Circuit design and simulation for the fractionalorder chaotic behavior in a new dynamical system
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Abstract
This paper presents a novel 3D fractionalordered 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 fractionalordered 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 StabilityIntroduction
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 wellknown 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 wellknown 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 fractionalorder 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 chaosbased information systems, and can 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 fractionalorder attractors, the circuit implementations for them are more important [24, 25, 26, 27, 28, 29, 30].
In this work, we construct a new 3D fractionalorder chaotic system. Through studying its dynamical behavior by numerical simulation based on the improved Adams–Bashforth–Moulton method [31] and designs chain ship fractionalorder chaotic circuit based on frequencydomain 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 \(C_{\mu },\,\mu \in {\mathbb {R}}\) if there exits a real number \(\lambda > \mu \), such that \( f(x)=x^{\lambda }g(x)\), where \(g(x)\in C[0,\infty )\) and it is said to be in the space \(C^{m}_{\mu }\) if and only if \(f^{(m)}\in C_{\mu }\) for \(m\in \mathbb {N}\).
Definition 2

\(J^{\alpha }J^{\beta }f(x)=J^{\alpha +\beta }f(x),\)

\(J^{\alpha }J^{\beta }f(x)=J^{\beta }J^{\alpha }f(x),\)

\(J^{\alpha }x^{\xi }= \frac{\varGamma (\xi +1)}{\varGamma (\alpha + \xi +1)}x^{\alpha + \xi }\).
Definition 3

\(D^{\alpha }J^{\alpha } f(x)= f(x),\)

\(J^{\alpha }D^{\alpha } f(x)= f(x)\displaystyle \sum \limits _{k=0}^{m1} f^{(k)}(0^{+})\frac{x^{k}}{k!},\) for \( x > 0, \)
Stability criterion
Theorem 1
Circuit implementation and numerical simulations
Adams–Bashforth (PECE) algorithm
Equilibrium points and corresponding eigenvalues
Equilibrium points  Eigenvalues 

\(\displaystyle E_0(0,0,0)\)  \(\quad \displaystyle \lambda _1 = 3,\quad \lambda _{2} = 7,\, \lambda _3 = 2\) 
\(E_1(0.923250,1.35886,0.889584)\)  \(\quad \displaystyle \lambda _1 = 5.478102,\quad \lambda _{2,3} = 0.418480 \pm 5.245549\text{ I }\) 
The fractional frequencydomain approximation
A new 3D fractionalorder chaotic system
Dynamical analysis
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 fractionalorder systems (16) to remain chaotic is keeping at least one eigenvalue \(\lambda _i\) in the unstable region, i.e., \(\displaystyle \mathrm{arg}(\lambda _i) > \frac{\alpha \pi }{2},\) It means that when \(\alpha > 0.949318\) 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 fractionalorder system (16) when \(\alpha > 0.94\) 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 \(\alpha =0.98\). On the other hand, when we take some values of \(\alpha \le 0.94 \), 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 fractionalorder 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 nonchaotic behaviors exist in the fractionalorder system (16), which verifies its existence and validity (Figs. 9, 10).
Conclusion
In this paper, we introduce a new threedimensional fractionalorder chaotic system and its existence and stability. By adopting a chain ship circuit form , the circuit experimental simulation of this fractionalorder system is presented. The derived results between numerical simulation and circuit experimental simulation are in agreement with each other.
Notes
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