Validated saddle-node bifurcations and applications to lattice dynamical systems

58. Evelyn Sander, Thomas Wanner:
    Validated saddle-node bifurcations and applications to lattice dynamical systems
    SIAM Journal on Applied Dynamical Systems 15(3), pp. 1690-1733, 2016.

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 for a continuum model. In the present paper, we show that computer-assisted proof techniques can be used to answer some of these questions. We apply these tools to the discrete Allen-Cahn equation, giving us results on the existence of branches of mosaic solutions and their robustness as it relates to grain size. We also demonstrate that there are situations in which classical continuation methods can fail to identify the correct branching behavior.

Links

The published version of the paper can be found at https://doi.org/10.1137/16M1061011.

Bibtex

@article{sander:wanner:16a,
   author = {Evelyn Sander and Thomas Wanner},
   title = {Validated saddle-node bifurcations and applications to lattice
            dynamical systems},
   journal = {SIAM Journal on Applied Dynamical Systems},
   volume = {15},
   year = {2016},
   number = {3},
   pages = {1690--1733},
   doi = {10.1137/16M1061011}
   }