The dynamic analysis of a chaotic system

Abstract

A three-dimensional chaotic system which contains different parameters is proposed in this article. By applying numerical and analytical methods, the basic properties of the system, such as dynamical behaviors, dissipation, sensitivity, power spectrum, Poincaré map, and Lyapunov exponents, are investigated. In addition, the existence of both Hopf bifurcation and topological horseshoe is presented. The obtained results clearly show that this is a new chaotic system.

Keywords

Chaotic system Poincaré mapping Hopf bifurcation Lyapunov exponent horseshoe

Introduction

In recent years, chaos has been intensively studied in many fields such as mathematics, science, neural network, and engineering communities. In a few literature,^1,2–11 the novel finance chaotic systems and the novel three-dimensional (3D) chaotic systems are studied. A novel four-dimensional hyperchaotic system with one equilibrium is proposed in Wan et al.¹² TJ Li et al.¹³ studied the dynamic load-sharing behavior of two-stage planetary gear train based on a nonlinear vibration model. XS Yang and Q Yuan¹⁴ studied the connection topology prohibit chaos in continuous time networks. A novel memristive time-delay chaotic system without equilibrium points is given in Pham et al.,¹⁵ and the topological horseshoe analysis is given in a few literature.^16–20 FH Min et al.²¹ proposed a new mixed-order chaotic system, and MY Qiao et al.²² studied the chaotic characteristics of workface gas emission time-series data.

In this article, based on nonlinear finance system,^1,2 the model for a chaotic system is proposed, where the autonomous system contains different parameters and one of the equation has nonlinear terms. The basic properties of the system are investigated. In addition, the existence of both Hopf bifurcation and topological horseshoe is presented.

The structure of this article is organized as follows. In section “The model of chaotic system,” the model of a novel chaotic system is presented. In section “Dynamical behavior of chaotic system,” the basic dynamical properties of this novel system, such as dynamical behaviors, are analyzed. The existence of both Hopf bifurcation and topological horseshoe is presented in section “Topological horseshoe in chaotic system,” and finally, some concluding remarks are given in section “Conclusion.”

The model of chaotic system

The model of chaotic system is shown as follows

{\begin{cases} \dot{x} = (- α + 1 / β) x + x y + z \\ \dot{y} = - x^{2} - β y \\ \dot{z} = - x - γ z \end{cases}

(1)

where ${[x (t), y (t), z (t)]}^{T} \in R^{3}$ are the state vectors, and the parameters a, b, and c are positive real constants.

The new chaotic attractor for the parameters $α = 2$ , $β = 0.1$ , and $γ = 1$ is shown in Figure 1.

Figure 1.

Phase diagram of system (1): (a) (x, y), (b) (x, z), (c) (y, z), and (d) (z, x, y).

Dynamical behavior of chaotic system

Dissipation and existence of attractors

As the dissipation is

\nabla V = \frac{\partial \dot{x}}{\partial x} + \frac{\partial \dot{y}}{\partial y} + \frac{\partial \dot{z}}{\partial z} = - α + 1 / β + y - β - γ

(2)

When $y < α + ((β^{2} - 1) / β) + γ$ , system (1) is dissipative.

Symmetry, equilibrium, Hopf bifurcation, and stability

The system has the symmetry, and when the relation of $(x, y, z) \to (- x, y, - z)$ is transformed, the system remains unchanged. The system trajectory in the x-z plane is symmetry of y-axis.

Set the right side of equation (1) equal to zero, and we get

{\begin{matrix} (- α + 1 / β) x + x y + z = 0 \\ - x^{2} - β y = 0 \\ - x - γ z = 0 \end{matrix}

(3)

Obviously, the system has one equilibrium point, $P_{0} = (0, 0, 0)$ , and the two saddle points

\begin{array}{l} P_{1} = (\sqrt{1 - α β - β / γ}, \sqrt{α - 1 / β - 1 / γ}, - \sqrt{1 - α β - β / γ}), \\ P_{2} = (- \sqrt{1 - α β - β / γ}, \sqrt{α - 1 / β - 1 / γ}, \sqrt{1 - α β - β / γ}) \end{array}

When $1 / 3 < β < 1 / 2$ , the Hopf bifurcation at $P_{0}$ is not degenerated; when the parameters $α = γ = 2$ , and if $0.162 < β < 0.4$ , $P_{1}$ is asymptotically stable point; when $β = 0.162$ , $P_{1}$ is a Hopf bifurcation point; and the Hopf bifurcation of point $P_{2}$ is similar to $P_{1}$ .

