡θ^1+b1)+1)(cot⁡θ^1+b1)B2+1} for μ1≤μ2<μ3, =cot−1⁡{−b2(B3(cot⁡θ^1+b1)+1)(cot⁡θ^1+b1)B1+1} for μ1<μ2=μ3 (2.27) and η12(θ^1)=−ω1(cos⁡θ^1+b1sin⁡θ^1)a32(μ3−μ1)(cos⁡θ2+b2sin⁡θ2) for μ1≤μ2<μ3=−ω1(cos⁡θ^1+b1sin⁡θ^1)a32(cos⁡θ2+b2sin⁡θ2) for μ1<μ2=μ3. Moreover, $$B_{1}:=\frac{\left(\lambda_{1}-a_{11}-\mu_{1}a_{21}\right) }{\omega_{1}\left(1+b_{1}^{2}\right) }, B_{2}:=\frac{\left(\lambda_{1}-a_{11}-\mu_{2}a_{21}\right) }{\omega_{1}\left(1+b_{1}^{2}\right) }$$, and $$b_{2}:=\cot \phi_{2}$$ as defined in Lemma 2.2 for eigenvalues with different multiplicities. 2. Let $$\mathbf{z}_{2}(t)$$ be a trajectory starting from $$\mathcal{S}_{2}\cap \mathcal{H}$$ with initial condition $$\mathbf{z}_{2}(0)=\eta_{12}\left(\hat{\theta}_{1}\right) e^{\sigma_{1}\tau_{1}} \mathbf{v}_{2}\left(\theta_{2}\right) $$. If $$\mathbf{z}_{2}(t)$$ is a transitive trajectory ($$\frac{\pi }{2}<\theta_{2}<\pi -\phi_{2}$$) (or equivalently outside of the cone $$\mathcal{C}^{-}$$), then z2(τ2)=η12(θ^1)eμ3τ2eσ1τ1η21(θ2)N2(x^1sin⁡θ11+y1cos⁡θ11), where θ11=cot−1⁡{−(cot⁡θ^2+b2)b2B3+B2cot⁡θ^2−b1} for μ1≤μ2<μ3,=cot−1⁡{−(cot⁡θ^2+b2)b2B3+B1cot⁡θ^2−b1}for μ1<μ2=μ3, (2.28) and η21(θ2)=−a32(μ3−μ1)(cos⁡θ^2+b2sin⁡θ^2)ω1(cos⁡θ11+b1sin⁡θ11) for μ1≤μ2<μ3,=−a32(ω1(cos⁡θ^2+b2sin⁡θ^2)(b12+1))ω12(1+b12)(cos⁡θ11+b1sin⁡θ11)for μ1<μ2=μ3. Furthermore, $$-\phi_{1}<\theta_{11}<{-}\psi_{1}$$ where $$\cot \psi_{1}=\frac{1}{B_1}+b_{1}$$ and $$-{\hat{s}}_{2}\simeq{v}_{1} {(-}\psi_{1})$$. Proof. The proof is given in Appendix. □ 2.7. One loop around $$\mathcal{H}$$ plane Let $$\mathbf{z}_{1}(t)$$ be a transitive trajectory with initial condition $$\mathbf{z}_{1}(0)=\mathbf{v}_{1}\left(\theta_{1}\right) $$ in $$\mathcal{S}_{1}\cap \mathcal{H}$$. Suppose that $$\mathbf{z}_{1}(\tau_{1})$$ is outside the cone $$\mathcal{C}^{-}.$$ Then, the movement of the trajectories on $$\mathcal{H}$$ is controlled by the following functions. F1(θ1)=θ^1, G1(θ^1)=θ2, F2(θ2)=θ^2, G2(θ^2)=θ11. Thus, we can define a function $$\mathcal{T}_{1}:\mathcal{S}_{1}\cap \mathcal{H\rightarrow S}_{1}\cap \mathcal{H}$$ as follows. T1(θ1)=G2(F2(G1(F1(θ1))))=θ11. (2.29) Since change of basis is continuous, it follows that all the maps described above are continuous. Therefore, $$\mathcal{T}_{1}(\theta_{1})=\theta_{11}$$ is continuous. Note that this map represents the following four steps in the dynamic behaviour of the trajectories. v1(θ1)→eσ1τ1v1(θ^1)→η12(θ^1)eσ1τ1v2(θ2)→η12(θ^1)eμ2τ2eσ1τ1N2v2(θ^2)→eμ2τ2eσ1τ1η21(θ2)η12(θ^1)N2v1(θ11). Thus, $$\mathcal{T}_{1}(\theta_{1})$$ can be interpreted as a Poincaré full map. Also note that $$\mathcal{T}_{1}(\theta_{1})$$ is defined only for trajectories which change mode at least two times. For trajectories which changes mode $$2k$$ times$$,$$ we define $$\mathcal{T}_{1}^{k} (\theta_{1})$$ as follows. T1k(θ1):=T1(T1(⋅⋅⋅T1(θ1)⋅⋅⋅))(k times) and T1k(θ1):=θ1k. For transitive trajectories starting from $$\mathcal{S}_{2}\cap \mathcal{H}$$, we can also define $$\mathcal{T}_{2}:\mathcal{S}_{2}\cap \mathcal{H\rightarrow S}_{2}\cap \mathcal{H}$$ as $$\mathcal{T}_{2}^{k}\left(\theta_{2}\right) :=\theta_{2k}$$ in a completely similar way. 3. Stability of fix directions In view of the development given in the previous section, some trajectories may change mode only finite number of times and enter a mode-invariant cone and some may change mode infinitely many times as $$t\rightarrow \infty.$$ The trajectories, which change mode only a finite number of times as $$t\rightarrow \infty$$, decay to the origin if and only if $$\mu_{3}<0.$$ Therefore, we investigate only GAS of the class of trajectories which change mode infinitely many times as $$t\rightarrow \infty.$$ Towards this end, we give the following definitions to be consistent with the terminology used in Eldem & Şahan (2014). Definition 3.1 Let $$\theta_{i}^{\ast }$$ be a fixed point of $$\mathcal{T}_{i}(\theta_{i})$$ or equivalently $$\mathcal{T}_{i}(\theta_{i}^{\ast })=\theta_{i}^{\ast }$$ for $$i=1,2$$. Then, $$\mathbf{v}_{i}(\theta_{i}^{\ast })$$ is called a fixed direction. A fixed direction $$\mathbf{v}_{i}(\theta_{i}^{\ast })$$ is said to be attractive in an interval$$ I_{i}$$containing$$\theta_{i}^{\ast }$$ if for any $$\theta_{i}$$ in $$I_{i}$$ and for every $$\varepsilon >0$$ there exists a positive integer $$k$$ such that $$\left\vert \mathcal{T}_{i}^{k}(\theta_{i})-\theta_{i}^{\ast }\right\vert <\varepsilon $$ for $$i=1,2$$. If $$I_{i}$$ consists of only one point $$\theta_{i}^{\ast }$$, then the fixed point is said to be repulsive. In view of this definition, let $$\mathbf{z}_{i}^{\ast }(t)$$ be a