NONAUTONOMOUS BIFURCATION OF BOUNDED SOLUTIONS: CROSSING CURVE SITUATIONS

Carathéodory differential equations naturally occur as path-wise realization of random differential equations and are amenable for deterministic calculus. In the setup of such nonautonomous differential equations with only measurable time-dependence, we present an approach to a bifurcation theory based on a topological change in the set of bounded entire solutions. In such a setting of at least planar equations, we provide sufficient criteria for a nonhyperbolic entire solution to bifurcate into two branches of bounded or homoclinic solutions. As opposed to transcritical or pitchfork bifurcations, no trivial solution branch is supposed to exist in advance. In particular, we discuss a degenerate fold bifurcation pattern, where the transversality assumption is replaced by a nondegeneracy condition on the second-order derivative. Both bifurcation patterns are intrinsically nonautonomous and do not occur for time-invariant equations. Our notion of a nonhyperbolic solution is based on the fact that the associate variational equation possesses exponential dichotomies on both semiaxes with compatible projectors. The resulting Fredholm theory allows one to apply recent abstract bifurcation results due to Liu, Shi and Wang (2007).