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

Motivated by decoupling effects in coupled oscillators, by viscous shock pr ofiles in systems of nonlinear hyperbolic balance laws, rind by binary osci llation effects in discretizations of systems of hyperbolic balance laws, w e 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 nonz ero eigenvalues which cross the imaginary axis transversely as we move alon g the equilibrium line. In normal form and under a suitable nondegeneracy condition, wt distinguish two cases of this Hopf-type loss of stability, hyperbolic and elliptic. Go ing beyond normal forms we present a rigorous analysis of both cases. In pa rticular. alpha- and omega -limit sets of nearby trajectories consist entir ely of equilibria on the line. (C) 2000 Academic Press.