Initial sensitivity, power spectrum, and Poincaré map

The chaotic attractors are shown in Figure 1(a)–(d). Its series are non-periodic, and it is sensitive to initial values. The time domain response is shown in Figure 2(a), and the initial value of the sensitivity characteristic is shown in Figure 2(b); even if the initial value is only a difference of 0.000001, the remaining initial values are unchanged after the wave completes about the same. System power spectrum is a continuous spectrum which is shown in Figure 2(c).

Figure 2.

Properties of system (1): (a) the time domain response of x, (b) the initial sensitivity of system x, and (c) the power spectrum.

Several Poincaré sections are shown in Figure 3(a)–(c). It can be seen that there are some stretches of a fractal structure of the dense point on the Poincaré section. The attractor blade is clearly visible. All these features express that system (1) is a chaotic system.

Figure 3.

Poincaré map of system (1): (a) x₀ = 0, (b) y₀= −6, and (c) z₀ = 0.

Lyapunov exponents and its dimension

Chaotic attractor between the adjacent orbits is mutually exclusive and separated by exponential rate. Today, there are many methods for calculating the maximum Lyapunov exponent (LE).²³ Using the singular value decomposition method,²³ the three LE of the system can be obtained as follows: $λ_{1} = 0.13,$ $λ_{2} = 0$ , and $λ_{3} = - 0.52$ . The Lyapunov dimension of the new chaotic system can be calculated as follows

\begin{array}{l} D_{L} = j + \frac{1}{| λ_{j + 1} |} \sum_{i = 1}^{j} λ_{i} = 2 + \frac{λ_{1} + λ_{2}}{| λ_{3} |} \\ = 2 + \frac{0.13 + 0}{| - 0.52 |} = 2.254 \end{array}

(4)

Thus, the Lyapunov dimension is the fractal dimension, and it shows that the system is a chaotic system.

System parameters

With the change of system parameters, the stability of equilibrium point will change, so the system will be in a different state. The various system parameters of the system state are analyzed by the LE spectrum and bifurcation diagrams.

1. The parameters $β = 0.1$ and $γ = 1$ are fixed, $a \in [0, 9]$ .

LE spectrum and Xmax bifurcation diagram are shown in Figure 4(a) and (b), respectively. When LE is greater than zero, the system is chaotic.²³

2. The parameters $α = 2$ and $γ = 1$ are fixed, $β \in [0, 0.25]$ (see Figure 5).

3. The parameters $α = 2$ and $β = 0.1$ are fixed, $γ \in [0, 5]$ (see Figure 6).

Figure 4.

When $α \in [0, 9]$ : (a) LE spectrum and (b) Xmax bifurcation diagram.

Figure 5.

When $β \in [0, 0.25]$ : (a) LE spectrum and (b) Xmax bifurcation diagram.

Figure 6.

When $γ \in [0, 5]$ : (a) LE spectrum and (b) Xmax bifurcation diagram.

Topological horseshoe in chaotic system

Choosing the parameter $α = 2$ and the initial value as $(1, - 6, 0.1)$ , the attractors of system (1) are shown in Figure 1. The cross-sectional M is taken as follows: M = { $(x, y, z) : - 3 \leq x \leq - 0.75$ , $- 8 \leq y \leq - 5, z = 0.7$ }, and its mapping under Poincaré map is illustrated in Figure 7.

Figure 7.

Topological horseshoe map.²

The above analysis and simulation results confirm that the proposed system is a nonlinear chaotic system; it has all the common features of chaotic systems such as uncertainty, bounded on the extreme sensitivity of initial and long-term unpredictability, the largest LE, a certain frequency range, and ergodicity continuous spectrum.

Conclusion

In this article, the detailed analysis such as the symmetry of this system, the basic dynamic characteristics of equilibrium points, the LE spectrum, phase diagrams, and Poincaré sections are given. As the parameters of this chaotic system change, the bifurcation diagram and the morphology change within the window period, and the calculated dual-parameter space distribution of the maximum Lyapunov exponent is discussed in the 3D space. Periodic bifurcation sequence and the characteristics of chaotic attractors enable the system to achieve robust control of chaos, and the existence of both Hopf bifurcation and topological horseshoe is presented. Thus, the system has a broad characteristic and deserves deep investigation.

Footnotes

Academic Editor: Crinela Pislaru

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China under Grant numbers 61471192 and 6137119 and also funded by the priority Academic Program Development of Jiangsu Higher Education Institutions.

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