Bifurcation and chaos in simple discontinuous systems separated by a hypersurface

Hany A. Hosham; Thoraya N. Alharthi; Hany A. Hosham; Thoraya N. Alharthi

doi:10.3934/math.2024826

AIMS Mathematics

2024, Volume 9, Issue 7: 17025-17038. doi: 10.3934/math.2024826

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Bifurcation and chaos in simple discontinuous systems separated by a hypersurface

Hany A. Hosham ^{1
,
,},
Thoraya N. Alharthi ²

1.
Department of Mathematics, Faculty of Science, Taibah University, Yanbu, 41911, Saudi Arabia
2.
Department of Mathematics, College of Science, University of Bisha, P.O. Box 551, Bisha 61922, Saudi Arabia

Received: 13 March 2024 Revised: 22 April 2024 Accepted: 09 May 2024 Published: 16 May 2024
MSC : 34A36, 34D23, 37G15, 34H10

This research focuses on a mathematical examination of a path to sliding period doubling and chaotic behaviour for a novel limited discontinuous systems of dimension three separated by a nonlinear hypersurface. The switching system is composed of dissipative subsystems, one of which is a linear systems, and the other is not linked with equilibria. The non-linear sliding surface is designed to improve transient response for these subsystems. A Poincaré return map is created that accounts for the existence of the hypersurface, completely describing each individual sliding period-doubling orbits that route to the sliding chaotic attractor. Through a rigorous analysis, we show that the presence of a nonlinear sliding surface and a set of such hidden trajectories leads to novel bifurcation scenarios. The proposed system exhibits period-$ m $ orbits as well as chaos, including partially hidden and sliding trajectories. The results are numerically verified through path-following techniques for discontinuous dynamical systems.
- discontinuous systems,
- bifurcations,
- period-doubling,
- sliding mode,
- chaos
Citation: Hany A. Hosham, Thoraya N. Alharthi. Bifurcation and chaos in simple discontinuous systems separated by a hypersurface[J]. AIMS Mathematics, 2024, 9(7): 17025-17038. doi: 10.3934/math.2024826

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Abstract

This research focuses on a mathematical examination of a path to sliding period doubling and chaotic behaviour for a novel limited discontinuous systems of dimension three separated by a nonlinear hypersurface. The switching system is composed of dissipative subsystems, one of which is a linear systems, and the other is not linked with equilibria. The non-linear sliding surface is designed to improve transient response for these subsystems. A Poincaré return map is created that accounts for the existence of the hypersurface, completely describing each individual sliding period-doubling orbits that route to the sliding chaotic attractor. Through a rigorous analysis, we show that the presence of a nonlinear sliding surface and a set of such hidden trajectories leads to novel bifurcation scenarios. The proposed system exhibits period-$ m $ orbits as well as chaos, including partially hidden and sliding trajectories. The results are numerically verified through path-following techniques for discontinuous dynamical systems.

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