Mega-Instability: Order Effect on the Fractional Order of Periodically Forced Oscillators

Abstract

The stability of differential equations is one of the most important aspects to consider in dynamical system theory. Chaotic systems were classified according to stability as multi-stable systems; systems with a single stable equilibrium; bi-stable systems; and, recently, mega-stable systems. Mega-stability refers to the infinity countable nested attractors of a periodically forced non-autonomous system. Many researchers attempted to present a simple mega-stable system. In this paper, we investigated the mega-stability of periodically damped non-autonomous differential systems with the following different order cases: integer and fractional. In the case of the integer order, we generalize the mega-stable system, such that the velocity is multiplied by a trigonometrical polynomial, and we present the necessary and sufficient conditions to generated countable infinity nested attractors. In the case of the fractional order, we obtained that the fractional order of periodically damped non-autonomous differential systems has infinity countable nested unstable attractors for some orders. The mega-instability was illustrated for two examples, showing the order effect on the trajectories. In addition, and to further recent work presenting simple high dimensional mega-stable chaotic systems, we introduce a 4D mega-stable hyperchaotic system, examining chaotic and hyperchaotic behaviors through Lyapunov exponents and bifurcation diagrams.

1. Introduction

The dynamics of differential systems are referred to as autonomous, defined in a smooth manifold, or non-autonomous, defined in a smooth fiber bundle by a continuous subjective map $p:E\to B_j\times F,$ where E is a smooth manifold, $B_j$ represents small regions of E, and F is the fiber. The theoretical foundation of dynamical systems is provided by offering an expanded understanding of the circumstances that result in stability or instability, especially when subjected to periodic damping. Limiting items that are typical of non-autonomous systems could not have real-world equivalents without an additional condition, in contrast to autonomous systems. This realization is especially important for mechanical engineering applications, where it is necessary to precisely characterize oscillatory system behavior under periodic damping in order to improve performance and safety. Non-autonomous systems are investigated by the following two methods: two-parameter semigroup operators defined on a Banach space or processes and skew product flows. The attractors of non-autonomous systems are described in terms of the families of sets that are projected onto each other under the dynamics, rather than a single set, as in autonomous systems. The two types of attraction that are now possible are forward attraction, which depends on the behavior of the system in the distant future, and pullback attraction, which depends on the behavior of the system in the distant past.

In particular, when Henri Poincaré “mathematically” studied the motion of a three-body problem in the early 1900s, he discovered that a nonlinear system displayed unexpected behavior. H. Poincaré noted that even a slight alteration to the beginning conditions would result in “unusual” behavior, similar to how wildly disparate orbits would arise. A formal definition of “chaos” as a nonlinear dynamical system that is sensitive to initial circumstances was provided by Edward Lorenz later in that century [1]. Numerous disciplines, including mathematics, physics, chemistry, biology, and others, include chaos [2]. This phenomenon has drawn much attention and is now a crucial factor to take into account when researching nonlinear dynamic systems [3,4,5,6]. For example, systems with no equilibrium [7], systems with stable equilibrium [8], and systems where the equilibrium points lie on a segmented straight line [9,10] are examples of new chaotic systems with spatial features like “algebraic features” that researchers have tried to create since Lorenz’s discovery.

Furthermore, chaotic systems can be categorized into multi-stable, extreme multi-stable, or mega-stable systems, based on the stability criteria. In dynamical systems, multi-stability is an extremely important phenomenon that is found in many different scientific fields, including physics, chemistry, biology, and economics. Apart from the normal sensitive dependency on initial conditions that prevents long-term predictability and defines chaotic systems, in multi-stable systems, the attractive state also depends on the initial conditions. Therefore, when developing an industrial system with specific criteria, such as the need to stabilize the proper state in the presence of noise, multi-stability might not be appropriate. However, when the proper control techniques are implemented, multi-stability can be used to create a switch between the coexisting states, and it offers flexibility in system performance when no changes to the parameters are needed [11]. In a dynamical system, extreme multi-stability is defined as the coexistence of an uncountable infinity of nested attractors [12]. Mega-stability occurs when a dynamical system has a countable infinity of nested attractors coexisting.

In addition, fractional orders of several chaotic systems have been investigated. Since a number of methods were developed in the 17th century, fractional calculus has continued to be an important topic. In this discipline, the Riemann–Liouville operator, the Caputo operator, and the Grunwald–Letnikov method are crucial tools. For more details, refer to [13,14]. Fractional calculus finds applications in diverse fields such as physics and chemical engineering [15], biology [16] and other areas of applied sciences [17,18,19]. Compared to integer-order models, fractional-order chaotic models display more complex behaviors. For more information, refer to [20,21].

This paper was structured as follows: In Section 2, we generalize a two-dimensional non-autonomous mega-stable system with velocity evolved to a trigonometric polynomial. In addition, in Section 3 we continue the recent investigation of exploring simple high-dimensional mega-stable chaotic systems by introducing a novel simple four-dimensional mega-stable hyperchaotic system. An examination of the dynamical characteristics of the proposed hyperchaotic mega-stable system and the complexity of its parameters is provided in Section 4. Section 5 presents the fractional order of mega-stable systems and shows that, in the case of the fractional order, this class of systems is mega-unstable. Section 6 illustrates the effect of order on the dynamics and the behavior of mega-unstable fractional order systems. Section 7 provides an implementation to the proposed systems using a microcontroller.

