A perturbation analysis of interactive static and dynamic bifurcations

The instability behavior of a nonlinear autonomous system in the vicinity of a coincident critical point, which leads to interactions between static and dynamic bifurcations, is studied. The critical point considered is characterized by a simple zero and a pair of pure imaginary eigenvalues of the Jacobian, and the system contains two independent parameters. The static and dynamic bifurcations and quasiperiodic motions resulting from the interaction of the bifurcation modes and the associated invariant tori are analyzed by a novel unification technique that is based on an intrinsic perturbation procedure. Divergence boundary, dynamic bifurcation boundary, secondary bifurcations, and invariant tori are determined explicitly. Two illustrative examples concerning control systems are presented.<>