Dynamic analysis and multistability of a discontinuous Jerk-like system

Thoraya N. Alharthi; Thoraya N. Alharthi

doi:10.3934/math.2025566

AIMS Mathematics

2025, Volume 10, Issue 5: 12554-12575. doi: 10.3934/math.2025566

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Dynamic analysis and multistability of a discontinuous Jerk-like system

Thoraya N. Alharthi ^,

Department of Mathematics, College of Science, University of Bisha, P. O. Box 551, Bisha 61922, Saudi Arabia

Received: 28 January 2025 Revised: 05 May 2025 Accepted: 20 May 2025 Published: 29 May 2025
MSC : 34A36, 34D23, 34H10, 37G15

This paper introduces a novel Jerk-like system characterized by discontinuous vector fields along a codimension-1 switching surface. This system is capable of exhibiting both the existence and non-existence of equilibria in response to defined arbitrary functions. Non-smooth properties are investigated, which are essential for identifying and allowing rapid transient responses to dynamic events that influence the system's behaviors. The proposed discontinuous systems, which include both crossing and sliding solutions, are examined specifically for the detection of self-excited and hidden attractors. The dynamic characteristics of these systems are examined utilising the criteria for discontinuous solutions, the Poincarè return map, the spectrum of Lyapunov exponents, and bifurcation diagrams. The analysis reveals that incorporating the switching surface into Jerk-like subsystems alters the dimensions of the generated attractors. This alteration provides the necessary basis for the formation of period-doubling orbits that culminate in chaotic attractors. In the scenarios involving sliding motion within the proposed system, various attractors are identified that exhibit sensitive dependence not only on the initial conditions and parameter variations, but also on the interaction of the flow with the discontinuity surface. An interesting result indicates that the proposed novel Jerk system, which is distinguished by the absence of both real and virtual equilibria, reveals a family of periodic orbits in the sliding region surrounding the pseudo-equilibrium. The use of analytical solutions and numerical techniques successfully identifies a wide range of attractors, including periodic orbits, period-doubling phenomena, and chaotic behavior in both crossing and sliding modes.
- Jerk-like chaotic system,
- discontinuous systems,
- self-excited and hidden attractors,
- multistability,
- sliding motion
Citation: Thoraya N. Alharthi. Dynamic analysis and multistability of a discontinuous Jerk-like system[J]. AIMS Mathematics, 2025, 10(5): 12554-12575. doi: 10.3934/math.2025566

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Abstract

This paper introduces a novel Jerk-like system characterized by discontinuous vector fields along a codimension-1 switching surface. This system is capable of exhibiting both the existence and non-existence of equilibria in response to defined arbitrary functions. Non-smooth properties are investigated, which are essential for identifying and allowing rapid transient responses to dynamic events that influence the system's behaviors. The proposed discontinuous systems, which include both crossing and sliding solutions, are examined specifically for the detection of self-excited and hidden attractors. The dynamic characteristics of these systems are examined utilising the criteria for discontinuous solutions, the Poincarè return map, the spectrum of Lyapunov exponents, and bifurcation diagrams. The analysis reveals that incorporating the switching surface into Jerk-like subsystems alters the dimensions of the generated attractors. This alteration provides the necessary basis for the formation of period-doubling orbits that culminate in chaotic attractors. In the scenarios involving sliding motion within the proposed system, various attractors are identified that exhibit sensitive dependence not only on the initial conditions and parameter variations, but also on the interaction of the flow with the discontinuity surface. An interesting result indicates that the proposed novel Jerk system, which is distinguished by the absence of both real and virtual equilibria, reveals a family of periodic orbits in the sliding region surrounding the pseudo-equilibrium. The use of analytical solutions and numerical techniques successfully identifies a wide range of attractors, including periodic orbits, period-doubling phenomena, and chaotic behavior in both crossing and sliding modes.

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