Qualitative Theory of Dynamical Systems

Wang, Lianwen; Zhang, Xingan; Liu, Zhijun

2017 Qualitative Theory of Dynamical Systems

doi: 10.1007/s12346-017-0231-6

The aim of this paper is to extend the incidence rate of an SEIR epidemic model with relapse and varying total population size to a general nonlinear form, which does not only include a wide range of monotonic and concave incidence rates but also takes on some neither monotonic nor concave cases, which may be used to reflect media education or psychological effect. By application of the novel geometric approach based on the third additive compound matrix, we focus on establishing the global stability of the SEIR model. Our analytical results reveal that the model proposed can retain its threshold dynamics that the basic reproduction number completely determines the global stability of equilibria. Our conclusions are applied to two special incidence functions reflecting media impact.

Bustos, M.; López, Miguel; Martínez, Raquel; Vera, Juan

2017 Qualitative Theory of Dynamical Systems

doi: 10.1007/s12346-017-0233-4

In this paper, using the averaging theory of first order, we obtain sufficient conditions for computing periodic solutions in the 3D Hill problem with oblateness.

Wang, Bingfeng; Shi, Yanling; Jiang, Shunjun

2017 Qualitative Theory of Dynamical Systems

doi: 10.1007/s12346-017-0236-1

This paper studies Gevrey smoothness of elliptic lower dimensional invariant tori in Hamiltonian systems under partial Melnikov’s conditions and Rüssmann’s nondegeneracy condition.

Corbera, Montserrat; Cors, Josep; Roberts, Gareth

2017 Qualitative Theory of Dynamical Systems

doi: 10.1007/s12346-017-0238-z

We prove that any four-body convex central configuration with perpendicular diagonals must be a kite configuration. The result extends to general power-law potential functions, including the planar four-vortex problem.

Kong, Fanchao; Luo, Zhiguo

2017 Qualitative Theory of Dynamical Systems

doi: 10.1007/s12346-017-0239-y

By using the continuation theorem due to Mawhin and Gaines, the sufficient conditions ensuring the existence of positive periodic solutions for a kind of first-order singular differential equation induced by impulses. Some recent results in the literature are improved.

Tigan, Gheorghe

2017 Qualitative Theory of Dynamical Systems

doi: 10.1007/s12346-017-0241-4

In this work we study degenerate fold–Hopf bifurcation of codimension three in a differential system of the form $$\begin{aligned} {\dot{x}}=-x-y-z,\ {\dot{y}}=x+ay+bxy,\ {\dot{z}}=bx-cz. \end{aligned}$$ x ˙ = - x - y - z , y ˙ = x + a y + b x y , z ˙ = b x - c z . We prove that the classical theory of the fold–Hopf bifurcation cannot be applied for studying the system’s behavior. However, using tools from averaging theory we prove the existence of periodic orbits generated by this bifurcation.

Boros, Balázs; Hofbauer, Josef; Müller, Stefan; Regensburger, Georg

2017 Qualitative Theory of Dynamical Systems

doi: 10.1007/s12346-017-0243-2pmid: 30636938

Chemical reaction networks with generalized mass-action kinetics lead to power-law dynamical systems. As a simple example, we consider the Lotka reactions and the resulting planar ODE. We characterize the parameters (positive coefficients and real exponents) for which the unique positive equilibrium is a center.

Perdomo, Oscar

2017 Qualitative Theory of Dynamical Systems

doi: 10.1007/s12346-017-0244-1

In this paper we describe a 1-dimensional family of initial conditions $$\Sigma $$ Σ that provides reduced periodic solutions of the spatial isosceles 3-body problem. This family $$\Sigma $$ Σ contains a bifurcation point that make it look like the union of two embedded smooth curves. We will explain how the trajectories of the bodies in the solutions coming from one of the embedded curves have two symmetries while those coming from the other embedded curve only have one symmetry. We give an explanation for the existence of this bifurcation point.

Valls, Claudia

2017 Qualitative Theory of Dynamical Systems

doi: 10.1007/s12346-017-0245-0

In this paper we completely characterize the existence of algebraic traveling wave solutions for the celebrated Kolmogorov–Petrovskii–Piskunov/Zeldovich equation. To do it, we find necessary and sufficient conditions in order that a polynomial linear differential equation has a polynomial solution and we classify all the Darboux polynomials of the planar system $$\dot{x} =y$$ x ˙ = y , $$\dot{y} =-c/d y +f(x)(f'(x)+r)$$ y ˙ = - c / d y + f ( x ) ( f ′ ( x ) + r ) where f is a polynomial with $$\deg f \ge 2$$ deg f ≥ 2 , $$c,d>0$$ c , d > 0 and r are real constants. All results are of interest by themselves.

Cai, Zhen; Kou, Kit

2017 Qualitative Theory of Dynamical Systems

doi: 10.1007/s12346-017-0246-z

The theory of quaternion differential equations (QDEs) has recently received a lot of attention. They have numerous applications in physics and engineering problems. In the present investigation, a new approach to solve the linear QDEs is achieved. Specifically, the solutions of QDEs with two-sided coefficients are studied via the adjoint matrix technique. That is, each quaternion can be uniquely expressed as a form of linear combinations of two complex numbers. By applying the complex adjoint representation of quaternion matrix, the connection between QDEs, with unilateral or two-sided coefficients, and a system of ordinary differential equations is achieved. By a novel specific algorithm, the solutions of QDEs with two-sided coefficients are fulfilled.

Zhou, Hui; Wang, Wen; Yang, Liu

2017 Qualitative Theory of Dynamical Systems

doi: 10.1007/s12346-017-0247-y

This paper is concerned with a class of impulsive prey–predator model with Beddington–DeAngelis functional response. By fixed point theorem and constructing a Lyapunov function, some sufficient conditions are established to ensure the permanence and uniformly asymptotic stability of positive almost periodic solutions for the concerned system.

Huzak, Renato

2017 Qualitative Theory of Dynamical Systems

doi: 10.1007/s12346-017-0248-x

It is well known that the slow divergence integral is a useful tool for obtaining a bound on the cyclicity of canard cycles in planar slow–fast systems. In this paper a new approach is introduced to determine upper bounds on the number of relaxation oscillations Hausdorff-close to a balanced canard cycle in planar slow–fast systems, by computing the box dimension of one orbit of a discrete one-dimensional dynamical system (so-called slow relation function) assigned to the canard cycle.