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 onedimensional 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 R, the lowest possible dimension in which such a heteroclinic cycle can occur, but are expected to be applicable to systems of higher dimension 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 Poincaré map in a full neighbourhood of the cycle in both phase and parameter space. Curves of orbits homoclinic to the equilibrium are located in a generic parameter plane, as are curves of homoclinic tangencies to the periodic orbit. Moreover, it is shown how curves of folds of periodic orbits, which have different asymptotics near the homoclinic bifurcation of the equilibrium and the homoclinic bifurcation of the periodic orbit, are glued together near the codimension-two point. A simple global assumption is made about the existence of a pair of codimension-two heteroclinic cycles corresponding to a first and last tangency of the stable manifold of the equilibrium and the unstable manifold of the periodic orbit. Under this assumption, it is shown how the locus of homoclinic orbits to the equilibrium should oscillate in the parameter space, a phenomenon known as homoclinic snaking. Finally, we present several numerical examples of systems that arise in applications, which corroborate and illustrate our theory.