Unfolding a Tangent Equilibrium-to-Periodic Heteroclinic Cycle

Abstract

The dynamics occurring near a heteroclinic cycle between a hyperbolic equilibrium and a hyperbolic periodic orbit is analyzed. The case of interest is when the equilibrium has a one-dimensional unstable manifold and a two-dimensional stable manifold while the stable and unstable manifolds of the periodic orbit are both two-dimensional. A codimension-two heteroclinic cycle occurs when there are two codimension-one heteroclinic connections, with the connection from the periodic orbit to the equilibrium corresponding to a tangency between the two relevant manifolds. The results are restricted to $\mathbb{R}^3$, the lowest possible dimension in which such a heteroclinic cycle can occur, but are expected to be applicable to systems of higher dimensions as well. A geometric analysis is used to partially unfold the dynamics near such a heteroclinic cycle by constructing a leading-order expression for the Poincare map in a full neighborhood of the cycle in both phase and parameter space. Curves of orbits homoclin...