A Chaos-Enhanced Fractional-Order Chaotic System with Self-Reproduction Based on a Memcapacitor and Meminductor

Abstract

Chaotic signals generated by chaotic oscillators based on memory elements are suitable for use in the field of confidential communications because of their very good randomness. But often their maximum Lyapunov exponent is not high enough, so the degree of randomness is not enough. It can be chaos enhanced by transforming it to fractional order using the Caputo differential definition. In this paper, based on the proposed hyperchaotic oscillator, it is extended to a fractional-order form to obtain a chaos-enhanced fractional-order memcapacitor meminductor system, in which several different styles of chaotic and hyperchaotic attractors are found. The dynamical behaviour of the system is studied using bifurcation diagrams, Lyapunov exponent spectrums and Lyapunov dimensions. The multistability of the system is explored in different initial orbits, and the spectral entropy complexity of this system is examined. Finally, a hardware implementation of the memcapacitor meminductor system is given, which demonstrates the effectiveness of the system. This study provides a reference for the study of chaos-enhanced.

1. Introduction

Research on memristors has been emerging for more than half a century since the concept of memristors was introduced [1]. The memristor has become a hot research topic due to its good memory properties and non-volatility. Subsequently, a memcapacitor and meminductor were proposed [2], and memory elements have been widely used to improve the performance of chaotic systems [3] and to construct various chaotic circuits [4] and neural networks [5]. Ma et al. greatly increased the complexity of the map by coupling a discrete memristor into a two-dimensional generalised map [6]. Bao et al. constructed a two-memristor dynamical system by adding two different memristors to a chaotic system to explore the multi-stability behaviour of the system [7]. Ma et al. constructed the Rulkov neural network model by taking into account local active memristor memory and local activity properties and using it to simulate biological synapses [8]. Guo et al. designed a Pavlovian associative memory circuit based on the proposed physical memristor model. The feasibility of the memristor as a synapse was verified by simulating various stimulus experiments [9]. Ma et al. verified the complex dynamical behaviour of the model by replacing neuronal synapses with two memristors in a Hopfield neural network [10].

Since chaotic systems have a high degree of randomness, they have a wide range of applications in the field of cryptography [11] and in pseudo-random number generators [12]. Echenausía-Monroy et al. realized Brownian motion, and the result has random Brownian behaviour [13]. L. G et al. constructed a pseudo random number generator based on chaotic mapping and a fixed-point algorithm [14]. Wang et al. proposed a colour-image-encryption algorithm by introducing the designed two-layer Joseph algorithm into a laser chaotic system [15]. Ma et al. proposed a hyper-chaotic image-encryption algorithm based on ring shrinkage with a variable modulo algorithm, which greatly reduces the time cost of encryption [16]. Sha et al. cleverly used the concept of three-input majority gate to design an image-encryption algorithm that can flexibly encrypt multiple images [17]. Zhu et al. optimised the artificial fish Swarms algorithm for use in a chaotic encryption scheme, resulting in a significant increase in key space and attack resistance [18]. Ma et al. introduced an improved Joseph algorithm into chaotic encryption in order to make the encryption algorithm more difficult to crack, making the algorithm very sensitive to the original image [19]. Gao et al. creatively combined BP neural networks with chaotic systems to achieve multi-size and multi-image encryption schemes [20]. Sha et al. designed an encryption algorithm using a hopfield neural network, and the results showed superior performance [21]. Sheng et al. proposed a multi-scale adjustment method for the amplitude and frequency of chaotic signals in order to better control chaotic systems [22]. However, as the research progresses, it was found that the maximum Lyapunov exponent of integer-order chaotic systems with the introduction of memory elements was not high enough, leading to the limitation of their chaotic properties. It has become an urgent problem to solve how to perform chaos-enhanced for chaotic systems.

Fractional-order(FO) calculation has excellent global memory and can more fully and accurately represent the essential relationships of natural models [23]. The extension of integer-order chaotic systems to FO makes them more free due to the introduction of the Caputo operator and thus can better reflect the properties of chaotic systems. Therefore, in recent years, researchers have developed a great interest in FO systems [24]. Xu et al. proposed a FO Hopfield neural network and designed a new image-encryption algorithm based on a multiple hash index chain [25]. Huang et al. used the FO Chen chaotic system for the fault detection of aircraft sensors based on the noise immunity of chaotic systems and the experimental verification of three types of fault detections under strong Gaussian noise, non-Gaussian noise and unknown disturbances [26]. A. Akgul et al. designed an FO chaotic system based on a memcapacitor and memristor to achieve secure communication in FO chaotic systems using a synchronisation method [27]. Ma et al. explored the dynamical behaviour of hopfield neural networks in the fractional order form [28]. Zhou et al. proposed a discrete FO map cross-coupled chaotic encryption scheme that can resist common attacks [29]. Liu et al. designed chaotic circuits based on a memcapacitor and meminductor with circuits consisting of only three elements, generalising them to the fractional order form. Unfortunately, the homogeneous multistability of the system was not explored [30].