trajectory starting from a fixed direction $$\mathbf{v}_{i}(\theta_{i}^{\ast })$$. To be more specific, $$\mathbf{z}_{i}^{\ast }(0)=K_{i}\mathbf{v}_{i}(\theta_{1}^{\ast })$$ where $$K_{i}>0$$ is a real constant. Then, in view of Lemma 2.4 $$K_{i}\mathbf{v}_{i}(\theta_{i}^{\ast })$$ is mapped as follows, Kivi(θi∗)→γ∗(θi∗)Kivi(θi∗), where γ∗(θ1∗):=eμ2τ2∗eσ1τ1∗η21(θ2∗)η12(θ^1∗)N2 Definition 3.2 In the sequel, $${\it {\gamma}} ^{\ast }(\theta_{i}^{\ast })$$ will be called convergence rate as in Eldem & Şahan (2014). It is clear from the development above that if $$\mathbf{z}_{1}^{\ast }(t)$$ is a trajectory starting from a fixed direction $$\mathbf{v}_{1}(\theta_{1}^{\ast })$$, then there exists a unique fixed direction in $$\mathcal{S}_{2}\cap \mathcal{H}$$ such that z1∗(τ1∗)=η12(θ1∗+ω1τ1∗)eσ1τ1∗v2(θ2∗). In view of the definition of $$\mathcal{T}(\theta_{1}^{\ast }),$$$$\mathbf{z}_{1}^{\ast }(\tau_{1}^{\ast })$$ is necessarily in the transitive region of $$ \mathcal{S}_{2}\cap \mathcal{H}$$ or equivalently is outside the cone $$ \mathcal{C}^{-}$$. This means that fixed directions exist as a pair $$\left(\theta_{1}^{\ast },\theta_{2}^{\ast }\right) $$. Therefore, we use the following notation for the convergence rate γ∗(θ1∗)=γ∗(θ1∗,θ2∗). In order to complete the picture, we also define below the attractiveness of mode-invariant cone $$\mathcal{C}^{-}$$ of $$\mathcal{S}_{2}\cap \mathcal{H}$$ as defined in Definition 2.5. Definition 3.3 $$\mathcal{C}^{-}$$ is said to be attractive in an interval$${\boldsymbol{I}}_{2}$$containing$$\mathcal{C}^{-}$$ if for any $$\theta_{2}$$$$\in $$$$I_{2}$$ there exists a finite non-negative integer $$k$$ such that $$-\phi_{2}< \mathcal{T}_{2}^{k}(\theta_{2})\leq \frac{\pi }{2}$$ or $$\mathcal{T}_{2}^{k}(\theta_{2})\rightarrow \frac{\pi }{2}$$ as $$k\rightarrow \infty $$. Lemma 3.1 Let $$\mathbf{v}_{i}(\theta_{i}^{\ast })$$ be a fixed direction which is attractive in an interval $$I_{i}$$. Then, the following hold. 1. If $$\mathbf{z}_{i}^{\ast }(t)$$ is a trajectory starting from the fixed direction $$\mathbf{v}_{i}^{\ast }(\theta_{i}^{\ast }),$$ then $$\mathbf{z}_{i}^{\ast }(t)\rightarrow 0$$ as $$t\rightarrow \infty $$ if and only if $${\it {\gamma}} ^{\ast }(\theta_{1}^{\ast },\theta_{2}^{\ast })<1$$. 2. If $$\mathbf{z}_{i}(t)$$ is a trajectory starting from $$\mathcal{S}_{i}\cap \mathcal{H}$$ with initial condition $$\mathbf{z}_{i}(0)=K_{i}\mathbf{v}_{i}(\theta_{i})$$ where $$\theta_{i}$$$$\in $$$$I_{i}$$, then $$\mathbf{z}_{i}(t)\rightarrow 0$$ as $$t\rightarrow \infty $$ if and only if $${\it {\gamma}}^{\ast }(\theta_{1}^{\ast },\theta_{2}^{\ast })<1$$. 3. Suppose that $$\mathcal{C}^{-}$$ is attractive in an interval $$I_{2}$$ containing $$\mathcal{C}^{-}$$. Let $$\mathbf{z}_{2}(0)=$$$$\mathbf{v}_{2}\left(\theta_{2}\right) $$ where $$\theta_{2}$$$$\in $$$$I_{2}$$. Then $$z_{2}(t)\rightarrow 0$$ as $$t\rightarrow \infty $$ if and only if $$\mu_{3}<0$$. Proof. The proof of this Lemma can be done by following the similar lines as in the proof of Lemma 5 in Eldem & Öner (2015). To avoid duplication, the proof is omitted here. □ Corollary 3.1 A general expression for $${\it {\gamma}} ^{\ast }(\theta_{1}^{\ast })$$ is γ∗(θ1∗,θ2∗)=eμ2τ2eσ1τ1cos⁡θ^1∗+b1sin⁡θ^1∗cos⁡θ1∗+b1sin⁡θ1∗cot⁡θ^2∗+b2cot⁡θ2∗+b2, and if $$\theta_{1}^{\ast }\neq 0,$$ we have γ∗(θ1∗,θ2∗)=eμ2τ2eλ1τ11+B2(cot⁡θ^1∗+b1)1+B2(cot⁡θ1∗+b1). (3.1) Proof. Using the definition of convergence rate, we have $${\it {\gamma}} ^{\ast }(\theta_{1}^{\ast }):=e^{\mu_{2}\tau_{2}^{\ast }}e^{\sigma_{1}\tau_{1}^{\ast }}\eta_{21}(\theta_{2}^{\ast })\eta_{12}\left(\hat{\theta}_{1}^{\ast }\right) N_{2}.$$ Substituting for $$\eta_{21}(\theta_{2}^{\ast })$$, $$\eta_{12}\left(\hat{\theta}_{1}^{\ast }\right) $$ and $$N_{2}$$ given in Lemma 2.3 and Lemma 2.4, we obtain γ∗(θ1∗)=eμ2τ2eσ1τ1cos⁡θ^1∗+b1sin⁡θ^1∗cos⁡θ1∗+b1sin⁡θ1∗cot⁡θ^2∗+b2cot⁡θ2∗+b2. Note that $$\frac{\cot \hat{\theta}_{2}^{\ast }+b_{2}}{\cot \theta_{2}^{\ast }+b_{2}}=\frac{\cot \theta_{1}^{\ast }+b_{1}}{\cot \hat{\theta}_{1}^{\ast }+b}\frac{1+B_{2}\left(\cot \hat{\theta}_{1}^{\ast }+b_{1}\right) }{1+B_{2}\left(\cot \theta_{1}^{\ast }+b_{1}\right) }\!.$$ Therefore, if $$\theta_{1}^{\ast }\neq 0$$, then $$\sin \theta_{1}^{\ast }$$$$\neq 0$$ and we get γ∗(θ1∗)=eμ2τ2eλ1τ11+B2(cot⁡θ^1∗+b1)1+B2(cot⁡θ1∗+b1). □ Let the interval $$I$$ be in the domain of $$\mathcal{T}_1 (.)