Coexisting infinitely many attractors in a new chaotic system with a curve of equilibria: Its extreme multi-stability and Kolmogorov–Sinai entropy computation

Abstract

A new five-dimensional chaotic system with extreme multi-stability is introduced in this article. The mathematical model is established, and numerical simulations are done. This dynamical system complicates incident of extreme multi-stability. Most significantly, relied on the mathematical model, the recently proposed system has a curve of equilibria that ends in the occurrence of hidden attractors. We examine the initial-condition-dependent dynamics of this system. We inspect that there is an unrestricted number of coexistent attractors, which signifies the occurrence of extreme multi-stability strictly. In addition, the extreme multi-stability according to initial condition is investigated consuming the Lyapunov exponent spectra and bifurcation diagrams. The existence of coexisting infinitely many attractors is displayed with phase portraits. In the end, we calculate and debate Kolmogorov–Sinai entropy in the chaotic system. We direct trying the Kolmogorov–Sinai technique of entropic inspection for the dynamics of the system.

Introduction

Recently, investigation of chaotic systems has been a hot topic. ¹ Many studies have been conducted on designing new chaotic attractors. ² The process of creating chaotic attractors is not clear. For many years, researchers think that the creation of chaotic attractors is related to saddle equilibria.^3–5 In the last decade, there are some examples which do not follow this rule. ⁶ Recently, many chaotic flows without saddle point equilibria have been proposed. ⁷ Studying chaotic systems with special properties helps us to find out the reason of creating chaotic attractors.

Chaotic systems with extreme multi-stability are a particular sort of nonlinear dynamical systems. ⁸ The dynamical stability of these chaotic systems is described related to their initial conditions, which signifies that there are numerous attractors in the system.^9,10 Coupled systems can display a strange form of multi-stability, specifically, the coexistence of infinitely many attractors for a specified group of parameters. This extreme multi-stability is revealed to be found in coupled systems with diverse kinds of coupling. In the coupled Rössler oscillator with partial synchronization, this case has been described as extreme multi-stability. ¹¹ The coexistence of a limitless quantity of attractors in dynamical systems has been recognized as extreme multi-stability in Bao et al. ¹² In other words, extreme multi-stability happens when there are distinct attractors for the identical category of system parameters. ¹³ If an inexhaustible number of attractors and points of equilibrium prevail concurrently and conditional on the initial conditions of the system, it means that the extreme multi-stability has been established.^14,15

An oscillation in a dynamical system can undoubtedly be specified mathematically if the initial conditions from its adjacent range end in a long-range functioning that moves toward the oscillation.^16,17 Such an oscillation (or collection of oscillations) is addressed an attractor, and its attracting set is described as the basin of attraction.^18,19 Consequently, from a computational viewpoint, the following compartmentalization of attractors is suggested based on the straightforwardness of basin of attraction determination in the phase space: an attractor is named a hidden attractor if its basin of attraction does not cross teeny vicinities of equilibria, in other respects, it is a self-excited attractor.^20–22 Over the last few years, hidden attractors have obtained considerable attentiveness in different dynamical fields.^23–27 Hidden attractors are particular attractors concealed in the arrangement of the system, and if disregarded (not discovered), they can stimulate important harms, as found out in reality. ²⁸

A predominant notion in the association between information theory and physics is entropy, which constitutes the aggregate of information extricated from the system by the spectator conducting assessments in a study.^29,30 Indeed, Jaynes’ principle of maximum entropy concedes to construct the relation between information entropy and entropy in statistical mechanics.^31–34 This procedure grants us to come up with worthwhile clarifications of information concepts in chaotic systems.

The remainder of this article is as follows. In section “Mathematical model, dynamical characteristic analysis, and numerical simulations,” a new chaotic system is designated, based on which the mathematical model is formulated, the curve equilibria are introduced, and chaotic attractors are showed. In section “Initial-condition-related extreme multi-stability,” coexisting event of infinitely many attractors with various initial conditions $v(0)$ is reported with disparate portraits of the phase and extreme multi-stability based on the initial condition is looked into taking advantage of the bifurcation diagrams and Lyapunov exponent spectra. In section “KSE computation,” we begin by describing the considered system, accompanied an explanation of an informational-theoretic strategy for dynamical systems, in which the concept Kolmogorov–Sinai entropy (KSE) of the dynamics is established. Finally, the endings of this article are offered in the final part.

