Circuit design and simulation for the fractional-order chaotic behavior in a new dynamical system

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.


Introduction
Recently, the study of fractional calculus have become a focus of interest [1][2][3][4][5][6][7][8][9][10][11][12]. Because the applications of fractional calculus were found in many scientific fields, such as rheology, diffusive transport, electrical networks, electromagnetic theory, quantum evolution of complex systems, colored noise, etc. Compared with the classical well-known models, it was found that fractional derivatives provide a better tool for modeling memory and heredity properties of various phenomena. Various types of fractional derivatives and their applications can be found in the literature, for instance, the Caputo derivative [13], the recently introduced fractional derivative without singular kernel (Caputo-Fabrizio derivative) [14] and the Atangana-Baleanu derivative which is based upon the well-known generalized Mittag-Leffler function [15,16].
Besides, many scientists and engineers have been attracted to the theory of chaos since the discovery of the Lorenz attractor [17]. It was found that fractional-order chaos has useful application in many field of science like engineering, physics, mathematical biology, psychological, and life sciences [18][19][20][21][22][23]. On the other hand, chaotic signal is a key issue for future applications of chaos-based information systems, and can B Z. Hammouch Hammouch.zakia@gmail.com T. Mekkaoui toufik_mekkaoui@yahoo.fr 1 E3MI, FSTE Moulay Ismail University, Errachidia, Morocco be applied to secure communication and control processing, e.g., the transmitted signals can be masked by chaotic signals in secure communications and the image messages can be covered by chaotic signals in image encryption. In addition, the circuit implementation can verify the chaotic characteristics of the chaotic systems physically, provide support for the application of chaos, and promote their technological application in the future. Therefore, the circuit implementation of the chaotic systems has also attracted more and more attention for engineering applications. Especially, for those fractional-order attractors, the circuit implementations for them are more important [24][25][26][27][28][29][30].
In this work, we construct a new 3D fractional-order chaotic system. Through studying its dynamical behavior by numerical simulation based on the improved Adams-Bashforth-Moulton method [31] and designs chain ship fractional-order chaotic circuit based on frequency-domain approximation method [28]. Besides, we realize the fractionalorder chaotic system through Multisim software 13.0 circuit simulation platform.

Preliminaries
In what follows, Caputo derivatives are considered, taking the advantage that this allows for traditional initial and boundary conditions to be included in the formulation of the considered problem.

Definition 1
A real function f (x), x > 0, is said to be in the space C μ , μ ∈ R if there exits a real number λ > μ, such

Definition 2
The Riemann-Liouville fractional integral operator of order α of a real function f (x) ∈ C μ , μ ≥ −1, is defined as The operators J α has some properties, for α, β ≥ 0 and ξ ≥ −1: and has the following properties for

Stability criterion
To investigate the dynamics and to control the chaotic behavior of a fractional-order dynamic system: we need the following indispensable stability theorem (Fig.  1).
Theorem 1 (See [32,33]) For a given commensurate fractional-order system (3), the equilibria can be obtained by calculating f (x) = 0. These equilibrium points are locally asymptotically stable if all the eigenvalues λ of the Jacobian
Consider the fractional-order initial value problem: It is equivalent to the Volterra integral equation: Diethelm et al. have given a predictor-corrector scheme (see [34]), based on the Adams-Bashforth-Moulton algorithm to integrate Eq. (6). By applying this scheme to the fractionalorder system (5), and setting Equation (6) can be discretized as follows: where and the predictor is given by The error estimate of the above scheme is in which p = min(2, 1 + α).

The fractional frequency-domain approximation
The standard definition of fractional differintegral does not allow the direct implementation of the fractional operators in In formula (10), s = jω, its complex frequency and the chain ship circuit unit is described in Fig. 2a. The transfer function between A and B can be obtained as follows: Taking C 0 = 1νF. Since H (s)C 0 = 1 s 0.98 , we can reach Similarly, for α = 0.9, we can reach that the approxima- . The chain ship circuit unit for this case is shown in Fig.  2b. The transfer function between A and B is we can reach R 1 = 62.84 MΩ, R 2 = 250 kΩ, R 3 = 2.5 kΩ, C 1 = 1.23 µF, C 2 = 1.83 µF, and C 3 = 1.1 µF. Fig. 9 Circuit simulation asymptotically stable orbits of the fractionalorder system (16) observed by the oscilloscope 1V/Div: a x − y, b x −z, c y − z, for α = 0.9

A new 3D fractional-order chaotic system
We introduce the following system: where the fractional-order α ∈ (0, 1]. 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

Dynamical analysis
To reveal dynamical properties of the nonlinear system (16), the equilibria should be considered at first The obtained equilibrium points from (17) and the corresponding eigenvalues are given in Table 1.
Hence, E 0 is unstable, and E 1 is a saddle point of index 2. With the aid of Theorem 1, a necessary condition for the fractional-order systems (16) to remain chaotic is keeping at least one eigenvalue λ i in the unstable region, i.e., |arg(λ i )| > απ 2 , It means that when α > 0.949318 system (16) exhibits a chaotic behavior.

Circuit designs and numerical simulations
Applying the improved version of Adams-Bashforth-Moulton numerical algorithm described above with a step size h = 0.01, system (16) can be discretized. It is found that chaos exists in the fractional-order system (16) when α > 0.94 with the initial condition (x 0 , y 0 , z 0 ) = (0.7, 0.1, 0). Figure 3a-c demonstrate that the systems has chaotic behavior for α = 0.98. On the other hand, when we take some values of α ≤ 0.94, the fractional system (16) can display the periodic attractors, and asymptotically stable orbits (see Figs. 4,5). Moreover, using Multisim software 13 to conduct simulations on the 3D fractional-order system (16), analog circuits are designed to realize the behavior of (16). Three state variables x, y and z are implemented by three channels, respectively. The implementations use resistors, capacitors, analog multipliers, and analog operational amplifiers, as shown in Figs. 6 and 7. A comparison of Figs. 3, 4, 5, 6, 7, and 8 (resp. 4-9 and 5-10) proves that analog circuit for system (16) is well coincident with numerical simulations. A conclusion can be made that the chaotic and non-chaotic behaviors exist in the fractional-order system (16), which verifies its existence and validity (Figs. 9, 10).

Conclusion
In this paper, we introduce a new three-dimensional fractionalorder chaotic system and its existence and stability. By adopting a chain ship circuit form , the circuit experimental simulation of this fractional-order system is presented.
The derived results between numerical simulation and circuit experimental simulation are in agreement with each other.
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, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.