2. Mega-Stability of Non-Autonomous Systems: General Case

Dynamical systems evolve along trajectories that converge toward an attractor, a subset of the phase space that characterizes the asymptotic behavior of the system. These attractors may manifest as fixed points, periodic (or quasi-periodic) orbits, tori, and strange attractors, among other structures. Notably, a strange attractor can be categorized based on its stability, as follows: multi-stable, extremely multi-stable, or mega-stable. A mega-stable system is defined as one that features an infinite number of coexisting hidden nested attractors of various types, including limit cycles, tori, and strange attractors. This section briefly introduces the concept of mega-stability, characterized by a countable infinity of nested attractors. The initial observation of this phenomenon was made by Dr. Sajad Jafari and their research group, culminating in numerous publications on this topic from 2017 to 2024, as detailed in references [22,23,24,25,26,27,28]. Consider the example of a basic damped pendulum in a three-dimensional flow, influenced by sinusoidal forcing, which is defined by the following:

$$\ddot{x}+x+\alpha \dot{x}\cos \left(x\right)=0$$

(1)

Mega-stability was first observed in the following non-autonomous system, defined by a continuous function that belongs to the space of continuous functions $F\left(t\right)\in C[a,b]$, for any $t\in [a,b]$ and $\alpha <0$, as

$$\ddot{x}+x+\alpha \dot{x}\phi \left(x\right)=F(\Omega ,t)$$

(2)

where $\phi \left(x\right)=\cos \left(x\right)$ and $F(t,\Omega )=\sin (\Omega t)$, with a set of initial conditions defined as $(x_0,y_0)=(n\pi ,0)$ for $n=5,7,9,11$. The countable infinity nested attractor is shown in [29]. Moreover, chaotic dynamics appeared in the system (2) where the third dimension was recognized only by the frequency $\Omega =0.73,\alpha =0.33, A=1$. For more details, see [29].

Its natural to ask is as follows: what is the reason behind the attractors overlapping? In mathematical language this translates to the following: what are the sufficient and necessary conditions for attractors to overlap? In [29], it was mentioned that the geometric series of cos function crates the nested infinite attractors, while the numerical examination showed that the main real role of the cos function is in creating a multi-scroll attractor. An answer is given in Theorem 1.

By $\Lambda$, we denote a set of convergent sequences and divergent sequences with convergent common ratio of geometric progressions as

$$\underset{n\to \infty }{\lim }{\displaystyle \frac{x_{n+1}}{x_n}}=\left|L\right|<\infty \mathrm{as}n\to \infty ,$$

(3)

where $x_n\ne 0$. For example, the Fibonacci sequence ${\mathcal{F}}_n\in \Lambda$, such that ${\mathcal{F}}_n$ is a divergent sequence with convergent common ration ${\lim }_{n\to \infty }{\displaystyle \frac{{\mathcal{F}}_{n+1}}{{\mathcal{F}}_n}}={\displaystyle \frac{1+\sqrt{5}}{2}}$ (The Golden Ratio).

Definition 1

([30]). Let A be a square matrix in Banach space $\mathcal{B}$, such that for any $x\in \mathcal{B},$ then, the norm of x is $\parallel x\parallel ={\sup }_t\left|x\left(t\right)\right|$. The associated logarithmic norm $\mathcal{L}\left(K\right)$ of matrix K is defined by

$$\mathcal{L}\left(K\right)=\underset{h\to 0}{\lim }{\displaystyle \frac{\parallel I+hK\parallel -1}{h}},$$

where $K={\displaystyle \frac{A+A^{*}}{2}}$, I is the identity matrix, and $h>0$.

Definition 2

([31]). Consider a dynamical system $\dot{x}=f\left(x\right)$, and let $x\left(t\right)$ be a trajectory with initial condition $x\left(0\right)$. To quantify the exponential divergence/convergence of nearby trajectories, we linearize the system around $x\left(t\right)$, yielding the following variational equation:

$$\dot{\delta x}=Df\left(x\left(t\right)\right)\delta x,$$

where $Df\left(x\right(t\left)\right)$ is the Jacobian of f evaluated along $x\left(t\right)$. Let $\Phi \left(t\right)$ denote the fundamental matrix solution to this linearized equation, initialized as $\Phi \left(0\right)=I$. The Lyapunov exponents $\left\{{\lambda }_j\right\}$ are defined as follows:

$${\lambda }_j=\underset{t\to \infty }{\lim }{\displaystyle \frac{1}{t}}\ln {\sigma }_j(\Phi \left(t\right)),$$

where ${\sigma }_j(\Phi \left(t\right))$ are the singular values of $\Phi \left(t\right)$. The largest Lyapunov exponent is as follows:

$${\lambda }_{\max }=\underset{j}{\max }{\lambda }_j.$$

Definition 3

([32]). Let $B_{{\lambda }_i}$ be a closed sphere with radius ${\lambda }_i$ in the phase space X of the corresponding system. The set $B_{{\lambda }_i\left(t\right)}$ is a closed sphere evolute for a time $t$. The global attractor is

$${\mathcal{A}}_{{\lambda }_i}={\cap }_{t>0}{\mathcal{A}}_{{\lambda }_i\left(t\right)},$$

(4)

where $\mathcal{A}$ is invariant and the distance of any solution from $\mathcal{A}$ approaches zero as $t\to \infty$. In other word, the global attractor $\mathcal{A}$ includes every trajectory that results from every initial condition.

Theorem 1.