Multistability is an interesting phenomenon in chaotic systems [31]. Ren et al. designed a new discrete memristor and constructed a chaotic map with multi-stability based on it and used it in the field of encryption [3]. Liu et al. proposed a new high-dimensional chaotic map with multiple stability based on a discrete memristor and meminductor and introduced a high-dimensional chaotic map into encryption schemes to greatly improve the effectiveness of encryption [32]. Generally, chaotic systems with a periodic function term control can often produce homogeneous extreme multistability. And in recent studies, it is referred to as the attractor self-reproduction phenomenon [33]. It provides multiple initial orbits with the same Lyapunov exponent, and this phenomenon offers a variety of possibilities in engineering applications [34]. Bao et al. performed the lossless control of chaotic sequences by varying the initial orbit in a simple two-dimensional sine map [35]. However, the attractor self-reproduction phenomenon has rarely been reported in FO hyperchaotic systems.

Based on the above problems, a new FO hyperchaotic system is constructed by extending the designed hyperchaotic oscillator based on the memcapacitor and meminductor to the FO form, which exhibits significant chaos-enhancing phenomena, and the system has a large Lyapunov exponent. The attractor self-reproduction phenomenon is also found and the direction of attractor growth can be controlled.

The rest of the paper is structured as follows. In Section 2, a dimensionless model of the hyperchaotic oscillator based on the memcapacitor and meminductor is presented and a new memcapacitor meminductor system(FOMMS) is obtained using the Caputo differential definition. The stability of the equilibrium point of the constructed FOMMS is analysed, and the numerical solution of the system is obtained using the Adomian decomposition method(ADM) algorithm. The dynamical behaviour of this system with different orders and parameters is analysed, and the rich multi-stability of this system is demonstrated in Section 3. The complexity of this system is examined using the spectral entropy(SE) algorithm in Section 4. The results of the hardware implementation of FOMMS are given in Section 5. Conclusions are given in the final section.

2. Circuit Models and System Equations

2.1. Caputo Differential Definition

The Caputo fractional differential is defined as [36]:

$$*D_t^{q}f(t)=\left\{\begin{array}{c}\frac{1}{\mathrm{\Gamma }(m-q)}{\displaystyle {\int }_0^t\frac{f(m)(\tau )\mathrm{d}\tau }{{(t-\tau )}^{q+1-m}},m-1<q<m}\\ \frac{{\mathrm{d}}^m}{\mathrm{d}{t}^m}f(t),q=m,q\in R^{+},m\in N\end{array}\right..$$

(1)

Fractional order nonlinear systems can be quickly solved with high accuracy and approximate numerical solutions using the ADM. Based on the ADM, the proposed fractional order differential equation can be expressed as

$$*D_{t_0}^{q}x(t)=Lx(t)+Nx(t)+c(t),$$

(2)

where *$D_{t_0}^q$ denotes the Caputo derivative operator of order q. L, N and c represent the linear, nonlinear and constant terms of the system, respectively. x(t) = [x₁(t), x₂(t), …, x_n(t)]^T is the state variable. Application of the R-L fractional integral operator $J_{t_0}^q$ representing q-order to both sides of Equation (2) will give

$$x(t)=J_{t_0}^{q}Lx(t)+J_{t_0}^{q}Nx(t)+J_{t_0}^{q}c+x(t_{t_0}^{+}),$$

(3)

where the initial value is given by x($t_{t_0}^{+}$). The integral operator $J_{t_0}^q$ has the following properties

$$J_{t_0}^{q}C=\frac{C}{\mathrm{\Gamma }(q+1)}{(t-t_0)}^q.$$

(4)

$$J_{t_0}^q{(t-t_0)}^{\xi }=\frac{\mathrm{\Gamma }(\xi +1)}{\mathrm{\Gamma }(\xi +1+q)}{(t-t_0)}^{q+\xi },(\xi >-1).$$

(5)

$$J_{t_0}^q{J}_{t_0}^{\alpha }x(t)=J_{t_0}^{q+\alpha }x(t),(\alpha \ge 0).$$

(6)

The nonlinear term can be decomposed by [37]:

$$\left\{\begin{array}{c}{B}_j^i=\frac{1}{i!}{\left(\frac{{\mathrm{d}}^i}{\mathrm{d}{\lambda }^i}N\left[v_j^i(\lambda )\right]\right)}_{\lambda =0}\hfill \\ v_j^i(\lambda )={\displaystyle \sum _{k=1}^i{(\lambda )}^k{x}_j^k}\hfill \\ i=0,1,2,\dots \infty ;j=1,2,3,\dots n\hfill \end{array}\right..$$

(7)

