Existence, Multiplicity, and Bifurcation in Systems of Ordinary Differential Equations

We prove new non-resonance conditions for boundary value problems for two dimensional systems of ordinary differential equations. We apply these results to the existence of solutions to nonlinear problems. We then study global bifurcation for such systems of ordinary differential equations Rotation numbers are associated with solutions and are shown to be invariant along bifurcating continua. This invariance is then used to analyze the global structure of the bifurcating continua, and to demonstrate the existence of multiple solutions to some boundary value problems.