Validated Saddle-Node Bifurcations and Applications to Lattice Dynamical Systems

Abstract

The use of rigorous verification methods is a powerful tool which permits progress in the analysis of dynamical processes that is not possible using purely analytical techniques. In this paper we develop a set of tools for branch validation, which allows for the rigorous verification of branch behavior, bifurcation, and solution index on branches generated through a saddle-node bifurcation. While the presented methodology can be applied in a variety of settings, we illustrate the use of these tools in the context of materials science. In particular, lattice models have been proposed as a more realistic reflection of the behavior of materials than traditional continuum models. For example, unlike their continuum counterparts, lattice models can account for phenomena such as pinning, and a significant body of work has been developed to study traveling waves. However, in a variety of other contexts such as bifurcation theory, questions about lattice dynamical systems are significantly harder to answer than t...