The nonlinear term N is denoted as

$$Nx={\displaystyle \sum _{i=0}^{\infty }{B}^i}(x^0,x^1,\dots ,x^i).$$

(8)

The numerical solution of the system is expressed as

$$x={\displaystyle \sum _{i=0}^{\infty }{x}^i}=J_{t_0}^q{\displaystyle \sum _{i=0}^{\infty }{x}^i}+J_{t_0}^q{\displaystyle \sum _{i=0}^{\infty }{B}^i+J_{t_0}^{q}c+x(t_0^{+})},$$

(9)

where the iterative relation for xⁱ is calculated as

$$\left\{\begin{array}{l}{x}^0=J_{t_0}^{q}c+x(t_0)\hfill \\ x^1=J_{t_0}^{q}L{x}^0+J_{t_0}^q{B}^0(x^0)\hfill \\ x^2=J_{t_0}^{q}L{x}^1+J_{t_0}^q{B}^1(x^0,x^1)\hfill \\ \cdots \hfill \\ x^i=J_{t_0}^{q}L{x}^{i-1}+J_{t_0}^q{B}^{i-1}(x^0,x^1,\dots ,x^{i-1})\hfill \\ \cdots \hfill \end{array}\right..$$

(10)

2.2. Numerical Solution for FOMMS

In this work, a memcapacitor and meminductor hyperchaotic oscillator was proposed. The circuit consists of a meminductor L_M, a memcapacitor C_M, a resistor R, a negative resistor G and a capacitor C. The circuit schematic diagram of the hyperchaotic oscillator is shown in Figure 1.

The equation for the state of the circuit can be expressed as [38]:

$$\left\{\begin{array}{l}\mathrm{d}{v}_{\mathrm{C}}/\mathrm{d}t=[(a+b\sin (\sigma ))q_{\mathrm{M}}-v_{\mathrm{C}}]/(RC)-(d+e\cos (\rho )){\phi }_{\mathrm{M}}/C\\ \mathrm{d}{q}_{\mathrm{M}}/\mathrm{d}t=v_{\mathrm{C}}/R+(G-1/R)(a+b\sin (\sigma ))q_{\mathrm{M}}\\ \mathrm{d}{\phi }_{\mathrm{M}}/\mathrm{d}t=v_{\mathrm{C}}\\ \mathrm{d}\sigma /\mathrm{d}t=q_{\mathrm{M}}\\ \mathrm{d}\rho /\mathrm{d}t={\phi }_{\mathrm{M}}\end{array}\right.,$$

(11)

let x₁ = v_C, x₂ = q_M, x₃ = φ_M, x₄ = σ, x₅ = ρ, the final dimensionless model of the circuit is obtained as

$$\left\{\begin{array}{l}{\dot{x}}_1=[(a+b\sin x_4)x_2-x_1]/(rk)-(d+e\cos x_5)x_3/k\\ {\dot{x}}_2=x_1/r+(g-1/r)(a+b\sin x_4)x_2\\ {\dot{x}}_3=x_1\\ {\dot{x}}_4=x_2\\ {\dot{x}}_5=x_3\end{array}\right..$$

(12)

By introducing the Caputo differential definition, the fractional order form of the dimensionless equation for the above circuit can be written:

$$\left\{\begin{array}{l}{D}_{t_0}^q{x}_1=[(a+b\sin x_4)x_2-x_1]/(rk)-(d+e\cos x_5)x_3/k\\ D_{t_0}^q{x}_2=x_1/r+(g-1/r)(a+b\sin x_4)x_2\\ D_{t_0}^q{x}_3=x_1\\ D_{t_0}^q{x}_4=x_2\\ D_{t_0}^q{x}_5=x_3\end{array}\right.,$$

(13)

where q is the order of the FOMMS. Using the ADM algorithm, the linear and nonlinear terms of the FOMMS are decomposed as

$$\begin{array}{l}\left[\begin{array}{l}L{x}_1\hfill \\ L{x}_2\hfill \\ L{x}_3\hfill \\ L{x}_4\hfill \\ L{x}_5\hfill \end{array}\right]=\left[\begin{array}{l}a{x}_2/(rk)-d{x}_3/k\hfill \\ x_1/r+(g-1/r)a{x}_2\hfill \\ x_1\hfill \\ x_2\hfill \\ x_3\hfill \end{array}\right],\left[\begin{array}{l}N{x}_1\hfill \\ N{x}_2\hfill \\ N{x}_3\hfill \\ N{x}_4\hfill \\ N{x}_5\hfill \end{array}\right]=\left[\begin{array}{l}-b{x}_2\sin (x_4)/(rk)-e{x}_3\cos (x_5)/k\hfill \\ (g-1/r)b{x}_2\sin (x_4)\hfill \\ 0\hfill \\ 0\hfill \\ 0\hfill \end{array}\right],\\ \left[\begin{array}{l}{c}_1\hfill \\ c_2\hfill \\ c_3\hfill \\ c_4\hfill \\ c_5\hfill \end{array}\right]=\left[\begin{array}{l}0\hfill \\ 0\hfill \\ 0\hfill \\ 0\hfill \\ 0\hfill \end{array}\right]\end{array}$$

