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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.

Since Lorenz found an atmosphere dynamical model which can generate butterfly-shaped chaotic attractor in 1963 [

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 [

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 [

Consider the following three-dimensional dynamical system:

Chaotic attractor of system (

Consider the symmetry and invariance; it is easy to get the invariance of system (

For a three-dimensional dynamical system, we know that it has distinct features in its Lyapunov exponents. If there is one positive Lyapunov exponent, the dynamics of this system is chaotic. If we assume that the largest Lyapunov exponent of system (

The largest Lyapunov exponent spectrum of system (

In this section, we firstly recall a result on horseshoes theory developed in [

Let

Let

Let

Next, we recall the semiconjugacy in terms of a continuous map and the shift map

Let

Suppose that there exists a

Let

In this section, a rigorous verification of chaos in the chaotic system (

Phase portrait of system (

Now in the plane

The Poincaré map

To prove this statement, we take two subsets

Let

Now let

In this paper, we proposed a new no-equilibrium system which can generate chaotic flow for the given parameters. Numerical simulation techniques, including phase portraits and Lyapunov exponents, illustrate its chaotic behavior. Moreover, numerical simulations show that the existence of horseshoe chaos in the proposed system is proven by means of topological horseshoe theory. The essence of the arguments is to choose a cross section and study the dynamics of the corresponding Poincaré map to which the topological horseshoe theory can apply. The horseshoe chaos in the Poincaré map shows that the proposed system does exhibit chaotic behavior.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported in part by the Shandong Natural Science Foundation of China under Grant no. ZR2014FQ019 and the Science Foundation of Binzhou University of China under Grant nos. BZXYG1618 and BZXYG1615.