Mathematical model, dynamical characteristic analysis, and numerical simulations

The proposed chaotic system dynamics can be reported by five coupled first-order autonomous nonlinear differential equations in terms of five states of x, y, z, w, and v. The state equations can be composed as

$$\begin{array}{c}\stackrel{·}{x}=y\hfill \\ \stackrel{·}{y}=z\hfill \\ \stackrel{·}{z}=w\hfill \\ \stackrel{·}{w}=1.64x^2-1.02w+v+0.28\mathrm{xy}\hfill \\ \stackrel{·}{v}=2.42\mathrm{xw}-0.3\mathrm{yz}\hfill \end{array}$$

(1)

The chaotic system modeled by equation (1) is studied in our following work. The system has been investigated with some change of variables, which is not like the previous chaotic systems. Assigning the left-hand side of equation (1) to zero, system (1) has a curve equilibrium, which can be stated by

$$P=\{(x,y,z,w,v)|y=z=w=0,v=-1.64x^2\}$$

(2)

To investigate the stability of curve of equilibria, characteristic equation of the system in equation (2) is calculated as ${\lambda }^5+1.02{\lambda }^4-2.42x{\lambda }^3-0.28x{\lambda }^2-3.28x\lambda =0$ . So, the system has a zero eigenvalue in all values of $x$ in the curve of equilibria. Figure 1 shows the real part (a) and imaginary part (b) of eigenvalues of system (1) at the curve of equilibria in the interval $x\in [-10,10]$ . It can be seen that equilibrium points in the interval $x\in [-10,-5.57]$ have four negative and one zero eigenvalues. So, the stability of curve of equilibria cannot be determined using the eigenvalues in this interval. Then, in the interval $x\in [-5.57,-0.1]$ , two equal positives, two equal negative, and one zero eigenvalues exist. So, the curve of equilibria is unstable. In the interval $x\in [-0.1,0]$ , the equilibria have two equal positives, two negatives, and one zero eigenvalues. So, the equilibria are unstable in that interval. After that in $x>0$ , there is one positive eigenvalue and so the equilibria are unstable.

Figure 1.

Real part (a) and imaginary part (b) of the eigenvalues of the curve of equilibria in the interval $x\in [-10,10]$ .

The phase portraits of the typical chaotic attractors obtained by simulating equation (1) in two different planes of $x-y$ and $z-w$ are portrayed in Figure 2(a) and (b), respectively, for the initial conditions $(-2.77,-0.53,2.7,-0.34,-15.36)$ . The time-domain waveforms of system (1) with initial conditions $(-2.77,-0.53,2.7,-0.34,-15.36)$ are reported in Figure 3. Evidently, time-domain waveforms are not periodic, which suggest that the attractor is chaotic.

Figure 2.

Numerically simulated phase portraits of typical chaotic attractors obtained from equation (1) in two different planes: (a) x–y plane and (b) z–w plane.

Figure 3.

Time-domain waveforms.

Initial-condition-related extreme multi-stability

In this part, we inspect that the dynamical behaviors of the system rely not only on its parameters but also a lot on initial conditions, and it causes an exceptional characteristic, named extreme multi-stability. In this phenomenon, many coexisting attractors arrive with different initial conditions, such as the coexistence of chaotic attractor and limit cycle.

To depict the extreme multi-stability phenomenon, the system parameters remain unchanged, and taking four initial conditions prepared as the form x(0)=−2.77, y(0)=−0.53, z(0)=2.7, w(0)=−0.34, the initial condition v(0) is changeable and adaptable in the district [–19, –15.36]. It can be seen in Figure 4 that the system’s dynamic shows a variety of behaviors just by changing the initial condition of one of its variables. The system shows a period 2 limit cycle in the interval $v_0\in [-19,-18.56]$ . In higher initial value $v_0$ , in the interval $[-18.56,-17.85]$ , the system has a jump from a period 2 limit cycle to a period 1 limit cycle. After that, the system has a period 4 limit cycle and continue with period doubling route to chaos $(v_0\in [-17.85,-17.46])$ . Then the system’s dynamic changes to a period 1 limit cycle by increasing initial value $v_0$ in the interval $[-17.46,-17.29]$ . The system continued by tolerating between chaotic behavior, periodic windows, and a period 1 limit cycle. If we have a glance with a general point of view to the bifurcation diagram of Figure 4(a), we can see that the system has two main routes by increasing initial value $v_0$ . One of them is the period doubling route to chaos, and another one is a period 1 limit cycle. So, we can conclude that the system is highly dependent on initial value $v_0$ , since it has a period doubling route to chaos and also jump to a period 1 limit cycle just by changing initial value $v_0$ . When $v(0)$ gently grows, the five Lyapunov exponents of the state variable $v$ are indicated in Figure 4(b), from which the dynamical behaviors of periodicity, and chaos is noticed. Lyapunov exponents of the system are calculated using Wolf’s method ³⁵ and run time of 20,000s.

Figure 4.

Dynamical behaviors when $v(0)$ increases for $x(0)=-2.77,y(0)=-0.53,z(0)=2.7,w(0)=-0.34$ : (a) bifurcation diagram and (b) Lyapunov exponent spectrum.