If $\phi \left(x\right)={\sum }_{m=1}^N{a}_m\cos \left(mx\right)+b_m\sin \left(mx\right)$ in (2) is a trigonometric polynomial defined on Banach space $\mathcal{B}$, at least one of the initial conditions is a sequence that belongs to $\mathcal{L}ambda$, and the other initial condition is zero, then,

1. 

The attractors of system (2) are countable infinity nested.

2. 

At least one of the attractors is strange with chaotic behavior.

Proof. 

1. Without losing generality, we assume that $x\left(0\right)\in \Lambda$ and $\dot{x}\left(0\right)=0$. We choose the initial conditions based on the fact that all generated attractors have the same velocity at $t=0$, with different displacement. Let $\mathcal{B}$ be a Banach space. The map $f:X\to X$ is called a Lipschitzian if ${\parallel f\left(x\right)-f\left(y\right)\parallel }_{\mathcal{B}}\le L{\parallel x-y\parallel }_{\mathcal{B}}$ for any $x,y\in X$ and constant $0<L<1$.

Its well known that the general solution of (2) contains two distinct parts named the transient state and steady state such as

$$x_G\left(t\right)=x\left(t\right)+Y\left(t\right),$$

where $x\left(t\right)$ is the transient solution of the autonomous case of (2) and $Y\left(t\right)$ is the steady solution. Due to $\phi \left(x\right)$, the solution $x\left(t\right)$ is perturbed such that, for any $\epsilon >0$, we have $-\epsilon <\phi \left(x\right)<\epsilon$ with period $T=[0,2\pi ]$.

If $\alpha <0$, the matrix A associated with the autonomous case of (2) is definite negatively. Using the logarithmic norm of the matrix $K={\displaystyle \frac{A+A^{*}}{2}}$ and the Gronwall–Bellman inequality (p. 293, [33]), we obtain the following:

$$\parallel x_G\left(t\right)\parallel \le \parallel x\left(0\right)\parallel \exp \left[{\int }_0^t{\lambda }_{Re}\left(\mathcal{L}\left(K(s,x\left(s\right))\right)\right)ds\right],$$

(5)

where ${\lambda }_{Re}\left(\mathcal{L}\left(K(s,x_G\left(s\right))\right)\right)=Re\lambda \left(\mathcal{L}\left(K(s,x_G\left(s\right))\right)\right)=-\phi \left(x\right)$.

From assumption ${\displaystyle \frac{x_{n+1}\left(0\right)}{x_n\left(0\right)}}=\left|L\right|$, we have the following:

$${\parallel x_{G_n}\left(t\right)-x_{G_{n+1}}\left(t\right)\parallel }_{\mathcal{B}}\le L{\parallel x_n\left(0\right)-x_{n+1}\left(0\right)\parallel }_{\mathcal{B}}$$

where $0\le L\le 1$. From this, it follows that the solutions belong to invariant subsets such that ${\mathcal{A}}_{x_{G_1}}\subset {\mathcal{A}}_{x_{G_2}}\cdots \subset {\mathcal{A}}_{x_{G_n}}$ and the global attractor is given by

$$\mathcal{A}={\cup }_{k=1}^n\left({\cap }_{s>0}{\mathcal{A}}_{{\lambda }_{max}\left(\mathcal{L}\left(K(s,x_k\left(s\right))\right)\right)}\right)$$

(6)

where ${\lambda }_{max}\left(K(s,x\left(s\right))\right)$ is the spectrum of $K(s,x(s\left)\right)$. Thus, the attractors of (2) are stable nested multi-scrolls with countable infinity. This way, the necessary condition is multiplying the $\dot{x}$ by a periodic function, and it is sufficient to have at least one of the initial conditions belong to $\Lambda$. □

Proof. 

2. From the computed ${\lambda }_{Re}\left(K(s,x_G\left(s\right))\right)=Re\left(K(s,x_G\left(s\right))\right)$, it is easy to verify the maximum Lyapunov exponent given by the following:

$${\lambda }_{\max }=\underset{t\to \infty }{\lim }{\displaystyle \frac{1}{t}}\log {\displaystyle \frac{\parallel x_G\left(t\right)\parallel }{\parallel x\left(0\right)\parallel }}=\underset{t\to \infty }{\lim }{\displaystyle \frac{1}{t}}{\displaystyle \frac{1}{\parallel x\left(0\right)\parallel }}{\int }_0^t\mathcal{L}\left(K(s,x_G\left(s\right))\right)ds>0$$

(7)

where $\mathcal{L}\left(K(t,x_G\left(t\right))\right)$ is the logarithmic norm of the Jacobian matrix $A$. From (7), it follows that the system (2) has a positive Lyapnouv exponent, which indicates that the attractor is chaotic. □

Corollary 1.

Let $\phi \left(x\right)$ in (2) be a trigonometric polynomial defined on Banach space $\mathcal{B}$. If at least one of the initial conditions is a divergent sequence and the other initial condition is zero, then, the system (2) has countable infinity separated attractors.

For example, Figure 1 illustrates the attractors and time series of the system (2) with two different sets of ICs as follows: (a) and (c) when the IC $x\left(0\right)\in \Lambda$ is such that $x\left(0\right),\dot{x}\left(0\right)=({\mathcal{F}}_n,0)$ for $n=3,4,5$ and $\phi \left(x\right)=2\cos \left(2x\right)+2\cos \left(2x\right)$; (b) and (d) when the IC is a convergent sequence $x\left(0\right),\dot{x}\left(0\right)=({\displaystyle \frac{1}{n}},0)$, where $n=2,3,4$ and $\phi \left(x\right)=2\cos \left(2x\right)+2\cos \left(2x\right)$.