$$ and $$\theta_1 \in I$$. We show below that the derivative of $$\mathcal{T}_1 (\theta_1)$$ with respect to $$\theta_1$$ is positive. Lemma 3.2 The derivative of $$\theta_{11}$$ with respect to $$\theta_{1}$$ is positive in domain of $$\mathcal{T}_{1}$$ and dT1dθ1=(sin⁡(θ11+ψ1)sin⁡(θ^1+ψ1))(sin⁡(θ11+ψ2)sin⁡(θ^1+ψ2))(sin⁡(θ11+ψ3)sin⁡(θ^1+ψ3))(sin⁡(θ1+ϕ1)sin⁡(θ11+ϕ1)) ×sin⁡θ^1sin⁡θ1 for μ1<μ2<μ3=(sin⁡(θ11+ψ2)sin⁡(θ^1+ψ2))2(sin⁡(θ11+ψ3)sin⁡(θ^1+ψ3))(sin⁡(θ1+ϕ1)sin⁡(θ11+ϕ1))sin⁡θ^1sin⁡θ1 for μ1=μ2<μ3=(sin⁡(θ11+ψ2)sin⁡(θ^1+ψ2))2(sin⁡(θ11+ψ1)sin⁡(θ^1+ψ1))(sin⁡(θ1+ϕ1)sin⁡(θ11+ϕ1))sin⁡θ^1sin⁡θ1 for μ1<μ2=μ3 where $$\cot \psi_{1}=\frac{1}{B_{1}}+b_{1},$$$$\cot \psi_{2}=\frac{1}{B_{2}}+b_{1}$$ and $$\cot \psi_{3}=\frac{1}{B_{3}}+b_{1}$$. Proof. The proof of the Lemma will be given in Appendix. □ Using Lemma 2.3 and Lemma 2.4, it is easy to show that v1(−ψ1)≃v2(3π/4),v1(−ψ2)≃v2(π)=−s^3,v1(−ψ3)≃v2(3π/2)=−s^2. (3.2) More precisely, the pairs $$\{\mathbf{v}_{1}(-\psi_{1}),\mathbf{v}_{2}(3\pi /4)\}$$, $$\{\mathbf{v}_{1}(-\psi_{2}),\mathbf{v}_{2}(\pi)\}$$, and $$\{\mathbf{v}_{1}(-\psi_{3}),\mathbf{v}_{2}(3\pi /2)\}$$ are equivalent directions as defined in Definition 2.4. Since $$B_{3}<B_{2}<B_{1}$$, it also follows that $$\psi_{3}<\psi_{2}<\psi_{1}<\phi_{1}$$. Suppose that a trajectory starting from $$\mathcal{S}_{1}\cap \mathcal{H}$$ along the direction $$\mathbf{v}_{1}(\theta_{1})$$ where $$-\phi_{1}<\theta_{1}<0,$$ hits $$\mathcal{S}_{2}\cap \mathcal{H}$$ outside the cone $$\mathcal{C}^{-}$$ and continues in the second mode. Then, in view of Lemma 2.3.2.b, this trajectory hits back $$\mathcal{S}_{1}\cap \mathcal{H}$$ along the direction $$\mathbf{v}_{1}(\theta_{11})$$ where $$-\phi_{1}<\theta_{11}<-\psi_{1}$$. In view of this observation, we can assume without loss of any generality that, $$\mathcal{T}_{1}(\theta_{1}) $$ is a continuous and differentiable function which maps $$[-\phi_{1},-\psi_{1}]$$ into itself. Thus, the existence of a fixed point of $$\mathcal{T}_{1}(\theta_{1}) $$ is clear. We have to show that the fixed point is unique and attractive in the interval $$[-\phi_{1},-\psi_{1}]$$. Towards this end, let ρ(θ1):=T1(θ1)−θ1. (3.3) Note that $$\rho (-\phi_1)>0$$ and $$\rho (-\psi_1)<0$$. Now we can state and prove the following results. Lemma 3.3 If $$\phi_{1}<2\psi_{2}$$, then $$\frac{d\rho \left(\theta_{1}\right) }{d\theta_{1}}<0 $$ whenever $$\rho \left(\theta_{1} \right) =0$$. Proof. We give the proof only for the case $$\mu_{1}<\mu_{2}<\mu_{3}.$$ The proof for the other cases can be done completely in a similar manner. If $$\rho (\theta_1)=\mathcal{T}_{1}(\theta_{1})-\theta_1=\theta_{11}-\theta_1=0$$, then $$\theta_{11}=\theta_1$$ and this implies that dT1(θ1)dθ1∣θ1=θ11=(sin⁡(θ1+ψ1)sin⁡(θ^1+ψ1))(sin⁡(θ1+ψ2)sin⁡(θ^1+ψ2))(sin⁡(θ1+ψ3)sin⁡(θ^1+ψ3))sin⁡θ^1sin⁡θ1. Since ddx(sin⁡(θ1+x)sin⁡(θ^1+x))=sin⁡(θ^1−θ1)(sin⁡(θ^1+x))2<0, we get sin⁡θ1sin⁡θ^1>sin⁡(θ1+ψ3)sin⁡(θ^1+ψ3)>sin⁡(θ1+ψ2)sin⁡(θ^1+ψ2)>sin⁡(θ1+ψ1)sin⁡(θ^1+ψ1). (3.4) This implies that sin⁡(θ1+ψ3)sin⁡(θ^1+ψ3)sin⁡θ^1sin⁡θ1<1. Since $$\frac{\sin \left(\theta_{1}+\psi_{2}\right) }{\sin \left(\hat{ \theta}_{1}+\psi_{2}\right) }>\frac{\sin \left(\theta_{1}+\psi_{1}\right) }{\sin \left(\hat{\theta}_{1}+\psi_{1}\right) }$$, it is enough to show that $$\frac{\sin \left(\theta_{1}+\psi_{2}\right) }{\sin \left(\hat{ \theta}_{1}+\psi_{2}\right) }<1.$$ Note that in view of Remark 2.1, we have $$\hat{\theta}_{1}=\pi +y_{1} $$ where $$y_{1}<\phi_{1}$$. Furthermore, since $$-\phi_{1}<-\theta_{11}<-\psi_{1}$$, we get sin⁡(θ1+ψ2)sin⁡(θ^1+ψ2)=sin⁡(|θ1|−ψ2)sin⁡(y+ψ2). Since $$\phi_{1}<2\psi_{2}$$, it follows that ϕ1<2ψ2⇔ϕ1−ψ2<ψ2<y+ψ2. If $$y+\psi_{2}\leq \frac{\pi }{2},$$ the inequalities given above imply that $$\frac{\sin \left(\theta_{1}+\psi_{2}\right) }{\sin \left(\hat{\theta}_{1}+\psi_{2}\right) }<\frac{\sin \left(\phi_{1}-\psi_{2}\right) }{\sin \left(y+\psi_{2}\right) }<1$$. On the other hand, if $$y+\psi_{2}>\frac{\pi }{2}$$ and $$\phi_{1}-\psi_{2}> \pi -y-\psi_{2}$$, then $$\phi_{1}+y>\pi.$$ This is a contradiction because $$\phi_{1}+y<\pi.$$ Therefore, if $$y+\psi_{2}>\frac{\pi }{2}$$ we must have $$\phi_{1}-\psi_{2}<\pi -y-\psi_{2}$$. This is equivalent to the fact that sin⁡(θ1+ψ2)sin⁡(θ^1+ψ2)<sin⁡(ϕ1−ψ2)sin⁡(y+ψ2)=sin⁡(ϕ1−ψ2)sin⁡(π−y−ψ2)<1. Therefore, if $$\phi_{1}<2\psi_{2},$$ using the last inequality and inequality (3.4) we have the following relations: 1>sin⁡(θ1+ψ2)sin⁡(θ^1+ψ2)>sin⁡(θ1+ψ1)sin⁡(θ^1+ψ1). Consequently, we get $$\frac{d\mathcal{T}_{1}}{d\theta_{1}}<1$$ whenever $$\theta_{11}=\theta_1$$ or equivalently whenever $$\rho (\theta_1)=0$$. □ Lemma 3.4 There is a unique and attractive fix point $$\theta_{1}^{\ast }$$ of $$ \mathcal{T}_{1}$$ if $$\frac{d\rho \left(\theta_{1}\right) }{d\theta_{1}}<0 $$ whenever $$\rho \left(\theta_{1} \right) =0.