Controlling Chaos for Multistep Iteration Process and Its Special Iterations in Discrete Dynamical Systems

Abstract

The close relationship between chaos and dynamical systems leads to naturally consider the iteration processes that are related to dynamical systems of fixed point theory. From this natural relationship, the control of chaos that occurs in fixed point iteration dynamics will be the main focus of the article. To achieve this goal, analytical solutions are obtained and used to control chaos that occurs at unstable fixed points of multistep iteration process. Later, we show an effective regime for the parameters of multistep iteration. To illustrate this claim, well-known special cases of multistep iteration process by Noor, Ishikawa, Mann, Krasnoselskij, Picard iteration processes are introduced. In particular, among these iterations, the Noor iteration process is studied in detail in terms of controlled chaos. The Lyapunov exponent is used to estimate the stability and unstability of fixed points and periods that generate chaos in iteration processes. Finally, with the help of MATLAB program, all these results are shown on logistic and cubic equations with chaotic properties.