3. Simple 4D Mega-Stable Oscillator

In 1970, Otto Rossler observed in [34] that there are some chaotic oscillators that exhibit increasingly complex behavior; two positive Lyapunov exponents are obtained, a phenomenon known as “hyperchaos”, usually involved in high-dimensional oscillators. Hyperchaotic oscillators are important in generating key encryption according to their high complexity; see [35,36].

Table 1. Comparison of related mega-stable chaotic systems with system (8).

Refs	Dim.	Term	Autonomous	Nonlin. Count Except Forcing Term
[29]	2D	4	No	1
[37]	3D	6	Yes	6
[28]	2D	4	Yes	3
[38]	3D	5	No	2
[27]	2D	4	Aut/No	3
[39]	4D	8	Yes	8
System (8)	4D	7	No	2

shows the comparison of the related mega-stable systems with (8).

With periodic function $F(t,\Omega )\in C[a,b]$, we propose a hyperchaotic oscillator with three nonlinear terms and the mega-stability property as follows:

$$\left\{\begin{array}{cc}\dot{x}=y;\hfill & \\ \dot{y}=z+y\cos \left(x\right);\hfill & \\ \dot{z}=\beta y+\alpha \cos \left(w\right)+AF(t,\Omega );\hfill & \\ \dot{w}=z.\hfill \end{array}\right.$$

(8)

where the forcing term is given by $F(t,\Omega )=\cos (\Omega t)$, and $\beta$ and $\alpha$ are control parameters.

Mega-Stability of Oscillator (8)

To recognize the mega-stability feature of the oscillator (8), we have to apply the definition in which varying one of the initial conditions will lead to infinitely countable nested attractors. Consequently, we vary $y\left(0\right)$ with fixing the others, such as $x\left(0\right)=0$, $y\left(0\right)=0.1n, z\left(0\right)=0.1, w\left(0\right)=0$. The phase portraits 2D, 3D and 4D projections of the system (8) with $A=1, \alpha =1, \beta =0.6, \Omega =0.7$ and the fixed ICs when $n=1,2,3,4.$ are shown in Figure 2 using the fourth-order Runge–Kutta method (ode45) on the MATLAB 2019 platform, while the error analysis was provided in Figure 3 present the error rate of the variables $y,z,w$ with the central variable x.

To demonstrate the attractor complexity, we have to find the Kaplan–Yorke dimension ${\mathcal{D}}_{KY}$ and the Lyapunov exponents. For any given Lyapunov exponents, we have the following:

$$\begin{array}{c}\hfill {\mathcal{D}}_{\mathrm{KY}}={\mathcal{D}}_L+{\displaystyle \frac{1}{|L_{\mathcal{D}+1}|}}\sum _{j=1}^{\mathcal{D}}{L}_j\end{array}$$

(9)

where ${\mathcal{D}}_L$ is the largest integer satisfying ${\sum }_{j=1}^{\mathcal{D}}{L}_j\ge 0$ and ${\sum }_{j=1}^{\mathcal{D}+1}{L}_j<0$.

By applying Wolfs method [40], the time length of $1500 s$, the four finite-time Lyapunov exponents, ${\mathcal{D}}_{KY}$, and the attractor classification are presented in

Table 2. Lyapunov exponents of system (8) when $\beta =0.6,\alpha =1$, and $ICs=(0,-0.1n,0,0)$.

n	${\mathit{L}}_1$	${\mathit{L}}_2$	${\mathit{L}}_3$	${\mathit{L}}_4$	${\mathcal{D}}_{\mathit{K}\mathit{Y}}$	Attractor Class.
1	0.101340	0.022871	0	−0.221613	3.554	Hyperchaotic
2	0.141520	0.029890	0	−0.184635	3.829	Hyperchaotic
3	0.089957	0.020787	0	−0.213151	3.213	Hyperchaotic
4	0.167079	0	−0.035889	−0.178175	3.737	Chaotic

. The maximum Lyapnouv spectrum for $\beta =\alpha =[0.1,3.5]$ is illustrated in Figure 4c,d, respectively. Note that the maximum Lyapnouv spectrum indicates the chaotic dynamics for a wide range of parameters, but it does not contain information on whether the system is hyperchaotic or not, since only one positive Lyapnouve exponent belongs to the maximum set. Table 2 shows more precise results.

Additionally, to evaluate the disspativity of system (8), we consider the Jacobian matrix’s trace $trac\left(\mathbf{J}\right)$, which is equivalent to the system’s divergence $▽V$. Particularly, the $\mathbf{J}$ of (8) is

$$\mathbf{J}=\left(\begin{array}{cccc}0& \alpha & 0& 0\\ -\beta y\sin x& \beta \cos x& 1& 0\\ 0& B& 0& -\sin \left(w\right)\\ 0& 0& 1& 0\end{array}\right)$$

(10)

where

$$▽V=\mathrm{trac}\left(\mathbf{J}\right)=\cos \left(x\right){\to }_{\mathrm{average}\mathrm{value}}0$$

(11)

The left side of Equation (11) implies that the energy dissipation is essentially dependent on the value of x. Thus, if we calculate the divergence for each attractor of the system separately, then no results will be obtained about the behavior of the overall system. The right side of Equation (11) present the average divergence value $▽V$, which is approaching zero as $t\to \infty$.

