Generic Hopf bifurcation from lines of equilibria without parameters: I. Theory

Motivated by decoupling e ects in coupled oscillators, by viscous shock proles in systems of nonlinear hyperbolic balance laws, and by binary oscillation e ects in discretizations of systems of hyperbolic balance laws, we consider vector elds 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 -/!-limit sets of nearby trajectories consist entirely of equilibria on the line.