Degenerate Hopf bifurcations, hidden attractors, and control in the extended Sprott E system with only one stable equilibrium

Authors

ZHOUCHAO WEI
Irene Moroz
Anping Liu

DOI

10.3906/mat-1305-64

Abstract

In this paper, we introduce an extended Sprott E system by a general quadratic control scheme with 3 arbitrary parameters for the new system. The resulting system can exhibit codimension-one Hopf bifurcations as parameters vary. The control strategy used can be applied to create degenerate Hopf bifurcations at desired locations with preferred stability. A complex chaotic attractor with only one stable equilibrium is derived in the sense of having a positive largest Lyapunov exponent. The chaotic attractor with only one stable equilibrium can be generated via a period-doubling bifurcation. To further suppress chaos in the extended Sprott E system coexisting with only one stable equilibrium, adaptive control laws are designed to stabilize the extended Sprott E system based on adaptive control theory and Lyapunov stability theory. Numerical simulations are shown to validate and demonstrate the effectiveness of the proposed adaptive control.

Keywords

Chaotic attractor, stable equilibrium, Sil'nikov's theorem, degenerate Hopf bifurcations, hidden attractor

First Page

672

Last Page

687

Recommended Citation

WEI, ZHOUCHAO; Moroz, Irene; and Liu, Anping (2014) "Degenerate Hopf bifurcations, hidden attractors, and control in the extended Sprott E system with only one stable equilibrium," Turkish Journal of Mathematics: Vol. 38: No. 4, Article 7. https://doi.org/10.3906/mat-1305-64
Available at: https://journals.tubitak.gov.tr/math/vol38/iss4/7