$$ Proof. Suppose that $$\theta_{1}^{1}$$ is the first and $$\theta_{1}^{2}>\theta_{1}^{1}$$ is the second fixed point of $$\mathcal{T}_{1}$$ such that $$-\phi_1<\theta_{1}^{1}< \theta_{1}^{2}<-\psi_1$$. Since $$\rho (-\phi_1)>0$$, it follows that $$\rho (\theta_1)>0$$ in the interval $$[-\phi_1, \theta_{1}^{1})$$. Since $$\frac{d\rho \left(\theta_{1}\right) }{d\theta_{1}}<0$$ at $$\theta_1=\theta_{1}^{1}$$ it also follows that $$\rho (\theta_1)<0$$ in the interval $$(\theta_{1}^{1}, \theta_{1}^{1}+\epsilon]$$ for some $$\epsilon>0$$ where $$\theta_{1}^{1}+\epsilon<\theta_{1}^{2}$$. Since $$\theta_{1}^{2}$$ is the second fixed point after $$\theta_{1}^{1}$$, we have $$\rho (\theta_1)<0$$ in the interval $$(\theta_{1}^{1}, \theta_{1}^{2})$$. Note that $$\rho (\theta_{1}^{2})=0$$ and $$\frac{d\rho \left(\theta_{1}\right) }{d\theta_{1}}<0$$ at $$\theta_1=\theta_{1}^{2}$$. This implies that again $$\rho (\theta_1)<0$$ in the interval $$[\theta_{1}^{2}-\epsilon) \cup (\theta_{1}^{2},\theta_{1}^{2}+\epsilon]$$ for some $$\epsilon>0$$. Consequently, $$\rho (\theta_{1}^{2})$$ is a local maximum. Since $$\rho (\theta_1)$$ is a differentiable function, it must hold that $$\frac{d\rho \left(\theta_{1}\right) }{d\theta_{1}}=0$$ at $$\theta_1=\theta_{1}^{2}$$. This contradicts the hypothesis that $$\frac{d\rho \left(\theta_{1}\right) }{d\theta_{1}}<0 $$ whenever $$\rho \left(\theta_{1} \right) =0.$$ Therefore, there is no second fixed point of $$\mathcal{T}_1$$ or equivalently the fixed point $$\theta_{1}^{1}$$ is unique. Attractivity follows from the fact that $$\mathcal{T}_1 (\theta_1)>\theta_1$$ in the interval $$[-\phi_1, \theta_{1}^{1})$$ and $$\mathcal{T}_1 (\theta_1)>\theta_1$$ in the interval $$(\theta_{1}^{1}, -\psi_1]$$. □ In view of the results given above, we have the following assumption. Assumption 3.1 $$\phi_{1}<2\psi_{2}.$$ 4. Main result In this section, we consider three different geometric structures given as follows. 1. $$-\mathbf{y}_{1}$$ is outside the cone $$\mathcal{C}^{-}$$$$\left(B_{3}>0\right) $$. 2. $$\mathbf{y}_{1}$$ and $$\hat{\mathbf{s}}_{2}$$ are on the same line $$\left(B_{3}=0\right) $$. 3. $$-\mathbf{y}_{1}$$ is in the cone $$\mathcal{C}^{-}$$$$\left(B_{3}<0\right) $$. For the first two cases above we provide the necessary and sufficient conditions for GAS of bimodal systems being considered. Furthermore, we demonstrate the effect of the discontinuity of the vector field on GAS of bimodal systems by two examples. In these examples, we basically show that the eigenvectors of one of the subsystems can be changed without changing the eigenvalues of the subsystem and this change can make bimodal system GAS or unstable. This change is achieved by changing the coupling constant given in Definition 2.6. The case where $$B_3<0$$ deserves a separate investigation. Our comments for this case is given in Remark 4.1. Theorem 4.1 Given the bimodal system $$\it{\Sigma}_{0}$$ with Assumptions 2.1, 2.2, 2.3 and 3.1, the following hold. If $$B_{3}\geq 0$$, then the unique fixed direction $$\theta_1^{\ast}$$ in the interval $$[-\phi_1,-\psi_1]$$ is attractive in the interval $$[-\phi_1, \bar{\theta}_1)$$ where $$\mathcal{G}_1 (\mathcal{F}_1 (\bar{\theta}_1))=\frac{\pi}{2}$$, and mode-invariant cone $$\mathcal{C}^{-}$$ is repulsive. If $$B_{3}\geq 0$$, then the system is GAS if and only if real eigenvalues of both modes are negative and the convergence rate $${\it {\gamma}} ^{\ast }(\theta_{1}^{\ast },\theta_{2}^{\ast })<1$$. Proof. The proof is given only for the case $$\mu_{1}<\mu_{2}<\mu_{3}.$$ The proof for the other cases can be done completely in a similar manner. Consider a trajectory starting from $$\mathcal{S}_{1}\cap \mathcal{H}$$. Suppose that $$B_{3}>0$$, i.e., $$-\mathbf{y}_{1}$$ is outside the cone $$\mathcal{C}^{-}$$. Since $$\mathcal{F}_{1}(\theta_{1})$$ is a continuous function which maps $$[0,\pi -\phi_{1})$$, to $$(\pi -\phi_{1},\pi]$$, there exists a unique $$\bar{\theta}_{1}\in \lbrack 0,\pi -\phi_{1})$$ such that $$ \mathcal{G}_{1}\mathcal{(F}_{1}\mathcal{(\bar{\theta}}_{1}))=\frac{\pi }{2}$$. Therefore, if $$\mathbf{z}_{1}(0)=\mathbf{v}_{1}(\theta_{1})$$ where $$ \theta_{1}\,\geq \,\bar{\theta}_{1}$$, then $$z_{1}(t)$$ will enter the cone $$ \mathcal{C}^{-}$$ after one mode change and stay in $$\mathcal{S}_{2}$$ for all $$t\geq \tau_{1}$$. On the other hand, for any $$\theta_{1}\in \lbrack -\phi_{1},\,\bar{\theta}_{1})$$, it follows that $$\mathbf{z}_{1}(\hat{\theta}_{1})$$ is outside the cone $$\mathcal{C}^{-}.