On the Poincare Index of Isolated Invariant Sets
M.R. Razvan, M. Fotouhi Firoozabad
TL;DR
This paper applies Conley index theory to analyze the Poincare index of isolated invariant sets, establishing conditions for critical points and demonstrating the existence of infinitely many homoclinic orbits when the index exceeds one.
Contribution
It introduces new limiting conditions for critical points to be isolated invariant sets and links the Poincare index to the existence of homoclinic orbits.
Findings
Critical points with Poincare index > 1 have infinitely many homoclinic orbits.
Derived conditions for a critical point to be an isolated invariant set.
Applied Conley index theory to planar vector fields.
Abstract
In this paper, we use Conley index theory to examine the Poincare index of an isolated invariant set. We obtain some limiting conditions on a critical point of a planar vector field to be an isolated invariant set. As a result we show the existence of infinitely many homoclinic orbits for a critical point with the Poincare index greater than one.
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Taxonomy
TopicsAdvanced Differential Equations and Dynamical Systems · Homotopy and Cohomology in Algebraic Topology · Advanced Topics in Algebra