A Lin's method approach to finding and continuing heteroclinic connections involving periodic orbits

We present a numerical method for finding and continuing heteroclinic connections of vector fields that involve periodic orbits. Specifically, we concentrate on the case of a codimension-d heteroclinic connection from a saddle equilibrium to a saddle periodic orbit, denoted EtoP connection for short. By employing a Lin's method approach we construct a boundary value problem that has as its solution two orbit segments, one from the equilibrium to a suitable section Sigma and the other from Sigma to the periodic orbit. The difference between their two end points in Sigma can be chosen in a d-dimensional subspace, and this gives rise to d well-defined test functions that are called the Lin gaps. A connecting orbit can be found in a systematic way by closing the Lin gaps one-by-one in d consecutive continuation runs. Indeed, any common zero of the Lin gaps corresponds to an EtoP connection, which can then be continued in system parameters. The performance of our method is demonstrated with a number of examples. First, we continue codimension-one EtoP connections and the associated heteroclinic EtoP cycles in the Lorenz system. We then consider a three-dimensional model vector field for the dynamics near a saddle-node Hopf bifurcation with global reinjection to show that our method allows us to complete a complicated bifurcation diagram involving codimension-one EtoP connections. With the example of a four-dimensional Duffing-type system we then demonstrate how a codimension-two EtoP connection can be found by closing two Lin gaps in succession. Finally, we show that our geometric approach can be used to find a codimension-zero heteroclinic connection between two saddle periodic orbits in a four-dimensional vector field.