$$ Consequently, $$\mathcal{T}_{1}(\theta_{1})$$$$\in \lbrack -\phi_{1},-\psi_{1})$$ by Lemma 2.4.2. Furthermore, for any $$\theta_{1} \in $$$$[-\phi_{1},-\psi_{1}]$$ it also follows that $$\mathcal{T}_{1}(\theta_{1}) \in $$$$[-\phi_{1},-\psi_{1})$$. Also note that if $$\theta_{1}>-\psi_{1},$$ we get $$\mathcal{T}_{1}(\theta_{1})<\theta_{1}$$ and if $$\theta_{1}=-\phi_{1}$$ we have $$\mathcal{T}_{1}(\theta_{1})>\theta_{1}.$$ This implies that there exists $$\theta_{1}^{\ast }$$ such that $$\mathcal{T}_{1}(\theta_{1}^{\ast })=\theta_{1}^{\ast }.$$ Then, in view of Lemmas 3.3 and 3.4 it follows that $$\theta_{1}^{\ast }$$ is a unique fix point which is attractive in $$[-\phi,\bar{\theta_{1}})$$ and and mode-invariant cone $$\mathcal{C}^{-}$$ is repulsive. Consequently, Lemma 3.1 implies that bimodal system is GAS if and only if real eigenvalues of both modes are negative and the convergence rate $${\it {\gamma}} ^{\ast }(\theta_{1}^{\ast },\theta_{2}^{\ast })<1$$. On the other hand, if $$B_{3}=0$$ then $$-\mathbf{y}_{1}$$ and $$\hat{\mathbf{s}}_{2}$$ are equivalent directions. This means that $$\bar{\theta}_{1}=0$$. As in the previous case, if $$\mathbf{z}_{1}(0)=\mathbf{v}_{1}(\theta_{1})$$ where $$\theta_{1}\in \lbrack 0,\pi -\phi_{1})$$, then $$\mathbf{z}_{1}(t)$$ will enter the cone $$\mathcal{C}^{-}$$ after one mode change and stay in $$\mathcal{S}_{2}$$ for all $$t\geq \tau_{1}$$. If $$\mathbf{z}_{1}(0)=\mathbf{v}_{1}(\theta_{1})$$ where $$\theta_{1}\in \lbrack -\phi_{1},0), $$ then $$\hat{\theta}_{1}\in (\pi,\pi +\phi_{1})$$ by equation (2.11) and this implies that $$\mathbf{z}_{1}\left(\tau_{1}\right) $$ is outside $$\mathcal{C}^{-}$$. Continuing along the similar lines as in the previous case, it follows that there exists a unique fixed point $$\theta_{1}^{\ast }$$ of $$\mathcal{T}_{1}(\theta_{1})$$ which is attractive in $$[-\phi,\bar{\theta_{1}})$$ and mode-invariant cone $$\mathcal{C}^{-}$$ is repulsive. Therefore, we again conclude that bimodal system is GAS if and only if real eigenvalues of both modes are negative and $${\it {\gamma}} ^{\ast }(\theta_{1}^{\ast },\theta_{2}^{\ast })<1$$. □ Remark 4.1 1. Suppose that $$B_{3}<0$$ and $$\mathcal{G}_1 (\mathcal{F}_1 (-\phi_1)) \leq \frac{\pi}{2}$$. This means that $$\mathcal{G}_1 (\mathcal{F}_1 (\theta_1))$$ maps $$[-\phi_1, \pi - \phi_1)$$ into mode-invariant cone $$\mathcal{C}^{-}$$. Therefore, $$\mathcal{C}^{-}$$ is attractive in $$\mathcal{S}_2 \cap \mathcal{H}$$ and there is no fixed direction. 2. If $$\mathcal{G}_1 (\mathcal{F}_1 (-\phi_1)) > \frac{\pi}{2},$$ then there exists $$\bar{\theta}_{1} \in \lbrack -\phi_{1},0) $$ such that $$\mathcal{G}_{1} (\mathcal{(F}_{1} (\bar{\theta}_{1}))=\frac{\pi }{2}$$ and the following points need a separate investigation. (a) If $$-\psi_{1}>\bar{\theta}_{1},$$ then there may or may not be a fixed point. Furthermore, the fixed point may or may not be attractive. Therefore, a separate analysis is necessary for these issues. (b) If $$-\psi_{1} \leq \bar{\theta}_{1}$$ and $$B_2 >0$$, then Lemmas 3.3 and 3.4 hold and intuitively we expect that bimodal system is GAS if only if real eigenvalues of both modes are negative and $${\it {\gamma}} ^{\ast }(\theta_{1}^{\ast },\theta_{2}^{\ast })<1$$. However, this needs a separate proof. The following examples are included to demonstrate the effect of the coupling constant on GAS of bimodal systems. Example 4.1 Consider the following bimodal system where $${A}_{2}$$ is in observable canonical form as described in (2.2) with eigenvalues $$\left\{ -4,-3,-2\right\} $$ and the eigenvalues of $${A}_{1}$$ (given below) are $$\left\{ -1,\frac{3}{10}+4i,\frac{3}{10}-4i\right\}\!.$$ A1=[−28391070−293130227269198811449162412782−410441391[9pt]0107130−23365]. (4.1) Additionally $$\phi_{1}<2\psi_{2}\left(1.2566<2.2196\right).$$ This system is unstable with $${\it {\gamma}} ^{\ast }\left(\theta_{1}^{\ast },\theta_{2}^{\ast }\right) =2.9111$$ and the coupling constant is $$B_{3}=4.0073$$. Note that we can change the coupling constant while keeping the eigenvalues of both modes fixed with $$\mathbf{A}_{1}$$ as given below. A1=[6791287−2295273436407943−58121930528289128702455607449364079430−6441219305015911818890800267521−2170413282890]. (4.2) We again have $$\phi_{1}<2\psi_{2}$$$$\left(1.2566<1.3912\right)$$. This system is stable with $${\it {\gamma}} ^{\ast }\left(\theta_{1}^{\ast },\theta_{2}^{\ast }\right) =0.4259$$ and $$B_{3}=$$$$0.6486$$. The trajectories of the systems above are depicted in Figs 1 and 2 (In figures, the arrows show the direction of the trajectories). Fig. 1. View largeDownload slide Unstable trajectories. Fig. 1. View largeDownload slide Unstable trajectories. Fig. 2. View largeDownload slide Stable trajectories. Fig. 2. View largeDownload slide Stable trajectories. Example 4.2 Consider the following bimodal system. The eigenvalues of $$\mathbf{A}_{1}$$ are $$\left\{-\frac{1}{3},-\frac{1}{40}+\frac{1}{5}i,\right.