Anti-control of Hopf bifurcation in the Shimizu-Morioka system using an explicit criterion

We consider anti-control of Hopf bifurcation for the Shimizu-Morioka system by using an explicit criterion. We first provide the two conditions for the existence of Hopf bifurcation, that is, eigenvalue assignment and transversality conditions, which could be formulated through the coefficients of characteristic equation, and the obtained conditions do not need to calculate the eigenvalue and eigenvalue's derivatives. The center manifold theory and normal form reduction are utilized to derive the nonlinear gains for controlling the stability of the created limit circle. In addition, we further improve the computing formulas of amplitude and frequency of Hopf limit cycle. Numerical analysis also verifies the effectiveness of the proposed results.