Applications Of Functional-Analytic Methods In Nonlocal And Dynamical System Problems

Abstract

This paper is concerned with functional analytic techniques in problems related to dynamical systems and contains two parts. In the first part we show bifurcation of localized spike solutions from spatially constant states in systems of nonlocally coupled equations in the whole space. The main assumptions are a generic bifurcation of saddle-node or transcritical type for spatially constant profiles, and a symmetry and second moment condition on the convolution kernel. Rather than relying on center manifolds methods, we pursue a more direct approach, deriving leading order asymptotics and Newton corrections for error terms. We are able to extend well-known results for spots, spikes, and fronts in locally coupled systems on the real line, and for radially symmetric profiles in higher space dimensions. In the second part, we revisit the classical problem of determining the asymptotic expansion of the solution near the passage of a fold point in a singularly perturbed system, where the theory of normally hyperbolic invariant manifold cannot be directly applied. While the standard remedy is the blow-up method first demonstrated by Krupa and Szmolyan, we will show how one can use a functional analytic approach to achieve the same goal.