4. Complexity of $\mathit{\beta }$ and $\mathit{\alpha }$

To thoroughly investigate the mega-stable dynamics of the proposed system, we analyze its behavior over a broad range of the two key parameters, $\beta$ and $\alpha$. Since multi-stability and sensitivity to the initial conditions play a crucial role in the system’s dynamics, it is essential to consider multiple initial conditions to capture the full range of possible behaviors.

For consistency, we fix the initial conditions as follows:

$$x\left(0\right)=0,y\left(0\right)=0.1n,z\left(0\right)=0.1,w\left(0\right)=0.$$

(12)

To visualize the impact of parameter variations, we construct bifurcation diagrams for different parameter settings. Figure 4a presents the bifurcation diagram for $\beta \in [0.1,2.51]$ while keeping $\alpha =1$ fixed. This allows us to observe how the system transitions between different dynamical regimes as $\beta$ varies. Similarly, in Figure 4b, we plot the bifurcation diagram for $\alpha \in [0.1,1.51]$ while keeping $\beta =0.6$, highlighting the influence of $\alpha$ on the system’s behavior.

To ensure numerical precision and capture the long-term dynamics, we employ a discrete time step of $\Delta t=0.001$. The analysis focuses on sufficiently large time intervals, specifically considering only values for $t>1000$, to eliminate transient effects and emphasize the system’s steady-state behavior.

These bifurcation diagrams provide valuable insight into the rich dynamical structures present in the system, including potential transitions between periodic, chaotic, and coexisting attractors. Understanding these transitions is crucial for applications that rely on stable or tunable dynamical regimes.

Furthermore, attraction basins offer an extra instrument for analyzing the behavior of the associated dynamic systems with respect to changes in initial conditions (ICs). Figure 5 shows numerical computation of the basins of the system (8) with $\beta =0.6,\beta =0.9$. To obtain the 2D basins, two of the IC must be zero with varying remains. The attraction basins are plotted in Figure 5a illustrate the $x\left(0\right)-y\left(0\right)$ plane with $x\left(0\right)\in [0,3]$ and $y\left(0\right)\in [-1,5]$ with ${\Delta }_t=0.1$. Figure 5b illustrates the $z\left(0\right)-w\left(0\right)$ plane with $z\left(0\right)\in [-2,5]$ and $w\left(0\right)\in [-1,3]$ with $\Delta t=0.1$.

The intervals of the initial conditions were selected to stay away from attractors of the torus. Turquoise, dark turquoise, and indigo represent periodic, chaotic, and hyperchaotic solutions, respectively.

5. Fractional-Order Mega-Stable Systems

This section examines Lyapunov exponents and phase portraits of a fractional-order chaotic model using the Grunwald–Letnikov derivative, focusing on how the fractional-order impacts system dynamics. It also introduces key concepts and tools for analyzing fractional-order systems, including their behavior and numerical solutions. In fractional calculus, for a continuous function $f\left(t\right)$, several fractional operators of order q exist. In this study, we employ the following two types of fractional operators:

The Caputo fractional derivative operator is defined for any continuous function $f:[0,\infty ]\to \mathbb{R}$ of order q as (see [41])

$$\begin{array}{c}\hfill D^{q}f\left(t\right)={\displaystyle \frac{1}{\Gamma (1-q)}}{\int }_0^t{\displaystyle \frac{df}{ds}}{(t-s)}^{-q}ds,\end{array}$$

(13)

where $t>0,$ the function $\Gamma (1-q)$ is a known gamma Euler function and of order $q\in (0,1)$.

In addition, the Grunwald–Letnikov derivative operator (see [42,43]) is defined as

$$\begin{array}{c}\hfill D_t^{q}f\left(t\right)=\underset{h\to 0}{\lim }{h}^{-q}\sum _{k=0}^{{\displaystyle \frac{t-a}{h}}}{(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{q}{k}}\right)f(k-l),\end{array}$$

(14)

where ${\displaystyle \frac{t-a}{h}}$ and ${(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{q}{k}}\right)={\displaystyle \frac{q!}{k!(q-k)!}}$ are the binomial coefficients $C_k^q$. The relationship between (13) and (14) is employed to simplify the numerical investigations. The foundation of this method is the equivalent definitions of the Captuo derivative operator and the Grunwald–Letnikov derivative operator for a large class of functions. The connection to the qth derivative’s stated numerical approximation at points $kh,(k=1,2,\dots )$ is expressed in the following form [42,43]:

$$(k-L_m/h)D_k^{q}f\left(t\right)\approx h^{-q}\sum _{j=0}^k{(-1)}^j\left({\displaystyle \genfrac{}{}{0pt}{}{q}{j}}\right)f\left(t_{k-j}\right),$$

(15)

where $L_m$ is the memory length, $t_k=kh$; h is the calculation time step; and $(j=0,1,\dots )$.