(14)

According to Equation (7), the nonlinear terms of the system can be decomposed as

$$\left\{\begin{array}{c}{B}_1^0=a+b\sin (x_1^0)\hfill \\ B_1^1=a+b{x}_1^1\cos (x_1^0)\hfill \\ B_1^2=a+b(x_1^2\cos (x_1^0)+0.5x_1^1{x}_1^1\cos (x_1^0))\hfill \\ B_1^3=a+b(x_1^3\cos (x_1^0)+x_1^2{x}_1^1\sin (x_1^0)-(x_1^1{x}_1^1{x}_1^1\cos (x_1^0))/6)\hfill \\ B_1^4=a+b(x_1^4\cos (x_1^0)+x_1^3{x}_1^1\sin (x_1^0)+0.5x_1^2{x}_1^2\sin (x_1^0)-\hfill \\ 0.5x_1^1{x}_1^1{x}_1^2\cos (x_1^0)-(x_1^1{x}_1^1{x}_1^1\sin (x_1^0))/24)\end{array}\right..$$

(15)

$$\left\{\begin{array}{c}{B}_2^0=d+e\cos (x_1^0)\hfill \\ B_2^1=d+e(-x_1^1\sin (x_1^0))\hfill \\ B_2^2=d+e(-x_1^2\sin (x_1^0)-0.5x_1^1{x}_1^1\cos (x_1^0))\hfill \\ B_2^3=d+e(-x_1^3\sin (x_1^0)-x_1^2{x}_1^1\cos (x_1^0)+(x_1^1{x}_1^1{x}_1^1\sin (x_1^0))/6)\hfill \\ B_2^4=d+e(-x_1^4\sin (x_1^0)-x_1^3{x}_1^1\cos (x_1^0)-0.5x_1^2{x}_1^2\cos (x_1^0)+\hfill \\ 0.5x_1^1{x}_1^1{x}_1^2\sin (x_1^0)+(x_1^1{x}_1^1{x}_1^1\cos (x_1^0))/24)\end{array}\right..$$

(16)

The initial condition is given as x⁰ = [x₁($t_0^{+}$), x₂($t_0^{+}$), x₃($t_0^{+}$), x₄($t_0^{+}$), x₅($t_0^{+}$)], and the first term can be written as

$$\left\{\begin{array}{l}{x}_1^0=x_1(t_0^{+})\\ x_2^0=x_2(t_0^{+})\\ x_3^0=x_3(t_0^{+})\\ x_4^0=x_4(t_0^{+})\\ x_5^0=x_5(t_0^{+})\end{array}\right..$$

(17)

Letting $c_0^1$ = $x_0^1$, $c_0^2$ = $x_0^2$, $c_0^3$ = $x_0^3$, $c_0^4$ = $x_0^4$, $c_0^5$ = $x_0^5$, it can be obtained that x⁰ = c⁰ = [$c_1^0$ $c_2^0$ $c_3^0$ $c_4^0$ $c_5^0$]. By means of the iteration relation and the properties of the integral operator, it follows that

$$\left\{\begin{array}{l}{x}_1^1=[(a+b\sin (c_4^0))c_2^0-c_1^0]/(rk)-\frac{1}{k}[d+e\cos (c_5^0)]c_3^0\frac{{(t-t_0)}^q}{\mathrm{\Gamma }(q+1)}\\ x_2^1=c_1^0/r+(g-1/r)[a+b\sin (c_4^0)]c_2^0\frac{{(t-t_0)}^q}{\mathrm{\Gamma }(q+1)}\\ x_3^1=c_1^0\frac{{(t-t_0)}^q}{\mathrm{\Gamma }(q+1)}\\ x_4^1=c_2^0\frac{{(t-t_0)}^q}{\mathrm{\Gamma }(q+1)}\\ x_5^1=c_3^0\frac{{(t-t_0)}^q}{\mathrm{\Gamma }(q+1)}\end{array}\right..$$

(18)

Let

$$\left\{\begin{array}{l}{c}_1^1=(a{c}_2^0+b{c}_2^0\sin c_4^0-c_1^0)/(rk)-(d{c}_3^0+e{c}_3^0\cos c_5^0)/k\\ c_2^1=c_1^0/r+(g-1/r)(a{c}_2^0+b{c}_2^0\sin c_4^0)\\ c_3^1=c_1^0\\ c_4^1=c_2^0\\ c_5^1=c_3^0\end{array}\right..$$

(19)

Thus, x¹ can be expressed as x¹ = c¹(t − t₀)^q/Γ(q + 1), and using the same method, the decomposition of the other term coefficients is shown in Appendix A.

