Hidden chaotic mechanisms for a family of chameleon systems

Xue Zhang; Bo Sang; Bingxue Li; Jie Liu; Lihua Fan; Ning Wang; Xue Zhang; Bo Sang; Bingxue Li; Jie Liu; Lihua Fan; Ning Wang

doi:10.3934/mmc.2023032

Mathematical Modelling and Control

2023, Volume 3, Issue 4: 400-415. doi: 10.3934/mmc.2023032

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Research article

Hidden chaotic mechanisms for a family of chameleon systems

1.
School of Mathematical Sciences, Liaocheng University, Liaocheng 252059, China
2.
School of Microelectronics and Control Engineering, Changzhou University, Changzhou 213159, China

Received: 27 May 2023 Revised: 27 August 2023 Accepted: 27 August 2023 Published: 27 December 2023

Chameleon chaotic systems are nonlinear dynamical systems whose chaotic attractors can transform between hidden and self-excited types by tuning system parameters to modify equilibrium points. This paper proposes a novel family of chameleon chaotic systems, which can exhibit three types of chaotic attractors: self-excited attractors with a nonhyperbolic equilibrium, hidden attractors with a stable equilibrium, and hidden attractors with no equilibrium points. Bifurcation analysis uncovers the mechanisms by which self-excited and hidden chaotic attractors arise in this family of chameleon systems. It is demonstrated that various forms of chaos emerge through period-doubling routes associated with changes in the coefficient of a linear term. An electronic circuit is designed and simulated in Multisim to realize a hidden chaotic system with no equilibrium points. It is demonstrated that the electronic circuit simulation is consistent with the theoretical model. This research has the potential to enhance our comprehension of chaotic attractors, especially the hidden chaotic attractors.
- Hopf bifurcation,
- period-doubling route,
- bifurcation diagram,
- hidden chaotic attractor
Citation: Xue Zhang, Bo Sang, Bingxue Li, Jie Liu, Lihua Fan, Ning Wang. Hidden chaotic mechanisms for a family of chameleon systems[J]. Mathematical Modelling and Control, 2023, 3(4): 400-415. doi: 10.3934/mmc.2023032

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Abstract

Chameleon chaotic systems are nonlinear dynamical systems whose chaotic attractors can transform between hidden and self-excited types by tuning system parameters to modify equilibrium points. This paper proposes a novel family of chameleon chaotic systems, which can exhibit three types of chaotic attractors: self-excited attractors with a nonhyperbolic equilibrium, hidden attractors with a stable equilibrium, and hidden attractors with no equilibrium points. Bifurcation analysis uncovers the mechanisms by which self-excited and hidden chaotic attractors arise in this family of chameleon systems. It is demonstrated that various forms of chaos emerge through period-doubling routes associated with changes in the coefficient of a linear term. An electronic circuit is designed and simulated in Multisim to realize a hidden chaotic system with no equilibrium points. It is demonstrated that the electronic circuit simulation is consistent with the theoretical model. This research has the potential to enhance our comprehension of chaotic attractors, especially the hidden chaotic attractors.

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