$$$$\left.-\frac{1}{40}-\frac{1}{5}i\right\} $$. $$\mathbf{A}_{2}$$ is in observable canonical form as described in (2.2) with eigenvalues $$\left\{ -4,-3,-2\right\}$$. A1=[−5511560−196513073202726019719559233153660−557260020617310472−6551576]. Both modes of this system are stable, but bimodal system is unstable with $${\it {\gamma}} ^{\ast }\left(\theta_{1}^{\ast },\theta_{2}^{\ast }\right) =1.4011. $$ The coupling constant is $$B=$$$$3.0212$$ and $$\phi_{1}<2\psi_{2}$$$$\left(0.5754<1.0319\right)$$. We can change the coupling constant while keeping the eigenvalues of both modes fixed. Consider the bimodal system given below. A1=[−4811832−289240−603802022852441−3217327480−170734580022910802120], where the eigenvalues of $$\mathbf{A}_{1}$$ are again $$\left\{ -\frac{1}{3},- \frac{1}{40}+\frac{1}{5}i,-\frac{1}{40}-\frac{1}{5}i\right\}$$ and $$\mathbf{A}_{2}$$ is in observable canonical form as described in (2.2) with the same eigenvalues $$\left\{ -4,-3,-2\right\} $$. Furthermore, we also have $$ \phi_{1}<2\psi_{2}$$$$\left(0.5754<0.8660\right)$$. This system is stable with $${\it {\gamma}} ^{\ast }\left(\theta_{1}^{\ast },\theta_{2}^{\ast }\right) =0.4641$$ and with the coupling constant $$B= 1.0375$$. 5. Conclusions In this article, we gave the necessary and sufficient conditions for GAS of a class of BPLS in $$\mathbb{R}^{3}$$ with discontinuous vector fields. Unlike the article Eldem & Öner (2015), we take into account three cases ($$\mu_{1}=\mu_{2}<\mu_{3},$$$$\mu_{1}<\mu_{2}=\mu_{3}$$ and $$\mu_{1}<\mu_{2}<\mu_{3}.$$) for the second mode. It is clear that the classification of the trajectories as $$(i)$$ the trajectories which change mode finite number of times as $$t \rightarrow \infty $$ and $$(ii)$$ the trajectories which change mode infinite number of times as $$t \rightarrow \infty $$ plays a crucial role in GAS. In addition, it is demonstrated by the example above that the coupling constant and the discontinuity of the vector field will play a crucial role in such cases, too. For future work, the other classes of BPLS in $$\mathbb{R}^{3}$$ can be investigated within the same framework. Moreover, finding the necessary and sufficient conditions for GAS of BPLS in $$\mathbb{R}^{n}$$ is still an open and non-trivial problem and can be considered as a challenging future work. References Branicky M. S. ( 1998 ) Multiple Lyapunov functions and other analysis tools for switched and hybrid systems . IEEE Trans. Automat. Contr., 43 , 475 – 482 . Google Scholar CrossRef Search ADS Çamlibel M. K. 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( 2014 ) Stability of bimodal systems in $$\mathbb{R}^{3}$$. Ph.D. Thesis , Gebze Technical University , Turkey . Şahan G. , & Eldem V. ( 2015 ) Well posedness conditions for bimodal piecewise affine systems , Syst. Control Lett. , 83 , 9 – 18 . Google Scholar CrossRef Search ADS Shorten R. , Mason O. , Cairbre F. O. & Curran P. ( 2004 ) A unifying framework for the SISO circle criterion and other quadratic stability criteria , Int. J. Control , 77 , 1 – 8 . Google Scholar CrossRef Search ADS Shorten R. N. , & Narendra K. S. ( 1999 ) Necessary and sufficient conditions for the existence of a common quadratic Lyapunov function for two stable linear second order systems . Proceedings of American Control Conference , San Diego, CA , June. 1999 . pp. 1410 – 1414 . Shorten R. , Wirth F. , Mason O. , Wulff K. , & King C. ( 2007 ) Stability criteria for switched and hybrid systems , SIAM Rev. , 49 , 545 – 592 . Google Scholar CrossRef Search ADS Spraker J. S. , & Biles D.C. ( 1996 ) A Comparison of the Carathéodory and Filippov ´ Solution Sets , J. Math. Anal. Appl. , 198 , 571 – 580 . Google Scholar CrossRef Search ADS Sun Z. ( 2010 ) Stability of piecewise linear systems revisited , Annu. Rev. Control , 34 , 221 – 231 . Google Scholar CrossRef Search ADS Thuan L. Q. , & Çamlibel M. K. ( 2014 ) On the existence, uniqueness and nature of Carathéodory and Filippov solutions for bimodal piecewise affine dynamical systems , Syst. Control Lett. , 68 , 76 – 85 . Google Scholar CrossRef Search ADS Appendix Proof of Lemma 2.4. We give the proof only for the case $$\mu_{1}<\mu_{2}<\mu_{3}.$$ The other cases can be proved using a similar approach. 