Hidden-like Attractors in a Class of Discontinuous Dynamical Systems

Hany A. Hosham (), Mashael A. Aljohani, Eman D. Abou Elela, Nada A. Almuallem and Thoraya N. Alharthi
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Hany A. Hosham: Department of Mathematics, Faculty of Science, Taibah University, Yanbu 41911, Saudi Arabia
Mashael A. Aljohani: Department of Mathematics, Faculty of Science, Taibah University, Yanbu 41911, Saudi Arabia
Eman D. Abou Elela: Department of Mathematics, Faculty of Science, Taibah University, Yanbu 41911, Saudi Arabia
Nada A. Almuallem: Department of Mathematics and Statistics, Faculty of Science, University of Jeddah, P.O. Box 80327, Jeddah 21589, Saudi Arabia
Thoraya N. Alharthi: Department of Mathematics, College of Science, University of Bisha, P.O. Box 551, Bisha 61922, Saudi Arabia

Mathematics, 2024, vol. 12, issue 23, 1-14

Abstract: In continuous dynamical systems, a hidden attractor occurs when its basin of attraction does not connect with small neighborhoods of equilibria. This research aims to investigate the presence of hidden-like attractors in a class of discontinuous systems that lack equilibria. The nature of non-smoothness in Filippov systems is critical for producing a wide variety of interesting dynamical behaviors and abrupt transient responses to dynamic processes. To show the effects of non-smoothness on dynamic behaviors, we provide a simple discontinuous system made of linear subsystems with no equilibria. The explicit closed-form solutions for each subsystem have been derived, and the generalized Poincaré maps have been established. Our results show that the periodic orbit can be completely established within a sliding region. We then carry out a mathematical investigation of hidden-like attractors that exhibit sliding-mode characteristics, particularly those associated with grazing-sliding behaviors. The proposed system evolves by adding a nonlinear function to one of the vector fields while still preserving the condition that equilibrium points do not exist in the whole system. The results of the linear system are useful for investigating the hidden-like attractors of flow behavior across a sliding surface in a nonlinear system using numerical simulation. The discontinuous behaviors are depicted as motion in a phase space governed by various hidden attractors, such as period doubling, period- m segments, and chaotic behavior, with varying interactions with the sliding mode.

Keywords: Filippov systems; grazing-sliding bifurcations; period doubling; sliding mode; chaotic behavior (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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