Thus, the six-term numerical solution of FOMMS is

$$\begin{array}{l}{\tilde{x}}_j(t)=c_j^0+c_j^1\frac{{(t-t_0)}^q}{\mathrm{\Gamma }(q+1)}+c_j^2\frac{{(t-t_0)}^q}{\mathrm{\Gamma }(2q+1)}+c_j^3\frac{{(t-t_0)}^q}{\mathrm{\Gamma }(3q+1)}\\ +c_j^4\frac{{(t-t_0)}^q}{\mathrm{\Gamma }(4q+1)}+c_j^5\frac{{(t-t_0)}^q}{\mathrm{\Gamma }(5q+1)},j=1,2,3,4,5\end{array}.$$

(20)

Setting the parameters q = 0.506, r = 0.2, g = 2, k = 1, a = 0.4, b = 2.8, d = 10, e = 3, the initial conditions are (1, 1, 0, 0, 0), and the iteration step h=0.001, it is obtained that the Lyapunov exponents of the system are LE1 = 7.3885, LE2 = 0.1046, LE3 = 0, LE4 = −6.0704 and LE5 = −171.3690, and the Lyapunov dimension is D_L = 4.0622. The system has two positive Lyapunov exponents and can generate hyperchaotic attractors. The phase trajectories of hyperchaotic attractors of the system are illustrated in Figure 2. To demonstrate the diversity of attractors, the three-dimensional(3D) phase trajectories of chaotic attractors of this system and their projection to the two-dimensional(2D) plane are given in the figure.

2.3. Dissipation and Stability of the System

The dissipation of the system can be solved by

$$\nabla V=\frac{\partial {\dot{x}}_1}{x_1}+\frac{\partial {\dot{x}}_2}{x_2}+\frac{\partial {\dot{x}}_3}{x_3}+\frac{\partial {\dot{x}}_4}{x_4}+\frac{\partial {\dot{x}}_5}{x_5}.$$

(21)

Taking the given system parameters into Equation (22)

$$\nabla V=-1/(rk)+(g-1/r)(a+b\sin (x_4))=-7.09<0.$$

(22)

To obtain the equilibrium point of the system, let the left-hand side of the five-dimensional equation be equal to 0. The equation can be expressed as

$$\left\{\begin{array}{l}[(a+b\sin x_4)x_2-x_1]/(rk)-(d+e\cos x_5)x_3/k=0\\ x_1/r+(g-1/r)(a+b\sin x_4)x_2=0\\ x_1=0\\ x_2=0\\ x_3=0\end{array}\right..$$

(23)

Clearly, the system has a planar equilibrium point set E(0, 0, 0, m, n), and the Jacobian matrix of the system is deduced as

$$J=\left[\begin{array}{ccccc}-1/(rk)& (a+b\sin (m))/(rk)& -(d+e\cos (n))/k& 0& 0\\ 1/r& (a+b\sin (m))(g-1/r)& 0& 0& 0\\ 1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\end{array}\right].$$

(24)

The characteristic equation for the equilibrium point is

$$-A{\lambda }^2(B{\lambda }^3+C{\lambda }^2+D\lambda +E)=0,$$

(25)

where

$$\left\{\begin{array}{l}A=1/(rk)\\ B=1\\ C=1/(rk)+(a+b\sin (m))/r-ag-bg\sin (m)\\ D=d/k-(ag+bg\sin (m))/(rk)+(e\cos (n))/k\\ E=(ad+bd\sin (m)+ae\cos (n)+be\cos (n)\sin (m))/(rk)\\ -(adg+aeg\cos (n)+bdg\sin (m)+beg\cos (n)\sin (m))/k\end{array}\right..$$

(26)

A fractional order system is stable if the eigenvalues of the Jacobian matrix satisfy [39]

$$\left|\arg (\lambda )\right|>\frac{\mu \pi }{2},\mu =\max (q_i),\left\{i\in \left.N\right|1\le i\le 5\right\},$$

(27)

obviously E(0, 0, 0, 0, 0) is an equilibrium point of the system. Substituting the system parameters into the characteristic equations, the eigenvalues are obtained as λ₁ = 0, λ₂ = 0, λ₃ = −5.3089 and λ_4,5 = −0.5055 ± 1.7253i. At this time, the system is stable. When the equilibrium point is E(0, 0, 0, −π/2, −π/2), the eigenvalues of the system are obtained as λ₁ = 0, λ₂ = 0, λ₃ = 2.104 and λ_4,5 = −0.012 ± 5.8009i. The eigenvalue λ₃ is a positive real number that falls on the real axis of the complex plane, so the equilibrium point is unstable.