In particular, we use the following expression:

$$c(q,0)=1,c(q,j)=\left(1-{\displaystyle \frac{1+j}{q}}\right)c(q,j-1).$$

(16)

Thus, the general form of the numerical solution is:

$$a{D}_t^{q}y\left(t\right)=f(y\left(t\right),t),$$

(17)

where

$$\begin{array}{c}\hfill y\left(t_k\right)=f\left(y\left(t_{k-1}\right)\right)h^q-\sum _{i=1}^k{C}_i^{q}y\left(t_{k-i}\right)\end{array}$$

(18)

and $f,y\in {\mathbb{R}}^n$ and $0<q<1$ with the “short memory” term represented by the sum in (18). By applying the Caputo derivative operator to the integer-order mega-stable oscillators (2) (when $\phi \left(x\right)=\cos \left(x\right)$) and (8), we get

$$\left\{\begin{array}{cc}{D}_t^{q}x=y;\hfill & \\ D_t^{q}y=-{\left(0.33\right)}^{2}x+y\cos \left(x\right)+\cos \left(0.73t\right).\hfill \end{array}\right.$$

(19)

and

$$\left\{\begin{array}{cc}{D}_t^{q}x=\alpha y;\hfill & \\ D_t^{q}y=z+\beta y\cos \left(x\right);\hfill & \\ D_t^{q}z=-0.6y+\cos \left(w\right)+\cos \left(0.7t\right);\hfill & \\ D_t^{q}w=z.\hfill \end{array}\right.$$

(20)

Applying the forward Grunwald–Letnikov derivative operator (14) to (2) (when $\phi \left(x\right)=\cos \left(x\right)$) and (8), the following fractional order nonautonomous system was obtained:

$$\begin{array}{cc}\hfill x_{\left(k\right)}& =\left(y_{(k-1)}\right)h^q-\sum _{k=1}^l{C}_k^q{x}_{\left(\right(k-l)},\hfill \\ \hfill y_{\left(k\right)}& =(-{\left(0.33\right)}^2{x}_{\left(k\right)}-y_{(k-1)}\cos \left(x_{(k-1)}\right)+\cos \left(0.73t_{\left(k\right)}\right))h^q-\sum _{k=1}^l{C}_k^q{y}_{\left(\right(k-l)}\hfill \end{array}$$

(21)

and

$$\begin{array}{cc}\hfill x_{\left(k\right)}& =\left(y_{(k-1)}\right)h^q-\sum _{k=1}^l{C}_k^q{x}_{\left(\right(k-l)},\hfill \\ \hfill y_{\left(k\right)}& =(z_{\left(k\right)}+y_{(k-1)}\cos \left(x_{k-1}\right))h^q-\sum _{k=1}^l{C}_k^q{y}_{\left(\right(k-l)},\hfill \\ \hfill z_{\left(k\right)}& =(-0.6y_{(k-1)}+\cos \left(w_{(k-1)}\right)+\cos \left(0.7t_{\left(k\right)}\right))h^q-\sum _{k=1}^l{C}_k^q{z}_{\left(\right(k-l)}.\hfill \\ \hfill w_{\left(k\right)}& =\left(z_{\left(k\right)}\right)h^q-\sum _{k=1}^l{C}_k^q{z}_{(k-l)}.\hfill \end{array}$$

(22)

Stability Analysis

To examine the stability of systems (19) and (20), we have to present the Matignon criterion and calculate the eigenvalues of the Jacobian matrices of unforced cases of oscillators (2) and (8).

Theorem 2

(Matignon, 1996 [44]). A commensurate fractional order is stable if and only if

$$|arg\left({\lambda }_i\right)|>q{\displaystyle \frac{\pi }{2}},foralli$$

with ${\lambda }_i$ representing the eigenvalues of the Jacobian matrix.

Theorem 3.

The fractional order of systems (2) and (8) are unstable.

Proof. 

Without loss of generality, we present the fractional order of system (2) using the Caputo derevitive operator and calculate the Jacobian matrix of the integer unforced case as follows:

$${\mathbf{J}}_{(\mathbf{0},\mathbf{0})}=\left(\begin{array}{cc}0& 1\\ \\ -{\left(0.33\right)}^2+y\sin x& \phi \left(x\right)\end{array}\right)$$

(23)

Thus, the eigenvalues of (23) at the only equilibrium, point $E^{*}=(0,0)$, are found to be ${\lambda }_{1,2}={\displaystyle \frac{-\phi \left(x\right)\pm \sqrt{{\phi }^2\left(x\right)+1}}{2}}$.

In addition, the oscillator (20) has infinitely many equilibrium points $E^{*}=(x,0,0,{\displaystyle \frac{\pi }{2}})$, where $\mu \in \mathbb{R}$ and

$${\mathbf{J}}_{{\mathbf{E}}^{\mathbf{*}}}=\left(\begin{array}{cccc}0& 1& 0& 0\\ \\ -y\sin x& \cos x& 1& 0\\ \\ 0& -\beta & 0& -\alpha \sin \left(w\right)\\ \\ 0& 0& 1& 0\end{array}\right)$$

(24)

Furthermore, the eigenvalues of (24) are ${\lambda }_{1,2}=0,{\lambda }_{3,4}=\pm 1.414i$.

