An Algebraic Criterion of Hopf Bifurcation with Relatively Reliability

The non-dominant characteristic root (NDCR) is defined. The possible damage to Hopf bifurcation caused by the NDCRs is put forward and confirmed by the simulations. The conception of the relative reliability of Hopf bifurcation is brought forward. In order to made the nth-order Hopf bifurcation relatively reliable, a new algebraic criterion is given, which can also help identifying the direction of bifurcation and the stability of the limit cycle. The example and simulation of the fourth-order controlled Lorenz system is presented to show the effectiveness and convenience of the criterion. Via adjusting of a distance parameter, the adverse influences induced by the NDCRs can be avoided.