3. Dynamical Analysis

3.1. Existence of Asymptotic Periodic Solutions for FO Systems

It has been shown that there are no exact periodic solutions for fractional order systems, but there are asymptotic periodic features in some fractional order systems.

Theorem 1

([40]). For fractional order differential equations

$$D^q{v}_i(t)=f_i(v_1(t),v_2(t),\dots v_n(t)),$$

(28)

where i = 1, 2, …, n, 0 < q < 1. When the trajectory of the system in phase space satisfies lim_t→∞v_i(t + T) − v_i(t) = 0, T > 0, it is called an asymptotic periodic oscillation(A-P-O).

In this section, three typical parameters q, e and b of the FOMMS are chosen to investigate the effect of different orders and parameters on the dynamical behaviour of the system.

3.2. Bifurcation Diagrams (BD) and Lyapunov Exponent Spectrums (LEs) for Relevant Parameters

FO systems introduce integral operators and have more parameter choices compared to integer-order systems, and the maximum Lyapunov exponent becomes very large, i.e., chaotic enhancement occurs. This section examines the BD, LEs and fractional dimension of the three typical parameters, q, e and b, of the system in their adjustable intervals.

Case I: When the parameters r = 0.2, g = 2, k = 1, a = 0.4, b = 2.8, d = 10, e = 3, initial conditions are (1, 1, 0, 0, 0) and q varies in [0.5, 1], the system undergoes a tangent bifurcation evolution, including hyperchaos, asymptotic periodic oscillations and chaos, as shown in Figure 3. When q is in [0.5, 0.521], the system shows hyperchaotic behaviour and has the maximum Lyapunov value in the system, at which point it is a hyperchaotic state of type (I). When q is in [0.521, 0.553], the system enters (I)-type asymptotic periodic oscillations. When q is in [0.553, 1], the system exhibits chaotic phenomena, and the maximum Lyapunov exponent shows a decreasing trend as the value of q increases, exhibiting a chaotic phase space trajectory of type (I-IV) in turn. It should be noted that since the system LE5 is a very small negative value, it has been removed from the LEs in order to show the other four Lyapunov exponents of the system more clearly, as is the case in all the LEs below. The three types of dynamical behaviour in the system are summarised in

Table 1. Dynamical behaviour with respect to order q.

q	Lyapunov Exponent	Lyapunov Dimension	Attractor Type
0.506	7.3885, 0.1046, 0, −6.0704, −171.3690	4.0622	Hyperchaos(I)
0.531	0, −0.4533, −7.6863, −7.6691, −140.4726	2	A-P-O(I)
0.572	3.7954, 0, −0.1267, −2.5675, −137.2317	4.0450	Chaos(I)
0.601	4.3579, 0, 0, −1.0012, −117.5874	4.0285	Chaos(II)
0.677	3.2477, 0, 0, −0.2873, 65.8004	4.0279	Chaos(III)
1	0.3535, 0, 0, −0.1014, −9.0481	4.0189	Chaos(IV)

, and the corresponding phase diagrams are depicted in Figure 4. As the value of q increases, the system completes the evolution from a hyperchaotic state to chaotic states. From the phase diagram, it can be found that the phase space trajectory of the hyperchaotic state is a superposition of the phase space trajectories of several chaotic states. It is clear from the LEs that the maximum Lyapunov exponent is much larger for FO systems than for integer-order systems.

Case II: When the parameters q = 0.7, r = 0.2, g = 2, k = 1, a = 0.4, b = 2.8, d = 10, initial conditions are (1, 1, 0, 0, 0), and the dynamical evolutionary behaviour of the system as e varies in [−5, 5] is given in Figure 5. Different system dynamical states are indicated in the bifurcation diagram by different zone colours, while arrows point out the bifurcation pattern when the bifurcation state of the system is changed. When e is in [−5, −4.14], the system exhibits asymptotic periodic oscillations of type (II). When e is in [−4.14, −3.84], the system exhibits (I)-type of asymptotic periodic oscillations. The jump switching between the two types of asymptotic periodic oscillations can be clearly seen in the BD. When e is in [−3.84, 0.7], the system transitions to a chaotic state via tangent bifurcation, exhibiting a (V)-type chaotic phase space trajectory during which there is a short asymptotic periodic window. When e is in [0.7, 1.04], the system exhibits (III)-type asymptotic periodic oscillations before transitions to a chaotic state via tangent bifurcation again. When e is in [1.04, 1.42], the system exhibits a (VI)-type chaotic phase space trajectory. When e is in [1.42, 5], the system exhibits a (III)-type chaotic phase space trajectory, and at e = 1.42, there is an interior crisis bifurcation that causes the chaotic state of the system to change in the range around this value. The resolution of typical states in the system is listed in

Table 2. Chaotic state with typical parameter e.

e	Lyapunov Exponent	Lyapunov Dimension	Attractor Type
−4.6	0, 0, −0.7315, −2.5151, −59.7031	2	A-P-O(II)
−3.42	1.6180, 0, 0, −0.3264, −68.7863	4.0245	Chaos(V)
0.79	0, 0, −0.7833, −9.3192, −41.8929	2	A-P-O(III)
1.36	1.0508, 0, 0, 0, −52.1427	4.0188	Chaos(VI)

, and the corresponding typical phase trajectory diagrams are shown in Figure 6. The (I)-type asymptotic periodic oscillations and (III)-type chaotic phase space trajectories are illustrated in Figure 3 and will not be shown repeatedly.

