We give new versions of the global Newton method and the Kellogg & Li & Yorke method for calculating zero points and fixed points of nonlinear maps, which are numerically stable, but do not require an extra homotopy dimension. In addition, regularity results are established so that predictor-corrector continuation methods will lead to solutions if appropriate boundary conditions are satisfied.

James Yorke won the 2003 Japan Prize for his work in the field of chaos theory. This book was compiled by four of his best-known collaborators in honor of his 60th birthday and contains papers by various authors on chaos and chaotic attractors. The papers are organized around the topics of “natural” invariant measure and fractal dimension of the associated chaotic attractors. A few of the paper...

We show that, for monotone graph map f , all the ω-limit sets are finite whenever f has periodic point and for monotone dendrite map, any infinite ω-limit set does not contain periodic points. As a consequence, monotone graph and dendrite maps have no Li-Yorke pairs. However, we built a homeomorphism on a dendroid with a scrambled set having nonempty interior.

Chaotic attractors in the two-dimensional border-collision normal form (a piecewise-linear map) can persist throughout open regions of parameter space. Such robust chaos has been established rigorously some regimes. Here we provide formal results for original regime [S. Banerjee, J.A. Yorke, C. Grebogi, Robust Chaos, Phys. Rev. Lett. 80(14):3049-3052, 1998]. We first construct a trapping region...

Chaos has been successfully applied in many fields to improve the performance of engineering systems, such as communication, vibration compact, and mixing. Generating chaos from originally non-chaotic systems is a relevant topic because potential applications. In this work, impulse control shown generate system. Using Chen system an example, we prove by analytical numerical methods that indeed ...

In this paper, some properties of the periodic shadowing are presented. It is shown that a homeomorphism has the periodic shadowing property if and only if so does every lift of it to the universal covering space. Also, it is proved that continuous mappings on a compact metric space with the periodic shadowing and the average shadowing property also have the shadowing property and then are chao...

Let Xn = {z ∈ C : z n ∈ [0, 1]}, n ∈ N, and let f : Xn → Xn be a continuous map such that f(0) = 0. In this paper we prove that f is chaotic in the sense of Li–Yorke iff there is a strictly increasing sequence of positive integers A such that the topological sequence entropy of f relative to A is positive.

In this paper, we derive new results for global chaos control of chaotic systems via sliding control. We also explore the analysis and global chaos control of the hyperchaotic Li system (2005) using sliding control. The global chaos control results via sliding control have been established using Lyapunov stability theory. Numerical simulations using MATLAB are shown to validate and depict the e...