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Constructing a chaotic system with any number of equilibria

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Abstract

In the chaotic Lorenz system, Chen system and Rössler system, their equilibria are unstable and the number of the equilibria are no more than three. This paper shows how to construct some simple chaotic systems that can have any preassigned number of equilibria. First, a chaotic system with no equilibrium is presented and discussed. Then a methodology is presented by adding symmetry to a new chaotic system with only one stable equilibrium, to show that chaotic systems with any preassigned number of equilibria can be generated. By adjusting the only parameter in these systems, one can further control the stability of their equilibria. This result reveals an intrinsic relationship of the global dynamical behaviors with the number and stability of the equilibria of a chaotic system.

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Acknowledgement

This research was supported by the Hong Kong Research Grants Council under the GRF Grant CityU1114/11E.

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Authors and Affiliations

  1. Department of Electronic Engineering, City University of Hong Kong, Hong Kong, SAR, China

    Xiong Wang & Guanrong Chen

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  1. Xiong Wang
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  2. Guanrong Chen
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Correspondence to Xiong Wang.

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Wang, X., Chen, G. Constructing a chaotic system with any number of equilibria. Nonlinear Dyn 71, 429–436 (2013). https://doi.org/10.1007/s11071-012-0669-7

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  • DOI: https://doi.org/10.1007/s11071-012-0669-7

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