Hopf bifurcation analysis and control of a new Lorenz-like system

Abstract

This paper is concerned with Hopf bifurcation behavior and bifurcation control of a new Lorenz-like system. Firstly, Hopf bifurcation type is studied based on the bifurcation stability norm, furthermore bifurcation period and characteristic exponent of system are given. Then the linear controller and the non-linear controller are applied to control the Hopf bifurcation of original system respectively. For linear control, the Routh-Hurwitz criterion is adopted to give parameter conditions making Hopf bifurcation dissapear; For non-linear control, the Hopf bifurcation Normal Form of controlled system is obtained by using the direct Normal Form method, besides, according to coefficient of Normal Form, the impact of selection criteria of non-linear parameter on amplitude of limit cycle and Hopf bifurcation type are discussed, discussions show that when the non-linear parameter satisfies certain condition, bifurcation type of original system will be changed, and the periodic solution amplitude will be changed with the parameter. Finally, theoretical results are verified by numerical simulations.