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.
Highlights
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]
For the dynamic systems described by differential equations, the Lorenz Poincaremap 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
Summary
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]. Lorenz system is approximated by the Shimizu-Morioka model (ẋ = y, ẏ = x − ay − xz, ż = −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). 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
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