# Lyapunov-Schmidt Reduction for Unfolding Heteroclinic Networks of Equilibria and Periodic Orbits with Tangencies - Mathematics > Dynamical Systems

Lyapunov-Schmidt Reduction for Unfolding Heteroclinic Networks of Equilibria and Periodic Orbits with Tangencies - Mathematics > Dynamical Systems - Download this document for free, or read online. Document in PDF available to download.

Abstract: This article concerns arbitrary finite heteroclinic networks in any phasespace dimension whose vertices can be a random mixture of equilibria andperiodic orbits. In addition, tangencies in the intersection of un-stablemanifolds are allowed. The main result is a reduction to algebraic equations ofthe problem to find all solutions that are close to the heteroclinic networkfor all time, and their parameter values. A leading order expansion is given interms of the time spent near vertices and, if applicable, the location on thenon-trivial tangent directions. The only difference between a periodic orbitand an equilibrium is that the time parameter is discrete for a periodic orbit.The essential assumptions are hyperbolicity of the vertices and transversalityof parameters. Using the result, conjugacy to shift dynamics for a generichomoclinic orbit to a periodic orbit is proven. Finally,equilibrium-to-periodic orbit heteroclinic cycles of various types areconsidered.

Author: ** Jens D.M. Rademacher**

Source: https://arxiv.org/