Critical Manifolds, Travelling Waves and an Example from Population Genetics.
Abstract
A generalized Morse index theory is used to study the existence of travelling wave solutions of a diffusion-reaction system of equations. The reaction system is assumed to be 'close' to one which admits an attracting manifold of critical points. A scaling argument is used to see that the equations for travelling waves of the full system are then close to a system with a normally hyperbolic manifold of critical points. Standard perturbation theorems are already available to study the behavior of solutions of the 'perturbed' system which lie near the critical manifold in terms of a (derived) system of 'slow' equations on the manifold itself. Here, another such theorem, dealing with aspects of the system which can be described in terms of isolated invariant sets, is proved. Specifically, it states that isolated invariant sets of slow equations correspond to isolated invariant sets of the full system, and that the Morse index of the latter set is an n-fold suspension of that of the former where n is the number of unstable normal directions.
Document Details
- Document Type
- Technical Report
- Publication Date
- Nov 01, 1980
- Accession Number
- ADA096659
Entities
People
- C. Conley
- P. Fife
Organizations
- University of Wisconsin–Madison