Generic Hopf Bifurcation from Lines of Equilibria without Parameters: I. Theory

Abstract

Motivated by decoupling effects in coupled oscillators, by viscous shock profiles in systems of nonlinear hyperbolic balance laws, and by binary oscillation effects in discretizations of systems of hyperbolic balance laws, we consider vector fields with a one-dimensional line of equilibria, even in the absence of any parameters. Besides a trivial eigenvalue zero we assume that the linearization at these equilibria possesses a simple pair of nonzero eigenvalues which cross the imaginary axis transversely as we move along the equilibrium line. In normal form and under a suitable nondegeneracy condition, we distinguish two cases of this Hopf-type loss of stability, hyperbolic and elliptic. Going beyond normal forms we present a rigorous analysis of both cases. In particular, α- and ω-limit sets of nearby trajectories consist entirely of equilibria on the line.