This paper presents a novel 3D fractional-ordered chaotic system. The dynamical behavior of this system is investigated. An analog circuit diagram is designed for generating strange attractors. Results have been observed using Electronic Workbench Multisim software, they demonstrate that the fractional-ordered nonlinear chaotic attractors exist in this new system. Moreover, they agree very well with those obtained by numerical simulations. Keywords Circuit design · Chaotic system · Fractional derivative · Stability Introduction be applied to secure communication and control processing, e.g., the transmitted signals can be masked by chaotic signals Recently, the study of fractional calculus have become a in secure communications and the image messages can be focus of interest [1–12]. Because the applications of frac- covered by chaotic signals in image encryption. In addition, tional calculus were found in many scientiﬁc ﬁelds, such the circuit implementation can verify the chaotic character- as rheology, diffusive transport, electrical networks, elec- istics of the chaotic systems physically, provide support for tromagnetic theory, quantum evolution of complex systems, the application of chaos, and promote their technological colored noise, etc. Compared with the classical well-known application in the future. Therefore, the circuit implemen- models, it was found that fractional derivatives provide a bet- tation of the chaotic systems has also attracted more and ter tool for modeling memory and heredity properties of var- more attention for engineering applications. Especially, for ious phenomena. Various types of fractional derivatives and those fractional-order attractors, the circuit implementations their applications can be found in the literature, for instance, for them are more important [24–30]. the Caputo derivative [13], the recently introduced fractional In this work, we construct a new 3D fractional-order derivative without singular kernel (Caputo–Fabrizio deriva- chaotic system. Through studying its dynamical behavior tive) [14] and the Atangana–Baleanu derivative which is by numerical simulation based on the improved Adams– based upon the well-known generalized Mittag–Lefﬂer func- Bashforth–Moulton method [31] and designs chain ship tion [15,16]. fractional-order chaotic circuit based on frequency-domain Besides, many scientists and engineers have been attracted approximation method [28]. Besides, we realize the fractional- to the theory of chaos since the discovery of the Lorenz attrac- order chaotic system through Multisim software 13.0 circuit tor [17]. It was found that fractional-order chaos has useful simulation platform. application in many ﬁeld of science like engineering, physics, mathematical biology, psychological, and life sciences [18– 23]. On the other hand, chaotic signal is a key issue for future Preliminaries applications of chaos-based information systems, and can In what follows, Caputo derivatives are considered, taking the advantage that this allows for traditional initial and boundary B Z. Hammouch Hammouch.zakia@gmail.com conditions to be included in the formulation of the considered problem. T. Mekkaoui touﬁk_mekkaoui@yahoo.fr Deﬁnition 1 A real function f (x ), x > 0, is said to be in the E3MI, FSTE Moulay Ismail University, Errachidia, Morocco space C ,μ ∈ R if there exits a real number λ>μ, such 123 252 Complex & Intelligent Systems (2018) 4:251–260 that f (x ) = x g(x ), where g(x ) ∈ C [0, ∞) and it is said to m (m) be in the space C if and only if f ∈ C for m ∈ IN. Deﬁnition 2 The Riemann–Liouville fractional integral oper- ator of order α of a real function f (x ) ∈ C ,μ ≥−1, is deﬁned as α α−1 J f (x ) = (x − t ) f (t )dt , Γ(α) α> 0, x > 0 and J f (x ) = f (x ). (1) The operators J has some properties, for α, β ≥ 0 and ξ ≥−1: α β α+β – J J f (x ) = J f (x ), α β β α – J J f (x ) = J J f (x ), Fig. 1 Stability region of the fractional-order system (3) Γ(ξ +1) α ξ α+ξ – J x = x . Γ(α+ξ +1) Circuit implementation and numerical Deﬁnition 3 The Caputo fractional derivative D of a func- simulations tion f (x ) of any real number α such that m − 1 <α ≤ m, m α m ∈ IN,for x > 0 and f ∈ C in the terms of J is −1 Adams–Bashforth (PECE) algorithm α m−α m D f (x ) = J D f (x ) x We recall here the improved version of Adams–Bashforth– m−α−1 (m) = (x − t ) f (t )dt (2) Moulton algorithm [31,34] for the fractional-order systems. Γ(m − α) Consider the fractional-order initial value problem: and has the following properties for m −1 <α ≤ m, m ∈ IN, m D x = f (x (t )) 0 ≤ t ≤ T , μ ≥−1 and f ∈ C : t (5) (k) (k) α α x (0) = x , k = 0, 1,... , m − 1. – D J f (x ) = f (x ), m−1 α α (k) + It is equivalent to the Volterra integral equation: – J D f (x ) = f (x ) − f (0 ) , for x > 0, k! k=0 [α]−1 k t t 1 (k) α−1 x (t ) = x + (t − s) f (s, x (s))ds. (6) Stability criterion k! Γ(α) k=0 To investigate the dynamics and to control the chaotic behav- Diethelm et al. have given a predictor–corrector scheme (see ior of a fractional-order dynamic system: [34]), based on the Adams–Bashforth–Moulton algorithm to integrate Eq. (6). By applying this scheme to the fractional- D x (t ) = f (x (t )), (3) order system (5), and setting we need the following indispensable stability theorem (Fig. h = , t = nh, n = 0, 1,..., N . 