Bifurcations Associated with a Double Zero and a Pair of Pure Imaginary Eigenvalues

Abstract

Interactive static and dynamic bifurcations associated with a nonlinear autonomous system and the stability properties of various solutions are explored. Attention is focused on the vicinity of a compound critical point where the Jacobian of the system exhibits a double zero eigenvalue of index one and a pair of pure imaginary eigenvalues. The system under consideration has three independent parameters, and depending on the route followed in the parameter space, the system may exhibit static bifurcations, Hopf bifurcations, secondary Hopf bifurcations, and bifurcations into two or three dimensional tori. The general analysis is based on a perturbation method which combines an intrinsic harmonic balancing technique with a certain unification procedure. This approach leads to bifurcation equations and simplified differential equations governing the local dynamics in the vicinity of the compound critical point. A control system is analyzed for illustration.