In particular, the Matignon criterion says that the equilibrium points $E^{*}$ of $f\left(y\right(t\left)\right)$ are locally asymptotically stable if every eigenvalue ${\lambda }_j$ of the Jacobian matrix $J={\displaystyle \frac{\partial f\left(y\right(t\left)\right)}{\partial y\left(t\right)}}$ is determined at the equilibrium points $E^{*}$ and satisfies $|arg\left({\lambda }_j\right)|>q{\displaystyle \frac{\pi }{2}}$. Applying the Matignon criterion to the Jacobian matrix (23) (when $\phi \left(x\right)=\cos \left(x\right)$) and (24) at the corresponding equilibrium points, we obtained the following cases for any $0<q<1$:

$$\left\{\begin{array}{cc}\left|arg\left(0.1244\right)\right|=0<{\displaystyle \frac{q\pi }{2}},\hfill & \\ \left|arg\left(0.8756\right)\right|=0<{\displaystyle \frac{q\pi }{2}}.\hfill \end{array}\right.$$

(25)

and

$$\left\{\begin{array}{cc}\left|arg\left({\lambda }_{1,2}\right)\right|=\left|arg\left(0\right)\right|=0<{\displaystyle \frac{q\pi }{2}},\hfill & \\ \left|arg\left({\lambda }_{3,4}\right)\right|=\left|arg(\pm 1.414i)\right|={\displaystyle \frac{\pi }{2}}<{\displaystyle \frac{q\pi }{2}}.\hfill \end{array}\right.$$

(26)

From (25) and (26), it follows that the fractional-order systems (19) and (20) are unstable.

In the end, mega-instability is infinitely countable for nested attractors of a fractional order for some orders. □

6. Order Effect

The Grunwald–Letnikov operator is used to evaluate the effect of the fractional order on the mega-stability of systems (2) and (8) by computing the Lyapnouv exponents and presenting the phase space of the converted fractional-order systems (21) and (21). We focus on the following setting:

• For system (21), all initial conditions are zero except $y\left(0\right)$, which is equal to ${\mathcal{F}}_5=3$, ${\mathcal{F}}_6=5$, and ${\mathcal{F}}_7=8$ of the Fibonacci sequence.
• For system (21), the initial conditions are $x\left(0\right)=0,y\left(0\right)=0.1n,z\left(0\right)=0.1$, and $w\left(0\right)=0$, such that the initial conditions are the sequence of $y\left(0\right)\in \Lambda$.

Basically, the Matlab platform coded based on the Grunwald–Letnikov operator obtained in [45] was used for numerical simulations to compute the order effect (integer and fractional).

The systems (21) and (22) display several attractors and various Lyapunov exponents when the order q is changed. It appears that the values of the Lyapunov exponents remain mostly the same or exhibit a small difference between different orders. However, despite this stability, the qualitative behavior of each system’s attractors undergoes significant changes. In particular, as the fractional order decreases from 1 to $0.91$, the dynamics of system (21) change from chaotic to regular. This is shown in the phase portraits in Figure 6a–l, with $q=0.80$ and $q=0.60$, where all cases showed nested three attractors ($y\left(0\right)=3$ (blue), $y\left(0\right)=5$ (red), $y\left(0\right)=8$ (green)). To better visualize the system’s attactor, we present the case when y(0) = 3 (blue) as a sub-figure. Similarly, Figure 7a–c illustrates the phase portraits of the fractional-order system (22) when $q=1.0,q=0.99$, and $q=0.97$, while (d) presents the case when $q=0.08 \left(y\right(0)=0.1(\mathrm{blue}),y(0)=0.2(\mathrm{red})$, and $y\left(0\right)=0.3\left(\mathrm{green}\right))$. All numerical simulations were obtained via $T_{simulation}$ with $k=1,2,3,\dots ,10$, $N=\left[{\displaystyle \frac{T_{simulation}}{h}}\right]$, and $h=0.002$.

These results underscore the sensitivity of the attractors in systems (21) and (22) to changes in fractional order. Even in cases where the Lyapunov exponents show minimal variation, as presented in

Table 3. Lyapunov exponent and $D_{KY}$ of system (21) with ICs $(0,{\mathcal{F}}_n,0)$ where $n=3,4,5$.

Fractional Order $\left(\mathit{q}\right)$	n	LEs	${\mathcal{D}}_{\mathit{K}\mathit{Y}}$	Attractor Class.
1	3	(0.1066, 0, −0.6683)	2.2605	Strange attractors
	5	(0.0169, 0, −0.7997)	2.2100	Strange attractors
	8	(0.1095, 0, −0.9054)	2.2302	Strange attractors
0.99	3	(0.0236, −0.4387, −0.0014)	2.1412	Strange attractors
	5	(0.0256, −0.7081, −0.0014)	2.1205	Strange attractors
	8	(0.0424, −0.9036, −0.0014)	2.1307	Strange attractors
0.98	3	(0.0601, −0.7521, −0.0029)	2.0928	Strange attractors
	5	(0.0591, −0.8213, −0.0029)	2.0805	Strange attractors
	8	(0.0691, −0.9932, −0.0029)	2.0857	Strange attractors
0.97	3	(0.0323, −0.6090, −0.0047)	2.1444	Strange attractors
	5	(0.0260, −0.6205, −0.0047)	2.1302	Strange attractors
	8	(0.0454, −0.9128, −0.0047)	2.1384	Strange attractors
0.96	3	(0.0174, −0.5381, −0.0067)	2.0834	Strange attractors
	5	(0.0247, −0.5633, −0.0067)	2.0725	Strange attractors
	8	(0.0376, −0.6902, −0.0067)	2.0775	Strange attractors
0.95	3	(0, −0.8030, −0.0090)	${\mathbb{R}}^2$	Limit cycle
	5	(0, −0.7133, −0.0090)	${\mathbb{R}}^2$	Limit cycle
	8	(0, −0.8186, −0.0090)	${\mathbb{R}}^2$	Limit cycle
0.94	3	(0, −0.5565, −0.0117)	${\mathbb{R}}^2$	Limit cycle
	5	(0, −0.9130, −0.0117)	${\mathbb{R}}^2$	Limit cycle
	8	(0, −0.6169, −0.0117)	${\mathbb{R}}^2$	Limit cycle
0.93	3	(0, −0.5473, −0.0146)	${\mathbb{R}}^2$	Limit cycle
	5	(0, −0.8067, −0.0146)	${\mathbb{R}}^2$	Limit cycle
	8	(0, −0.8307, −0.0146)	${\mathbb{R}}^2$	Limit cycle
0.92	3	(0, −0.6295, −0.0180)	${\mathbb{R}}^2$	Limit cycle
	5	(0, −0.7152, −0.0180)	${\mathbb{R}}^2$	Limit cycle
	8	(0, −0.7772, −0.0180)	${\mathbb{R}}^2$	Limit cycle
0.91	3	(0, −0.7660, −0.0218)	${\mathbb{R}}^2$	Limit cycle
	5	(0, −0.9177, −0.0218)	${\mathbb{R}}^2$	Limit cycle
	8	(0, −0.8605, −0.0218)	${\mathbb{R}}^2$	Limit cycle
0.90	3	(0, −0.9419, −0.0262)	${\mathbb{R}}^2$	Limit cycle
	5	(0, −1.0126, −0.0262)	${\mathbb{R}}^2$	Limit cycle
	8	(0 −1.0294, −1.0294, −0.0262)	${\mathbb{R}}^2$	Limit cycle

