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ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Stabilization of Unstable Periodic Orbits in a Three-Dimensional Chaotic System Using Time-Delay Autosynchronization Control Method

Journal of Applied Nonlinear Dynamics 12(3) (2023) 453--464 | DOI:10.5890/JAND.2023.09.003

Abdul Hussain Surosh$^{{1,2}} $, Reza Khoshsiar Ghaziani$ ^{{1}} $, Javad Alidousti$ ^{{1}} $

$ ^{mathrm{1}} $ Department of Applied Mathematics, Shahrekord University, Shahrekord, P.O.Box 115, Iran

$ ^{mathrm{2}} $ Department of Mathematics, Baghlan University, Pol-e-Khomri, Baghlan, Afghanistan

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Abstract

This paper is concerned with controlling complex dynamics of a three-dimensional chaotic system consisting of two quadratic cross-products and one square term. We use Pyragas' time-delayed feedback control (TDFC) known as time delay autosynchronization (TDAS) method to stabilize the unstable equilibrium point and unstable periodic orbits of the system. An explicit formula is derived to determine the critical value of time delay $ \tau_{0} $ for which when the delay passes through a certain threshold critical value, the chaotic dynamical system undergoes a Hopf bifurcation. Furthermore, by choosing the appropriate range of feedback strength $ K $ and control parameter $ \tau $ as a free parameters, existence of Hopf bifurcation is investigated theoretically and numerically. Finally, some numerical simulations are presented to verify the analytical results. \vspace{0.2cm}

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