When different initial conditions are utilized in this system, some common phase portraits of coexisting infinitely many attractors on the z–w plane are visible in Figure 5. Generally, Figure 5 displays the coexistence of four attractors with different dynamical behaviors, which entails the appearance of extreme multi-stability in system (1). Other simulations demonstrate that the system can have infinitely many attractors. It shows the multi-stability of the system.

Figure 5.

Numerically simulated coexisting attractors’ projections with different $v(0)$ on z–w plane: (a) Limit cycle with period 1 at $v(0)=-18$ , (b) limit cycle with period 2 at $v(0)=-19$ , (c) limit cycle with period 4 at $v(0)=-17.75$ , and (d) chaotic attractor at $v(0)=-16.5$ .

The results prove that the long-term dynamical behaviors related to different initial conditions result in the existence of extreme multi-stability of the system (1).

KSE computation

In this part, we discuss the widely known KSE. We ought to start with the description of the KSE. The impression of the KSE was deliberated by Kolmogorov in 1958 for the first time on the troubles emerging from the dimension of functional spaces and information theory, that evaluates the hesitancy of the dynamical systems.^29,30 Employing the mathematical expression prepared by Shannon, Kolmogorov established a speculative instrument, presently recognized as KSE, conceding to examine the accidental actions of dynamical systems.^36–38 Seeing that chaotic attractors have innumerous amount of states, an additional kind of entropy is required to compute their arbitrariness.^32,39 KSE is a guideline of the dynamical system which plans a standard to outline chaos, for the reason that positive KSE is a characteristic of chaotic manners.^32,40 It is proved that the indication of the KSE is as shown in equation (3)

$$H_{\mathrm{ks}}\left(\beta \left[\epsilon \right]\right)=\frac{1}{{\tau }_{\min }\left(\beta \left[\epsilon \right]\right)}\sum _{\tau }\rho (\tau ,\beta \left[\epsilon \right])\log \left(\frac{1}{\rho (\tau ,\beta \left[\epsilon \right])}\right)$$

(3)

It is determined to bring the first Poincaré recurrence times (FPRs) designated by ${\tau }_i$ into play. A D-dimensional carton in the state space with edge ${\epsilon }_1$ where the FPRs are watched over is $\beta$ . The probability distribution of ${\tau }_i$ is $\rho (\tau ,\beta )$ . $H_{\mathrm{ks}}$ for a smooth chaotic system is equivalent to the entirety of all positive Lyapunov exponents.^32,41,42 In other words, KSE is a measure of chaotic dynamics. The positive KSE shows the existence of chaotic manners.^32,40 The KSE of system (1) looks up to the floating parameter $v(0)$ and is shown in Figure 6. As Figure 6 demonstrates, in large values of parameter $v(0)$ , the system volatility is soaring, and it has a chaotic attractor. This entropy indicates a more actual perspective on the complexity of the system. When the entropy is zero, there are not any positive Lyapunov exponents. In the bifurcation points, the system’s transient time is added, and its state turns into a slower one. That is the sanity for the augmenting entropy in the bifurcation points.

Figure 6.

Kolmogorov–Sinai entropy as a function of $v(0)$ for system (1).

Well-nigh bifurcation points, the system goes back more sluggish to its attractor under a minor fluster. ⁴³ So, within reach of the bifurcation points, the system has a greater extent of transient time, and it takes a while as the system extends to its attractor. As an instance, Figure 7 is taken into consideration. In this figure, the eventual state by detaching 0.9 of time series as transient time appears in orange and the entire changing of the system’s state is displayed in green. The frame expresses that when the system is far from its bifurcation point, the transient time is slight (Figure 7(a)). When the system is at hand of a bifurcation point (Figure 7(b)), the system is long-drawn-out, and the transient time grows. This is critical slowing down incident. ⁴⁴ In brief, by approaching to the bifurcation points, the dynamics of the system became slower. So, its transient time to reach to the final state increases. It can be seen in the green part of Figure 7.

Figure 7.

Numerically simulated attractor projections with different $v(0)$ on z–w plane: (a) at $v(0)=-18$ (distant from tipping point) and (b) at $v(0)=-17.85$ (close to the tipping point). The uncut journey of system’s state from initial conditions to the attractor (green) and ultimate state by dismissing transient time (orange).

Conclusion

In this article, a chaotic system has been designed that demonstrates the phenomenon of extreme multi-stability, which implies that there are infinitely many coexistent attractors. The dynamical qualities of this system were considered with consideration for the varieties of the initial conditions. Then, the compound dynamic features were proved. Equilibrium points of the system were analyzed, and their stabilities were investigated. Coexisting attractors and extreme multi-stability of this system were studied in this article. The most important characteristics of this system were extreme multi-stability. Eventually, the KSE of the system was stated. Critical slowing down of the system was investigated to study bifurcation points of the system.