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Topological Entropy of One Type of Nonoriented Lorenz-Type Maps
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.
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 . 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.
Because the system is symmetric, we consider the case of and obtain the mapConsidering the truncated map,The above formula is the Lorenz Poincaré map as shown in Figure 1(b).
It is similar to the condition of an existing strange attractor about oriented Lorenz map in . When , ; that is, and ; the invariant set is a nonoriented Lorenz attractor as shown in Figure 4. Obviously, if , then in the set .
Being similar to , the dynamical property of 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 .
3. The Calculation of the Topological Entropy of Nonoriented Lorenz-Type Map
As , if the map satisfies the condition where(1) and are continuous and monotone,(2) and , ,(3)the set is dense on the interval ,then is called a nonoriented Lorenz-type map.
The topological entropy of one-dimensional Lorenz-type map is defined in .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 . Here, the discontinuous points correspond to the critical points in . This paper will calculate topological entropy of the map . In order to facilitate the calculation, for a sequence , we consider the power serieswhere .
Lemma 1. As , the sequence is convergent.
Proof. Firstly, we show thatwhere 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. Let be a fixed natural number; for , such that , where . By the formula,we haveIf , then , and is bounded. Thus, we have and . Therefore, Since is arbitrary,we can prove the sequence is convergent.
Let be the number of points which fall at the discontinuous points firstly when and , .
Proof. It is clear that and that exists; this also holds for . Besides, . Thus, we have .
Let denote the power series associated with the sequence . By Lemma 2, the convergence radius of the series and is equal to .
Now, we give the symbolic dynamics model of the one-dimensional nonoriented Lorenz maps. For , Let denote its symbol sequence, whereThere is also a sequence , whereThus, we can obtain a sequence which is called the invariant coordinates of , whereTherefore, we obtain a map , where is a one-sided symbol space associated with the symbols −1, 0, and 1. It is clear that , where is the shift map. The order of invariant coordinates is as follows: if , , then .
Lemma 3. For , if , then ; that is, is a monotone increasing function on the interval .
Proof. If and lie in the left-hand side and right-hand side of the discontinuous point 0, respectively, then the conclusion is obvious. Suppose that for . Let be the interval . Then there is or determined by increasing or decreasing of . When , is a monotone interval of . For , by the chain rule, . If , then ; that is, .
If , then ; that is, . For both cases, we havewhen ; the above equal signs hold. Obviously, if , then . Let and denote the left and right limit points of , respectively. By Lemma 1, and exist. For discussion, the sequence of point is signed by a power series in complex field; that is, .
Lemma 4. .
Proof. The function is the series , which is to be truncated from . For fixed , is a step function with a finite number of discontinuous points; then the statement of the lemma follows.
We call and the kneading sequences of the map . The result of Lemma 4 can be simplified by kneading sequences.
Theorem 5. .
Proof. If and , , thenSubstituting the above two equations into Lemma 4, we can obtain For one-dimensional nonoriented Lorenz-type maps, that is,Substituting (24) and (25) into (23),Because the series is absolutely convergent, when , the series is an analytic function in convergent circle .
The series has poles in convergent circle. The series of is positive, so and the point in convergent circle is the pole of the series . By (26), is the minimum positive zero of the polynomial function ; that is, is the minimum positive root of the model of the equationSubstituting into (9), we obtain a formula of topological entropy of the map:
Example 6. Calculate the topological entropy of the map: is an eventually periodic sequence; then . By the symmetry of the map , we have , Substituting the expressions of and into (27), we obtain and . Thus, .
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.
The author declares that there are no competing interests regarding the publication of this paper.
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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.