1). Theorem 1 (See [32,33]) For a given commensurate fracti- Equation (6) can be discretized as follows: onal-order system (3), the equilibria can be obtained by [α]−1 calculating f (x ) = 0. These equilibrium points are locally k α t h (k) p asymptotically stable if all the eigenvalues λ of the Jacobian x (t ) = x + f (t , x (t )) h n+1 n+1 n+1 0 h k! Γ(α + 2) ∂ f k=0 matrix J = at the equilibrium points satisfy ∂ x α + a f (t , x (t )), (7) j ,n+1 j h j π Γ(α + 2) j =0 |arg(λ)| > α. (4) where 123 Complex & Intelligent Systems (2018) 4:251–260 253 1 1 Fig. 2 Chain ship unit of: a and b 0.98 0.9 s s Table 1 Equilibrium points and corresponding eigenvalues Equilibrium Eigenvalues points E (0, 0, 0)λ = 3,λ =−7,λ = 0 1 2 3 −2 E (−0.923250, 1.35886, −0.889584)λ =−5.478102, 1 1 λ = 0.418480 ± 5.245549I 2,3 α+1 α n − (n − α)(n + 1) , j = 0, α+1 α+1 α+1 a = (n − j +2) +(n − j ) −2(n − j + 1) , 1 ≤ j ≤ n j ,n+1 1, j = n + 1, (8) and the predictor is given by [α]−1 n t 1 (k) x (t ) = x + b f (t , x (t )), n+1 j ,n+1 j h j h 0 k! Γ(α) Fig. 3 Chaotic attractors of the fractional-order system (16) obtained k=0 j =0 by numerical simulations: a x − y, b y − z, c x − z,for α = 0.98 (9) α α time-domain simulations. To study such systems, it is nec- where b = ((n + 1) − j ) − (n − j ) . j ,n+1 essary to develop approximations to the fractional operators The error estimate of the above scheme is using the standard integer order operators. According to cir- cuit theory, the approximation formulation of α,from0.1to max |x (t ) − x (t )| = O(h ), j =0,1,...,N j h j 0.9, in reference [30], bode plot approximation chart, can be realized by the complex-frequency domain of the chain ship in which p = min(2, 1 + α). equivalent circuit. When α = 0.98, it can be worked out that the approximation formula of is The fractional frequency-domain approximation 0.98 The standard deﬁnition of fractional differintegral does not 1 1.2974(s + 1125) = . (10) allow the direct implementation of the fractional operators in 0.98 s (s + 1423)(s + 0.01125) 123 254 Complex & Intelligent Systems (2018) 4:251–260 Fig. 4 Asymptotically stable orbits of the fractional-order system (16) by numerical simulations: a x − y, b x − z, c y − z,for α = 0.9 Fig. 5 Time series of the fractional-order system (16) by numerical simulations: a x, b y, c z for α = 0.9 In formula (10), s = j ω, its complex frequency and the chain ship circuit unit is described in Fig. 2a. The transfer R = 91.1873 MΩ, R = 190.933 ω, 1 2 function between A and B can be obtained as follows: C = 975.32 nF, and C = 3.6806 µF. (12) 1 2 R R 1 2 H (s) = + 0.98 Similarly, for α = 0.9, we can reach that the approxima- sR C + 1 sR C + 1 1 1 2 2 1 1 1 R R C C tion formula of is 0 0 1 2 + s + 0.9 C C C +C 1 2 1 2 = . (11) 1 1 s + s + R C R C 1 2.2675(s + 1.292)(s + 215.4) 1 1 2 2 = . (13) 0.9 s (s + 0.01292)(s + 2.154)(s + 359.4) Taking C = 1νF. Since H (s)C = , we can reach 0 0 0.98 123 Complex & Intelligent Systems (2018) 4:251–260 255 Fig. 6 Circuit diagram for the realization of the fractional-order chaotic system (16)for α = 0.98 123 256 Complex & Intelligent Systems (2018) 4:251–260 Fig. 7 Circuit diagram for the realization of the fractional-order chaotic system (16)for α = 0.9 123 Complex & Intelligent Systems (2018) 4:251–260 257 Fig. 9 Circuit simulation asymptotically stable orbits of the fractional- order system (16) observed by the oscilloscope 1V/Div: a x − y, b x − z, Fig. 8 Chaotic attractors of the fractional-order system (16) observed c y − z,for α = 0.9 by the oscilloscope 1V/Div: a x − y, b y − z, c x − z with α = 0.98 A new 3D fractional-order chaotic system The chain ship circuit unit for this case is shown in Fig. 2b. The transfer function between A and B is We introduce the following system: 1 1 1 C C C 1 2 3 H (s) = + + , (14) 0.9 ⎧ 1 1 1 α 2 s + s + s + D x =−2x − y , R C R C R C 1 1 2 2 3 3 α 2 (16) D y =−4xz + 3y − z , we can reach D z = 4xy − 7z + yz, R = 62.84 MΩ, R = 250 kΩ, R = 2.5kΩ, 1 2 3 C = 1.23 µF, C = 1.83 µF, and C = 1.1 µF. (15) 1 2 3 where the fractional-order α ∈ (0, 1]. 123 258 Complex & Intelligent Systems (2018) 4:251–260 Fig. 10 Time series of the fractional-order system (16) observed by the oscilloscope 1V/Div: a x, b y, c z,for α = 0.9 123 Complex & Intelligent Systems (2018) 4:251–260 259 Dynamical analysis The derived results between numerical simulation and circuit experimental simulation are in agreement with each other. To reveal dynamical properties of the nonlinear system (16), the equilibria should be considered at ﬁrst Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecomm ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, −2x − y = 0, ⎪ and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative −4xz + 3y − z = 0, (17) Commons license, and indicate if changes were made. 4xy − 7z + yz = 0. The obtained equilibrium points from (17) and the corre- References sponding eigenvalues are given in Table 1. Hence, E is unstable, and E is a saddle point of index 0 1 1. Caponetto R, Dongola G, Fortuna L (2010) Fractional order 2. With the aid of Theorem 1, a necessary condition for the systems: modeling and control application. 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Published: May 29, 2018
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