Chaos and Coexisting Bifurcations in a Novel 3D Autonomous System with a Non-Hyperbolic Fixed Point: Theoretical Analysis and Electronic Circuit Implementation

Abstract

A 3D autonomous chaotic system with the distinguishing feature of having a couple of fixed points, one of which is non-hyperbolic, is proposed. Interestingly, these fixed points become all hyperbolic in other parameters’ regions yielding diverse modes of oscillations in the system. The stability of the equilibria is discussed based on the Routh–Hurwitz stability criterion. The complex dynamics of the proposed system is numerically explored by using phase space trajectory plots, bifurcation diagrams, graphs of Lyapunov exponents, and basins of attraction. It is found that the system experiences period-doubling bifurcation, coexisting bifurcations, periodic windows, and chaos when monitoring its parameters. When the system is tuned to develop non-hyperbolic chaos, the basin of attraction of the strange attractor (coexisting with one of the fixed points) intersects with neighborhood of equilibria which is typical of self-excited oscillations. The coexistence between periodic and chaotic behaviors is found for specific parameter values. The analysis of the basins of attraction for the coexisting attractors reveals extremely complex structures. PSpice simulations based on a suitably designed electronic analog of the system confirm the results of theoretical analysis. The model proposed in this work shows “elegant” mathematical simplicity (i.e., only quadratic nonlinearities) and extremely rich modes of oscillations, and thus may be regarded as a prototypal member of the recently discovered and very restricted class of nonlinear systems developing non-hyperbolic chaos.

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Kengne, J., Abdolmohammadi, H., Signing, V.F. et al. Chaos and Coexisting Bifurcations in a Novel 3D Autonomous System with a Non-Hyperbolic Fixed Point: Theoretical Analysis and Electronic Circuit Implementation. Braz J Phys 50, 442–453 (2020). https://doi.org/10.1007/s13538-020-00758-7

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Keywords

  • 3D autonomous chaotic system
  • Non-hyperbolic fixed point
  • Coexisting bifurcations
  • Basins of attraction
  • PSpice simulations