and

Table 4. Lyapunov exponent and ${\mathcal{D}}_{KY}$ of system (22) with ICs $(0,0.1n,0.1,0)$, where $n=1,2,3$.

Fractional Order $\left(\mathit{q}\right)$	IC	LEs	${\mathit{D}}_{\mathit{K}\mathit{Y}}$	Attractor Class.
1	1	(0.1119, 0.0575, 0, −0.1140)	2.2605	Hyperchaotic
	2	(0.1320, 0.0439, 0, −0.1191)	2.2100	Hyperchaotic
	3	(0.1281, 0.0816, 0, −0.1211)	2.2302	Hyperchaotic
0.99	1	(0.0239, 0, −0.0304, −0.0456)	2.1412	Chaotic
	2	(0.0350, 0, −0.0235, −0.0611)	2.1205	Chaotic
	3	(0.0297,0, −0.0288, −0.0550)	2.1307	Chaotic
0.97	1	(0.0392, 0 −0.0221, −0.2845)	2.0928	Chaotic
	2	(0.0387, 0, −0.0246, −0.2579)	2.0805	Chaotic
	3	(0.0391, 0, −0.0217, −0.2699)	2.0857	Chaotic
0.80	1	(0, −0.3733, −0.9666, −0.9752)	${\mathbb{R}}^2$	Limit cycle
	2	(0, −0.3718, −0.9651, −0.9742)	${\mathbb{R}}^2$	Limit cycle
	3	(0, −0.3611, −0.9585, −0.9675)	${\mathbb{R}}^2$	Limit cycle

, the systems’ attractors exhibit significant qualitative transformations. The method used to calculate Lyapunov exponents of fractional-order systems was coded in [45].

7. Microcontroller Implementation

Following a numerical analysis of chaotic, hyperchaotic systems of fractional order, an experimental investigation is necessary to confirm the findings. One of two methods is typically used to conduct this experimental study. The first involves building integrator circuits with electronic components to integrate the different equations that make up the chaotic system in realization. Combining usual programming and electronic performance is made possible by a digital processing board. It enables the possibility to program electronic systems to be more precise. The primary benefit of programmed electronics is that they significantly simplify electronic circuits, which reduces production costs and the amount of work required to build an electronic board. It should be mentioned that every board has a microcontroller with unique features. The intricacy of the chaotic system that is being implemented greatly influences the board selection. Microcontroller implementation has many advantages over electrical circuit implementation, such as fast computation time, great flexibility, and outstanding stability and precision. It is also easy to change the initial circumstances and settings of the system. Because of these characteristics, we used a microcontroller named “Arduino Due” to create the four-dimensional hyperchaotic oscillator. One benefit of this card is its built-in digital-to-analog converter, which makes installation easier.

The application of the experimental results with a Voltcraft D50-1062D by programming the software with the 4th order Runge–Kutta method for $\Delta t=0.001$ is shown in Figure 8a, implementing system (8) for $\alpha =1$, $\beta =0.6$, and $x\left(0\right)=0,y\left(0\right)=0.1,z\left(0\right)=0.1$, $w\left(0\right)=0.1$. The attractors of the fractional order chaotic system (21) are shown in Figure 8b,c when the order $q=0.99,q=0.93$, and $q=0.8$. It is clear that the numerical and experimental results match.

8. Conclusions

Recently, many scholars attempted to introduce a simple chaotic system with the mega-stability property. This paper investigated mega-stable phenomena and introduced a simple general case of a two-dimensional mega-stable system when the velocity is multiplied by a trigonometric polynomial. The natural reason behind the overleaping of the attractors was obtained. Moreover, a 4D simple example of a mega-stable system was introduced, examining its dynamics through the most typically used tools like Lyapnouv exponents and bifurcation diagram. In addition, the fractional order of mega-stable systems was studied analytically by the Caputo derivative operator and numerically by the Grunwald–Letnikov derivative operator. The stability analysis proved that in the case of a fractional order, the systems are infinity countable unstable nested attractors for some orders.