Chaos control in delayed phase space constructed by the Takens embedding theory

• Published in: Communications in nonlinear science & numerical simulation Vol. 54; pp. 453 - 465
• Main Authors: Hajiloo, R., Salarieh, H., Alasty, A.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.01.2018; Elsevier Science Ltd
• Subjects: Chaos; Chaos theory; Controllers; Delayed feedback; Delayed phase space; Discrete element method; Discrete time systems; Dynamic models; Effectiveness; Embedding; Feedback control; Least squares method; Lorenz Curve; Lorenz system; Reconstructed model; Recursive methods; Studies; Time series; Chaos; Time-series; Delayed phase space; Reconstructed model; Delayed feedback
• ISSN: 1007-5704; 1878-7274
• DOI: 10.1016/j.cnsns.2017.05.022

•An algorithm is proposed to control chaos in unknown discrete-time chaotic systems.•Time-series of one measurable state is employed to reconstruct a delayed phase space.•A dynamic model is identified using RLS method to estimate time-series data.•An appropriate delayed feedback controller is obtained for stabilizing UFPs.•The proper gains of the Pyragas control are computed using a systematic approach. In this paper, the problem of chaos control in discrete-time chaotic systems with unknown governing equations and limited measurable states is investigated. Using the time-series of only one measurable state, an algorithm is proposed to stabilize unstable fixed points. The approach consists of three steps: first, using Takens embedding theory, a delayed phase space preserving the topological characteristics of the unknown system is reconstructed. Second, a dynamic model is identified by recursive least squares method to estimate the time-series data in the delayed phase space. Finally, based on the reconstructed model, an appropriate linear delayed feedback controller is obtained for stabilizing unstable fixed points of the system. Controller gains are computed using a systematic approach. The effectiveness of the proposed algorithm is examined by applying it to the generalized hyperchaotic Henon system, prey-predator population map, and the discrete-time Lorenz system.