Advances in Mathematical Physics

Volume 2016, Article ID 3142068, 6 pages

http://dx.doi.org/10.1155/2016/3142068

## A New No-Equilibrium Chaotic System and Its Topological Horseshoe Chaos

^{1}Department of Information Engineering, Binzhou University, Binzhou 256600, China^{2}Department of Electrical Engineering, Binzhou University, Binzhou 256600, China^{3}College of Aeronautical Engineering, Binzhou University, Binzhou 256600, China^{4}Department of Product Design, Tianjin University of Science and Technology, Tianjin 300457, China^{5}School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, China

Received 13 September 2016; Revised 11 November 2016; Accepted 6 December 2016

Academic Editor: Xavier Leoncini

Copyright © 2016 Chunmei Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A new no-equilibrium chaotic system is reported in this paper. Numerical simulation techniques, including phase portraits and Lyapunov exponents, are used to investigate its basic dynamical behavior. To confirm the chaotic behavior of this system, the existence of topological horseshoe is proven via the Poincaré map and topological horseshoe theory.

#### 1. Introduction

Since Lorenz found an atmosphere dynamical model which can generate butterfly-shaped chaotic attractor in 1963 [1], chaos theory in the past five decades has attracted a lot of attention and hence triggered the emergence of a huge literature in this area. Since then, many kinds of chaotic or hyperchaotic systems governed by nonlinear ordinary differential equations (ODEs), including autonomous and nonautonomous chaotic systems [2–4], continuous and discrete chaotic systems [5–7], integer-order and fractional-order chaotic systems [1, 2, 7, 8], and chaotic systems with self-excited attractor and hidden attractor [9–11], were developed, and continuous chaotic systems governed by nonlinear partial differential equations (PDEs) [12–14] were also investigated.

To our knowledge, we summarize four criteria for the existence of chaos in the investigation of dynamical systems. The first one is the well-known Lyapunov exponents [15]. If there is at least one positive Lyapunov exponent in a dynamical system, the dynamics of this system is chaotic. The second one is Sil’nikov’s criterion for the existence of chaos [16, 17]. The main steps are as follows: calculate equilibrium points of a dynamical system; find a homoclinic or heteroclinic orbit connecting equilibrium points by using the undetermined coefficient method; and prove the convergence of the homoclinic or heteroclinic orbit series expansion obtained before. If the convergence can be proved, horseshoe chaos occurs. The third one is Melnikov’s criterion which is a powerful approximate tool for investigating chaos occurrence in near Hamiltonian systems and has been successfully applied to the analysis of chaos in smooth systems by calculating the distance between the stable and unstable manifold [18]. For dynamical systems, when the stable and unstable manifolds of their fixed points in the Poincaré map intersect transversely for sufficiently small parameter, there exists chaos in the sense of Smale horseshoe. The last one is the topological horseshoes theory which is based on the geometry of continuous maps on some subsets of interest in state space [19–23]. It is more applicable for computer-assisted verifications for the existence of chaotic behavior in dynamical systems in theory. A comparative analysis of these methods shows that the calculation of Lyapunov exponents and the topological horseshoes theory can be widely applied, but the Sil’nikov criterion is suitable for these systems where there is a homoclinic or heteroclinic orbit. Obviously, the Sil’nikov criterion cannot be used in no-equilibrium systems.

Very recently, hidden attractor in dynamical systems has been an important research topic because it has properties different from self-excited attractor. An attractor is called the hidden attractor if its basin of attraction does not intersect with small neighborhoods of the unstable fixed point; that is, the basins of attraction of the hidden attractors do not touch unstable fixed points and are located far away from such points [9]. The hidden attractors have been observed in these systems without fixed points, with no unstable fixed points, or with one stable fixed point, which motivate further construction and study of various artificial chaotic systems without equilibria. So far, hidden chaotic attractor has not been studied by the topological horseshoe theory. The main contribution of this research is to propose a new no-equilibrium system and verify the existence of its chaotic behavior by topological horseshoes theory. The proposed system is an artificial chaotic system having hidden attractor and has simple structure. The objective of this study is that we try to use the topological horseshoe theory to verify the existence of chaotic behavior. The rest of this paper is organized as follows. In Section 2, we introduce a no-equilibrium chaotic system and analyze its basic dynamics. In Section 3, we present rigorous arguments on existence of chaos in the new no-equilibrium system via topological horseshoe theory and computer computations. The conclusion is presented in the last section.

#### 2. The Proposed No-Equilibrium System

##### 2.1. Mathematical Model

Consider the following three-dimensional dynamical system:where , , and make up the system variables, and the constant term is an external DC excitation, while are the system’s parameters. When , , , and , system (1) can generate a chaotic attractor under the initial condition , as shown in Figure 1.