Topological Entropy of One Type of Nonoriented Lorenz-Type Maps

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 (?̇? = y, ?̇? = x − ay − xz, ?̇? = −bz + x2) when the parameter r is large. This model has nonoriented Lorenz attractors for certain parameters, (e.g., a ≈ 0.59 and b ≈ 0.45). The bifurcations and chaos of themodel 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.


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][3][4].In this paper, the nonoriented situation is discussed.Lorenz system is approximated by the Shimizu-Morioka model ( ẋ = , ẏ =  −  − , ż = − +  2 ) when the parameter  is large.This model has nonoriented Lorenz attractors for certain parameters, (e.g.,  ≈ 0.59 and  ≈ 0.45).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.

A Geometric Model of the Nonoriented Lorenz-Type Attractors
The differential equation with a single parameter Ẋ = (, ),  ∈  3 , is considered, and it is symmetric about axis.The butterfly homoclinic orbit exists when the parameter  is zero and the equilibrium point is  = (0, 0, 0).The eigenvalue of the linearization matrix at  satisfies 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  ≥ 0. The sections Σ 0 :  =  and Σ 1 :  =  (where  > 0) are taken near the equilibrium .The sections are shown in Figure 1(a).The solution of the linear equations ẋ = , ẏ = −, and ż = − are  =  0   ,  =  0  − , and  =  − with the initial point ( 0 ,  0 , ) on Σ 0 .The time of the flow with the initial point from Σ 0 to Σ 1 is  = (1/) ln(/ 0 ) with  =  0   .Then the map  0 : Σ 0 → Σ 1  0 is defined by that is,  where ] = / is called the saddle point index and  = /.Let Γ(0) be the homoclinic orbit of the unperturbed system (when  = 0), and let Γ() be its perturbed solution; see Figure 1(b).Then the flows adjacent Γ() determine a map  1 : Σ 1 → Σ 0 .In the small neighborhood of the origin  on Σ 1 , the map  1 : Σ 1 → Σ 0 can be written as When the determinant of a matrix (  11  12  21  22 ) is less than zero, the manifold by the map  1 defined is nonoriented. is a splitting parameter, and the coordinate of  0 () = Γ() ∩ Σ 0 is (, ()).The map  =  1 ∘  0 is obtained on Σ 0 : Let  = 1 by scaling variable, and remove the under index zero.Thus, we can obtain where  =  11 is called the boundary line.This paper discusses the nonoriented Lorenz-type map corresponding to the situation  < 0 and  > 0, and let  12 ,  21 , and  22 be all greater than zero.
Because the system is symmetric, we consider the case of  ≤ 0 and obtain the map Considering the truncated map, The above formula is the Lorenz Poincaré map as shown in Figure 1(b).
Being similar to [2], the dynamical property of | Λ can be described by the shift map of the inverse limit pace of the map |  .So it is very important to study the dynamic behavior of this kind of attractor by studying the one-dimensional map ().

The Calculation of the Topological Entropy of Nonoriented Lorenz-Type Map
As The topological entropy of one-dimensional Lorenz-type map is defined in [7].
where   is the number of discontinuous points of   ().
The definition in (9) is similar to the one about piecewise monotone continuous map given by de Melo and van Strien [8].Here, the discontinuous points correspond to the critical points in [8].This paper will calculate topological entropy of the map .In order to facilitate the calculation, for a sequence  1 ,  2 ,  3 , . .., we consider the power series where  ∈ .
Proof.Firstly, we show that where  and  are Lorenz-type maps; () is the number of discontinuous points of .Because, in each interval on which  is continuous, the number of the discontinuous points of  is at most (), the inequality ( ∘ ) ≤ ()() holds.
For one-dimensional nonoriented Lorenz-type maps, that is, Substituting  0 into (9), we obtain a formula of topological entropy of the map:

Conclusion
For the dynamic systems described by differential equations, the Lorenz Poincaré map is a mostly used method to study the structure of strange attractors of the systems.The dynamical behavior of the Lorenz-type attractor can be described by one-dimensional Lorenz-type map.The topological entropy of the Lorenz map can be calculated by using the symbol sequence of the boundary points of the invariant interval, and we can know the system is chaos or not.