1. Suppose that $$\mathbf{z}_{1}(t)$$ be a trajectory starting from $$\mathcal{S}_{1}\cap \mathcal{H}$$ with initial condition $$\mathbf{z}_{1}(0)= \mathbf{v}_{1}\left(\theta_{1}\right) $$ where $$-\phi_{1}\leq \theta_{1}<\pi -\phi_{1}$$. In view of Remark 2.1 there exists $$\tau_{1}>0 $$ such that $$c^{T}z_{1}(\tau_{1})=0$$ and the trajectory changes mode. Using equation (2.9) we get z1(τ1)=eσ1τ1v1(θ^1), where $$\hat{\theta}_{1}$$ in $$(\pi -\phi_{1},\pi +\phi_{1})$$. Then, in view of Definition 2.4 there exists a unique $$\theta_{2}$$ in $$\left[-\phi_{2},2\pi -\phi_{2}\right] $$ and $$\eta_{12}(\hat{\theta}_{1})>0$$ such that $$\mathbf{v}_{1}(\hat{\theta}_{1})=\eta_{12}(\hat{\theta}_{1})\mathbf{v}_{2}(\theta_{2}).$$ Therefore, we get eσ1τ1v1(θ^1)=η12(θ^1)eσ1τ1(s^2sin⁡θ2+s^3cos⁡θ2) which implies that x^1sin⁡(θ^1)+y1cos⁡(θ^1)=η12(θ^1)(s^2sin⁡θ2+s^3cos⁡θ2) or equivalently Γ1[sin⁡θ^1cos⁡θ^1]=η12(θ^1)Γ2[sin⁡θ2cos⁡θ2]. The equation above can be written more explicitly as follows. Γ1[sin⁡θ^1cos⁡θ^1]=η12Γ2[sin⁡θ2cos⁡θ2]. Let $Q_{1}=\left[\begin{array}{@{}cc}1 & 0 \\b_{1} & 1\end{array}\right] $, $Q_{2}=\left[\begin{array}{@{}cc}1 & 0 \\ \frac{\mu_{2}-\mu_{1}}{\mu_{3}-\mu_{1}} & 1\end{array}\right] $ and $$Q_{3}=$$ $\left[\begin{array}{@{}cc}1 & \mu_{2} \\0 & 1\end{array}\right].$ Then, we get Q3Γ1Q1−1Q1[sin⁡θ^1cos⁡θ^1]=η12Q3Γ2Q2−1Q2[sin⁡θ2cos⁡θ2] and we obtain, η12[sin⁡θ2cos⁡θ2+b2sin⁡θ2]=Γ2−1Γ1[sin⁡θ^1cos⁡θ^1+b1sin⁡θ^1] where $$b_{2}=\cot \phi_{2}=\frac{\mu_{2}-\mu_{1}}{\mu_{3}-\mu_{1}}$$ and Γ2−1Γ1=ω1a32a21(μ2−μ1)(μ3−μ2)[−ω1(b12+1)−(λ1−a11−μ2a21)0−b2a21(μ3−μ2)]. More explicitly, we have η12[sin⁡θ2cos⁡θ2+b2sin⁡θ2]=−ω1a32(μ2−μ1)(B2−B3)[1B20b2(B2−B3)][sin⁡θ^1cos⁡θ^1+b1sin⁡θ^1]. Using the equations above, we get the following solutions. η12sin⁡θ2=−ω1(sin⁡θ^1+B2(cos⁡θ^1+b1sin⁡θ^1))a32(μ2−μ1)(B2−B3)η12(cos⁡θ2+b2sin⁡θ2)=−ω1b2(cos⁡θ^1+b1sin⁡θ^1)a32(μ2−μ1)θ2=cot−1⁡{(B2−B3)b2(cot⁡θ^1+b1)(cot⁡θ^1+b1)B2+1−b2}=cot−1⁡{−b2(B3(cot⁡θ^1+b1)+1)(cot⁡θ^1+b1)B2+1} 2. Let $$\mu_{1}<\mu_{2}<\mu_{3}$$ and $$\mathbf{z}_{2}(t)$$ be a trajectory starting from $$\mathcal{S}_{2}\cap \mathcal{H}$$ with initial condition $$\mathbf{z}_{2}(0)=\eta_{12}\left(\hat{\theta}_{1}\right) e^{\sigma_{1}\tau_{1}}\mathbf{v}_{2}\left(\theta_{2}\right) $$. If $$\mathbf{z}_{2}(t)$$ is a transitive trajectory ($$\frac{\pi }{2}<\theta_{2}<\pi -\phi_{2}$$) (or equivalently outside of the cone $$\mathcal{C}^{-}$$), then there exists a $$\tau_{2}>0$$ such that $$\mathbf{c^{T}}\mathbf{\ z}_{2}(\tau_{2})=0$$ and the trajectory changes mode at $$t=\tau_{2}$$. Note that $$\mathbf{\ z}_{2}(\tau_{2})$$ can be expressed as follows. z2(τ2)=η12(θ^1)eσ1τ1eμ2τ2N2{s^2sin⁡θ^2+s^3cos⁡θ^2}. Then, in view of Definition 2.4 there exists a unique $$\theta_{11}$$ in $$(-\phi_{1},-\psi_{1})$$ and $$\eta_{21}(\theta_{2})>0$$ such that $$ \mathbf{v}_{2}(\hat{\theta}_{2})=\eta_{21}(\hat{\theta}_{2})\mathbf{v}_{1}(\theta_{11}).$$ Therefore, we have z2(τ2)=η12(θ^1)eσ1τ1eμ2τ2N2{s^2sin⁡θ^2+s^3cos⁡θ^2}=η12(θ^1)eμ2τ2eσ1τ1η21(θ^2)N2(x^1sin⁡θ11+y1cos⁡θ11) where s^2sin⁡θ^2+s^3cos⁡θ^2:=η21(θ^2){x^1sin⁡θ11+y1cos⁡θ11}. This equation can also be written as follows. Γ2[sin⁡θ^2cos⁡θ^2]=η21Γ1[sin⁡θ11cos⁡θ11]. Using the same approach as in the previous case, let $Q_{1}=\left[\begin{array}{@{}cc} 1 & 0 \\ b_{1} & 1 \end{array} \right] $, $Q_{2}=\left[\begin{array}{@{}cc} 1 & 0 \\ \frac{\mu_{2}-\mu_{1}}{\mu_{3}-\mu_{1}} & 1 \end{array} \right] $ and $Q_{3}=\left[\begin{array}{@{}cc} a_{21} & \left(\lambda_{1}-a_{11}\right) \\ 0 & 1 \end{array} \right] $. Then, we obtain Q3Γ2Q2−1Q2[10b21][sin⁡θ^2cos⁡θ^2]=η21Q3Γ1Q1−1Q1. which yields η21[sin⁡θ11cos⁡θ11+b1sin⁡θ11]=Γ1−1Γ2[sin⁡θ^2cos⁡θ^2+b2sin⁡θ^2] where Γ1−1Γ2=(μ3−μ1)a32ω1[b2(B3−B2)B20−1]. More explicitly, we have η21[sin⁡θ11cos⁡θ11+b1sin⁡θ11]=(μ3−μ1)a32ω1[b2(B3−B2)B20−1][sin⁡θ^2cos⁡θ^2+b2sin⁡θ^2]. The equation above implies that η21sin⁡θ11=(μ3−μ1)a32ω1[b2B3sin⁡θ^2+B2cos⁡θ^2]η21(cos⁡θ11+b1sin⁡θ11)=−(μ3−μ1)a32ω1(cos⁡θ^2+b2sin⁡θ2)θ11=cot−1⁡{−(cot⁡θ^2+b2)b2B3+B2cot⁡θ^2−b1}. Furthermore, using Lemma 2.3.2.b we get $$\hat{\theta}_{2}\rightarrow \frac{3\pi }{4}$$ and $$\tau_{2} \rightarrow \infty$$ as $$\theta_{2}\rightarrow \frac{\pi }{2}$$. Substituting $$ \hat{\theta}_{2}\rightarrow \frac{3\pi }{4}$$, in equation (2.28) and using the equality $$b_{2}B_{3}-B_{2}=B_{1}\left(b_{2}-1\right),$$ we get $$ \cot \theta_{11}=-\frac{1}{B_{1}}-b_{1}= \cot(-\psi_{1})$$. This implies that $$-\phi_{1}<\theta_{11}<-\psi_{1}.