On the reasonability of linearized approximation and Hopf bifurcation control for a fractional-order delay Bhalekar–Gejji chaotic system

Advances in Difference Equations

article

Shi, Jianping¹  Ruan, Liyuan¹ 

[1] Department of System Science and Applied Mathematics, Kunming University of Science and Technology;Center for Nonlinear Science Studies, Kunming University of Science and Technology

关键词: Fractional-order BG system;    Hopf bifurcation;    Delayed feedback control;    Linearization;    Stability;   

DOI  :  10.1186/s13662-020-02908-2

学科分类：航空航天科学

来源: SpringerOpen

【 摘 要 】

In this paper, we study the reasonability of linearized approximation and Hopf bifurcation control for a fractional-order delay Bhalekar–Gejji (BG) chaotic system. Since the current study on Hopf bifurcation for fractional-order delay systems is carried out on the basis of analyses for stability of equilibrium of its linearized approximation system, it is necessary to verify the reasonability of linearized approximation. Through Laplace transformation, we first illustrate the equivalence of stability of equilibrium for a fractional-order delay Bhalekar–Gejji chaotic system and its linearized approximation system under an appropriate prior assumption. This semianalytically verifies the reasonability of linearized approximation from the viewpoint of stability. Then we theoretically explore the relationship between the time delay and Hopf bifurcation of such a system. By introducing the delayed feedback controller into the proposed system, the influence of the feedback gain changes on Hopf bifurcation is also investigated. The obtained results indicate that the stability domain can be effectively controlled by the proposed delayed feedback controller. Moreover, numerical simulations are made to verify the validity of the theoretical results.

【 授权许可】

CC BY