Studying a Hidden Bifurcation and Finding Hopf Bifurcation with Generated New Saturated Function Series

Résumé

In this article, a hidden bifurcation of the multispiral chaotic attractor generated by the new saturated function series has been considered. The general shape of the chaotic attractors is described in terms of the number of spirals (also reffered to as multiscroll attractor) governed by integer parameters p and q. Due to the integer nature of the parameter, it is not possible to observe bifurcations from M spirals when the parameter is increased by two. However, by using the method of hidden bifurcations, an additional real parameter ε was introduced to observe such bifurcations. Additionally, this added parameter allowed us to find the Hopf bifurcation of the multispiral attractor generated by the new saturated function series transitioning from a stable state to a chaotic state. Furthermore, the Routh-Hurwitz criterion was used to study the stability of the original equilibrium point of the system.

Keywords: Saturated function series, hidden bifurcation, Hopf bifurcation, multiscroll.

MSC: 39A21, 39A28, 39A33.

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Communicated Editor: Khaled Zennir

Manuscript received Feb 20, 2024; revised Apr 17, 2024; accepted Mai 01, 2024; published Dec 07, 2024.