$$ □ Proof of Lemma 3.2. We calculate the derivative of $$\theta_{11}$$ with respect to $$\theta_{1},$$ for all cases and we get following results: dθ11dθ1=dθ11dθ^2dθ^2dτ2dτ2dθ2dθ2dθ^1dθ^1dθ1 1. Firstly, we give the proof the case $$\mu_{1}<\mu_{2}<\mu_{3}$$. Using the equation (2.28),we calculate $$\frac{d\theta_{11}}{d\hat{\theta}_{2}},$$ dθ11dθ^2=b2sin2⁡θ11(B2−B3)sin2⁡θ^2(b2B3+B2cot⁡θ^2)2 and using equations (2.24) and (2.26), we get −cot⁡θ^2=e(μ3−μ2)τ2(e(μ2−μ1)τ2−1)(e(μ3−μ1)τ2−1). Calculating the derivative of above equation with respect to $$\tau_{2},$$ we get the following equation 1sin2⁡θ^2dθ^2dτ2=e(μ3−μ2)τ2(e(μ3−μ1)τ2−1){(μ3−μ1)e(μ2−μ1)τ2+(μ3−μ1)e(μ3−μ1)τ2cot⁡θ2−(μ3−μ2)}. Since −cot⁡θ2=(e(μ2−μ1)τ2−1)(e(μ3−μ1)τ2−1)⇒e(μ3−μ1)τ2cot⁡θ2+e(μ2−μ1)τ2=cot⁡θ2+1 we get dθ^2dτ2=sin2⁡θ^2e(μ3−μ2)τ2(μ3−μ1)(cot⁡θ2+b2)(e(μ3−μ1)τ2−1). Calculate the derivative of $$\cot \theta_{2}$$ with respect to $$\hat{\theta}_{1},$$ we get 1sin2⁡θ2dθ2dθ^1=−b2B3((cot⁡θ^1+b1)B2+1)sin2⁡θ^1+b2B2(B3(cot⁡θ^1+b1)+1)sin2⁡θ^1((cot⁡θ^1+b1)B2+1)2dθ2dθ^1=b2(B2−B3)sin2⁡θ2sin2⁡θ^1((cot⁡θ^1+b1)B2+1)2 Additionally, we calculate $$\frac{d\tau_{2}}{d\theta_{2}},$$ using the equation (2.24) 1sin2⁡θ2=dτ2dθ2[(μ2−μ1)e(μ2−μ1)τ2(e(μ3−μ1)τ2−1)−(μ3−μ1)e(μ3−μ1)τ2(e(μ2−μ1)τ2−1)](exp⁡((μ3−μ1)τ2)−1)21sin2⁡θ2=dτ2dθ2[(μ2−μ1)e(μ2−μ1)τ2+(μ3−μ1)e(μ3−μ1)τ2cot⁡θ2](exp⁡((μ3−μ1)τ2)−1)dτ2dθ2=(e(μ3−μ1)τ2−1)(μ3−μ1)sin2⁡θ2e(μ2−μ1)τ2(b2+cot⁡θ^2) and then dθ11dθ1=dθ11dθ^2dθ^2dτ2dτ2dθ2dθ2dθ^1dθ^1dθ1=(b2(B2−B3)b2B3+B2cot⁡θ^2)2(e(μ3−μ2)τ2(b2+cot⁡θ2)e(μ2−μ1)τ2(b2+cot⁡θ^2))(dF1dθ1sin2⁡θ11sin2⁡θ^1((cot⁡θ^1+b1)B2+1)2) Since $$-\left(\cot \theta_{11}+b_{1}\right) =\frac{\cot \hat{\theta}_{2}+b_{2}}{b_{2}B_{3}+B_{2}\cot \hat{\theta}_{2}}$$ and $$\cot \theta_{2}+b_{2}=\frac{\left(B_{2}-B_{3}\right) b_{2}\left(\cot \hat{\theta}_{1}+b_{1}\right) }{\left(\cot \hat{\theta}_{1}+b_{1}\right) B_{2}+1}$$ we get dθ11dθ1=dF1dθ1e(μ3−μ2)τ2sin2⁡θ11e(μ2−μ1)τ2sin2⁡θ^1(cot⁡θ11+b1cot⁡θ^2+b2)2(cot⁡θ2+b2cot⁡θ^2+b2)(cot⁡θ2+b2cot⁡θ^1+b1)2=dF1dθ1e(μ3−μ2)τ2sin2⁡θ11e(μ2−μ1)τ2sin2⁡θ^1(cot⁡θ11+b1cot⁡θ^1+b1)2(cot⁡θ2+b2cot⁡θ^2+b2)3 We know that $$e^{\left(\mu_{3}-\mu_{2}\right) \tau_{2}}=\frac{\cot \hat{ \theta}_{2}}{\cot \theta_{2}}=\left(\frac{\left(\cot \theta_{11}+b_{1}\right) B_{3}+1}{\left(\cot \theta_{11}+b_{1}\right) B_{2}+1} \right) \left(\frac{\left(\cot \hat{\theta}_{1}+b_{1}\right) B_{2}+1}{ \left(\cot \hat{\theta}_{1}+b_{1}\right) B_{3}+1}\right) $$, e(μ1−μ2)τ2=1+cot⁡θ^21+cot⁡θ2=((cot⁡θ11+b1)B1+1(cot⁡θ^1+b1)B1+1)((cot⁡θ^1+b1)B2+1(cot⁡θ11+b1)B2+1) and $$\frac{\cot \theta_{2}+b_{2}}{\cot \hat{\theta}_{2}+b_{2}}=\frac{\frac{ \left(B_{2}-B_{3}\right) b_{2}\left(\cot \hat{\theta}_{1}+b_{1}\right) }{ \left(\cot \hat{\theta}_{1}+b_{1}\right) B_{2}+1}}{\frac{\left(B_{2}-B_{3}\right) b_{2}\left(\cot \theta_{11}+b_{1}\right) }{\left(\cot \theta_{11}+b_{1}\right) B_{2}+1}}$$ then dθ11dθ1=sin2⁡θ11e(μ2−μ1)τ2sin2⁡θ^1(cot⁡θ1+b1)(cot⁡θ11+b1)((cot⁡θ11+b1)B3+1(cot⁡θ^1+b1)B3+1)((cot⁡θ11+b1)B2+1(cot⁡θ^1+b1)B2+1)2=e(μ1−μ2)τ2(sin⁡(θ11+ψ2)sin⁡(θ^1+ψ2))2(cot⁡θ1+b1)(cot⁡θ11+b1)((cot⁡θ11+b1)B3+1(cot⁡θ^1+b1)B3+1)=(sin⁡(θ11+ψ1)sin⁡(θ^1+ψ1))(sin⁡(θ11+ψ2)sin⁡(θ^1+ψ2))(sin⁡(θ1+ϕ1)sin⁡(θ11+ϕ1))(sin⁡(θ11+ψ3)sin⁡(θ^1+ψ3))sin⁡θ^1sin⁡θ1 where $$\cot \psi_{2}=\frac{1}{B_{2}}+b_{1}$$ and $$\cot \psi_{3}=\frac{1}{B_{3}}+b_{1}$$. 2. In view of the above development, we have dθ11dθ1=e(μ1−μ2)τ2(sin⁡(θ11+ψ2)sin⁡(θ^1+ψ2))2(cot⁡θ1+b1)(cot⁡θ11+b1)((cot⁡θ11+b1)B3+1(cot⁡θ^1+b1)B3+1). Using this equation and $$\mu_{1}=\mu_{2},$$ (and $$B_{1}=B_{2}$$), then we get dθ11dθ1=(sin⁡(θ11+ψ2)sin⁡(θ^1+ψ2))2(cot⁡θ1+b1)(cot⁡θ11+b1)((cot⁡θ11+b1)B3+1(cot⁡θ^1+b1)B3+1)=(sin⁡(θ11+ψ2)sin⁡(θ^1+ψ2))2(sin⁡(θ1+ϕ1)sin⁡(θ11+ϕ1))(sin⁡(θ11+ψ3)sin⁡(θ^1+ψ3))sin⁡θ^1sin⁡θ1. 3. We get following equation for the first case. dθ11dθ1=sin2⁡θ11e(μ2−μ1)τ2sin2⁡θ^1(cot⁡θ1+b1)(cot⁡θ11+b1)((cot⁡θ11+b1)B3+1(cot⁡θ^1+b1)B3+1)((cot⁡θ11+b1)B2+1(cot⁡θ^1+b1)B2+1)2 For $$\mu_{1}<\mu_{2}=\mu_{3}$$, we know that $$B_{3}=B_{2}$$, then we get dθ11dθ1=sin2⁡θ11e(μ3−μ1)τ2sin2⁡θ^1(cot⁡θ1+b1)(cot⁡θ11+b1)((cot⁡θ11+b1)B3+1(cot⁡θ^1+b1)B3+1)3 and for this case we know that e(μ3−μ1)τ2=cot⁡θ^2cot⁡θ2=((cot⁡θ11+b1)B3+1(cot⁡θ11+b1)B1+1)((cot⁡θ^1+b1)B1+1(cot⁡θ^1+b1)B3+1). Therefore dθ11dθ1=sin2⁡θ11sin2⁡θ^1(cot⁡θ1+b1cot⁡θ11+b1)((cot⁡θ11+b1)B3+1(cot⁡θ^1+b1)B3+1)2((cot⁡θ11+b1)B1+1(cot⁡θ^1+b1)B1+1)=(sin⁡(θ11+ψ3)sin⁡(θ^1+ψ3))2(cot⁡θ1+b1cot⁡θ11+b1)((cot⁡θ11+b1)B1+1(cot⁡θ^1+b1)B1+1)=(sin⁡(θ11+ψ3)sin⁡(θ^1+ψ3))2(sin⁡(θ1+ϕ1)sin⁡(θ11+ϕ1))(sin⁡(θ11+ψ1)sin⁡(θ^1+ψ1))sin⁡θ^1sin⁡θ1. □ © The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

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IMA Journal of Mathematical Control and InformationOxford University Press

Published: Apr 10, 2017

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