Discrete Dynamics in Nature and Society

Volume 2016 (2016), Article ID 6987471, 5 pages

http://dx.doi.org/10.1155/2016/6987471

## Topological Entropy of One Type of Nonoriented Lorenz-Type Maps

Basic Subject Department, Shandong Women’s University, Jinan 250300, China

Received 16 June 2016; Accepted 29 September 2016

Academic Editor: Massimiliano Ferrara

Copyright © 2016 Guo Feng. 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

Constructing a Poincaré map is a method that is often used to study high-dimensional dynamical systems. In this paper, a geometric model of nonoriented Lorenz-type attractor is studied using this method, and its dynamical property is described. The topological entropy of one-dimensional nonoriented Lorenz-type maps is also computed in terms of their kneading sequences.

#### 1. Introduction

The Lorenz attractor is usually divided into three types, which are called oriented, semioriented, and nonoriented Lorenz attractor, respectively, and the existence condition of Lorenz attractor of planar map is given in [1]. The oriented dynamical property is studied in detail in [2–4]. In this paper, the nonoriented situation is discussed. Lorenz system is approximated by the Shimizu-Morioka model (, , ) when the parameter is large. This model has nonoriented Lorenz attractors for certain parameters, (e.g., and ). The bifurcations and chaos of the model are discussed using numerical method in [5, 6]. In this paper, a geometric model of the type attractor is described, and a formula for computation of the topological entropy of one-dimensional nonoriented Lorenz maps is given.

#### 2. A Geometric Model of the Nonoriented Lorenz-Type Attractors

The differential equation with a single parameter , , is considered, and it is symmetric about -axis. The butterfly homoclinic orbit exists when the parameter is zero and the equilibrium point is . The eigenvalue of the linearization matrix at satisfies . Through simple coordinate transformations, the equation can be reduced to where , , and are high-order items. In the neighborhood of , the dynamical property of the equations can be described by its linear part. Due to the symmetry of the system, we only discuss the situation . The sections and (where ) are taken near the equilibrium . The sections are shown in Figure 1(a). The solution of the linear equations , , and are , , and with the initial point on . The time of the flow with the initial point from to is with . Then the map is defined bythat is, where is called the saddle point index and . Let be the homoclinic orbit of the unperturbed system (when ), and let be its perturbed solution; see Figure 1(b). Then the flows adjacent determine a map . In the small neighborhood of the origin on , the map can be written asWhen the determinant of a matrix is less than zero, the manifold by the map defined is nonoriented. is a splitting parameter, and the coordinate of is . The map is obtained on : Let by scaling variable, and remove the under index zero. Thus, we can obtainwhere is called the boundary line. This paper discusses the nonoriented Lorenz-type map corresponding to the situation and , and let , , and be all greater than zero.