Hopf bifurcation Control Using Nonlinear Feedback with Polynomial Functions

A general explicit formula is derived for controlling bifurcations using nonlinear state feedback. This method does not increase the dimension of the system, and can be used to either delay (or eliminate) existing bifurcations or change the stability of bifurcation solutions. The method is then employed for Hopf bifurcation control. The Lorenz equation and Rössler system are used to illustrate the application of the approach. It is shown that a simple control can be obtained to simultaneously stabilize two symmetrical equilibria of the Lorenz system, and keep the symmetry of Hopf bifurcations from the equilibria. For the Rössler system, a control is also obtained to simultaneously stabilize two nonsymmetric equilibria and meanwhile stabilize possible Hopf bifurcations from the equilibria. Computer simulation results are presented to confirm the analytical predictions.