Case III: The dynamical evolution of the system is shown in Figure 7; when the parameters are q = 0.7, r = 0.2, g = 2, k = 1, a = 0.4, d = 10, e = 3, the initial conditions are (1, 1, 0, 0, 0), and b varies in [2, 10]. When b is in [2, 2.252], the system exhibits (IV)-type asymptotic periodic oscillations. When b is in [2.252, 2.336], the system undergoes a very brief chaotic state, exhibiting (VII)-type chaotic phase space trajectories, and b enters (V)-type asymptotic periodic oscillations again when it is in [2.336, 2.476]. Then, the system transitions to a chaotic state through a tangent bifurcation, with the phase diagram evolving from (III) to (II) to (I)-type chaotic states when b is in [2.476, 3.696], and finally the system enters a hyperchaotic state when b is in [3.696, 10], exhibiting a (II)-type hyperchaotic phase space trajectory. A typical resolution of the system with respect to the parameter b is given by

Table 3. Dynamical behaviour of parameter b.

b	Lyapunov Exponent	Lyapunov Dimension	Attractor Type
2.16	0, 0, −0.4545, −0.7457, −48.3074	2	A-P-O(IV)
2.28	2.7585, 0, 0, −0.3099, −51.3781	4.0346	Chaos(VII)
2.42	0, 0, −0.7500, −0.6785, −37.9772	2	A-P-O(V)
8.63	7.6442, 0.4123, 0, 0, −128.7118	4.0687	Hyperchaos(II)

, while the corresponding phase diagram is plotted in Figure 8.

3.3. Multiple Stability

3.3.1. Self-Reproduction of Attractors

The presence of the sine and cosine functions in the system makes the system inherently self-reproducing. The parameters are set to q = 0.7, r = 0.2, g = 2, k = 1, a = 0.4, b = 2.8, d = 10, e = 3. In this self-reproducing system, an infinite number of self-reproducing coexisting attractors can be generated, and only five sets of initial conditions are chosen symmetrically in the analysis. The state of the system controlled by the sine function manifested in x₄ dimensions is shown in Figure 9. The initial conditions are x₄(0) = kπ, k = (−8, −4, 0, 4, 8). The phase trajectory of the self-reproducing attractor in 3D space is shown in Figure 9A, while its projection in the 2D plane is plotted in Figure 9C, where the attractor can be seen to be periodically aligned along the x₄ dimension. The corresponding BD for x₄(0) in the (−8π, 8π) interval is shown in Figure 9B, where the dynamical evolution of the BD is linearly conditioned upwards in the intercepted interval. The corresponding time-domain waveforms are also depicted in Figure 9D, and the comparison shows that the mode of attractor reproduction and its time series signal are consistent with the performance of the BD. And the self-reproducing attractor controlled by the cosine function manifested in x₅ dimensions is shown in Figure 10. The initial conditions are x₅(0) = kπ, k = (−4, −2, 0, 2, 4). Similar to the x₄ dimensional case, the combined analysis of Figure 10A–D, one by one, corresponds to the reproduction in the x₅ dimensional system.

3.3.2. Coexisting Attractors

Setting parameters q = 0.7, r = 0.2, g = 2, k = 1, a = 0.4, b = 2.8, d = 10, e = 3, and initial conditions of (1, x₂(0), 0, 0, 0), the system has conditional coexistence behaviour, at which point the system exhibits coexistence behaviour of attractors different from the self-reproducing mode. The particular initial parameters chosen for the system in x₂ dimensions cause the system to produce conditional homogeneous and heterogeneous multistability in phase space, with attractors advancing in either consistent or unequal steps. The coexistence of unequal-step heterogeneous attractors is shown in Figure 11A. Style (I) chaotic attractors alternate with style (II) chaotic attractors, and the attractors advance in the x₂ direction in unequal steps. The coexistence of homogeneous style (I) chaotic attractors is shown in Figure 11B, where the system produces style (I) chaotic attractors and the attractors advance in the x₂ direction in equal steps. The coexistence of homogeneous style (II) attractors is illustrated in Figure 11C, where this system produces style (II) chaotic attractors, and the attractors advance in the x₂ dimension with unequal steps. The initial value of x₂ at this point is indicated at the corresponding position of each attractor in the figure. Chaotic attractors with different initial conditions are shown in different colours, and the direction of advance of the chaotic attractor is indicated by the corresponding coloured arrow.

