Bifurcation analysis with self-excited and hidden attractors for a chaotic jerk system

Volume 35, Issue 3, pp 319--335 https://dx.doi.org/10.22436/jmcs.035.03.05

Publication Date: May 22, 2024 Submission Date: January 28, 2024 Revision Date: April 12, 2024 Accteptance Date: April 21, 2024

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Authors

T. I. Rasul - Department of Mathematics, Faculty of Science, Soran University, Soran, Kurdistan Region, Iraq. R. H. Salih - Department of Mathematics, College of Basic Education, University of Raparin, Rania, Kurdistan Region, Iraq.

Abstract

This paper is devoted to investigating the local bifurcation of a chaotic jerk system. The local stability of equilibrium points is analyzed, as well as the existence of transcritical bifurcation at the origin. For the proposed jerk system, the Hopf and Zero-Hopf bifurcations are investigated at the origin. Moreover, a zero-Hopf equilibrium point at the origin is characterized for the system. By using the averaging theory of first order, a limit cycle can be bifurcated from the zero-Hopf equilibrium located at the origin. Liapunov quantities techniques are used to investigate the cyclicity of the system. It is shown that three limit cycles can be bifurcated from the origin. Finally, both self-excited chaotic attractors and hidden chaotic attractors are studied for special cases of the chaotic jerk systems using bifurcation diagrams, Lyapunov exponents, and cross-sections. All results reported in this study have been obtained using Maple software.

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Keywords

• Transcritical bifurcation
• zero hopf
• Hopf bifurcation
• jerk system
• self-excited attractors
• hidden attractors

MSC

•  34C23
•  34C28
•  37D45

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