Anti-control of Hopf bifurcation at zero equilibrium of 4D Qi system

Abstract

Anti-control of Hopf bifurcation at the zero equilibrium of the 4D Qi system is achieved by the linear and nonlinear state feedback. First, aimed at the emergence of the Hopf bifurcation, linear control terms are determined and the conditions which the linear control gains should satisfy are presented. Then, with respect to the nonlinear control terms, which are associated with the stability of Hopf bifurcation, the nonlinear feedback applied directly to the Jordan form of the Qi system is investigated. By the stability analysis and a proper transformation, the nonlinear control terms which play roles on the stability of Hopf bifurcation are collected and the condition restricting the nonlinear control gain isobtained. The control strategy is different as the bifurcation parameter concerned is altered. Finally, the numerical results are presented that confirm the analytical predictions.