3.4. State Transition

State transition is a special and complex phenomenon that occurs in chaotic systems. As time changes, the system state manifests itself as a transient or stable periodic or chaotic state, as a transition from one state to another. The process may be a transition from one periodic state to another periodic or chaotic (hyperchaotic) state or from one chaotic state to another periodic or chaotic (hyperchaotic) state. Given the initial conditions of (1, 1, 0, 0, 0), parameters q = 0.63, a = 0.32, b = 2.8, d = 10, e = 3, r = 0.2, g = 2, k = 1, the system state transition relationship is shown in Figure 12 and is composed of the time-domain waveform and x₃-x₄ phase diagram of the system. It is interesting to note that three chaotic state evolutions of the system occur during t at [0, 100]. When t is in [0, 42.263), this system is in chaotic state (I), indicated by light red. When t is in [42.263, 77.456), this system is in chaotic state (II), indicated by light blue. When t is in [77.456, 100], this system is in chaotic state (III), indicated in dark green. Therefore, the final chaotic attractor produced by the system is a light red double-scrolls chaotic attractor first, followed by a light blue double-scrolls chaotic attractor immediately based on it, which eventually manifests as a dark green four-scrolls chaotic attractor.

4. Complexity Analysis

In the above analysis of a range of dynamical characteristics of the system, it can be seen that dynamical behaviour of the system is very complex. The complexity of FOMMS can be obtained using the SE algorithm. It provides convenience for the selection of system parameters in engineering applications. The darker the colour in the complexity diagram, the higher the complexity, indicating that the chaotic signal generated by the system in this parameter interval is closer to a random signal. Fixed parameters r = 0.2, g = 2, k = 1, a = 0.4, d = 10 with initial values of (1, 1, 0, 0, 0) were investigated for parameters (q, b, e) in the adjustable parameter interval with the two-parameter complexity diagram shown in Figure 13A–C. The initial complexity of the parameters r = 0.2, g = 2, k = 1, a = 0.4, d = 10, with initial conditions x₄ in the interval [−8π, 8π] and x₅ in the interval [−4π, 4π] was also examined as shown in Figure 13D. The 3D complexity gives an intuitive view of the variation in the high and low complexity of the system, while the 2D complexity diagram is better for the details of the different parameter intervals. Therefore, all things considered, the 3D complexity diagram of the system and its projection in the 2D phase space is given. It can be seen that the system has a relatively high complexity and a wide range of high complexity parameters.

5. Hardware Implementation Based on DSP

The hardware implementation of chaotic systems is a prerequisite for their practical engineering application. In this section, the DSPF28335 is used as the core controller to implement the acquisition and transmission of FO chaotic systems. The DAC8552 is selected as the digital-to-analogue converter, and the corresponding phase trajectories are displayed via an oscilloscope. The equations of the FO chaotic system are written into the main function of the CCS integrated development environment. Setting parameters q = 0.7, r = 0.2, g = 2, k = 1, a = 0.4, b = 2.8, d = 10, e = 3 and initial conditions of (1, 1, 0, 0, 0), the processed chaotic signal is converted to analogue voltage through a digital-to-analogue converter and output to a two-channel oscilloscope. The hardware connection schematic of the experiment is shown in Figure 14, and the DSP hardware implementation platform is shown in Figure 15. The final experimental results obtained are shown in Figure 16 and Figure 17, which can prove that the simulation results are consistent with the results of the DSP hardware implementation.

6. Conclusions

The proposed memcapacitor meminductor hyperchaotic oscillator circuit is extended to a fractional order form and solved using the ADM algorithm. There are five types of A-P-O attractors, two types of hyperchaotic attractors and seven types of chaotic attractors that can be generated in the newly obtained FOMMS. The generation of hyperchaotic phenomena depends on the order of this system and the system parameters. Compared to the integer order system, the degrees of freedom of the system are extended and the maximum Lyapunov exponent is enhanced very significantly, verifying the chaos-enhancing behaviour of the fractional order system. Secondly, this system is multi-stable, and the system has the attractor coexistence behaviour in a self-reproducing manner. At the same time, there is a state transition phenomenon in the system; in a certain time interval, the system state has a chaotic state (I) to chaotic state (II) and finally to chaotic state (III), and chaotic state (I) and (II) exhibit double-scolls chaotic attractors, while chaotic state (III) exhibits four-scolls chaotic attractors, revealing the formation pattern of chaotic attractors. It is shown that FOMMS has a very complex dynamical behaviour. Finally, hardware implementation of the FOMMS based on DSP is carried out. The FOMMS proposed in this paper has good applications in the field of cryptography, and it is sincerely hoped that it can provide guidance for the study of high-dimensional FO chaotic systems. Next, it will be considered for application to the field of confidential communication.