Death of period-doublings: locating the homoclinic-doubling cascade

This paper studies a natural mechanism, called a homoclinic-doubling cascade, for the disappearance of period-doubling cascades in vector fields. Simply put, an entire period-doubling cascade collides with a saddle-type equilibrium. Homoclinic-doubling cascades are known to have self-similar structure. In contrast to the well-known Feigenbaum constant, the scaling constants for homoclinic-doubling depend on the eigenvalues of the saddle equilibrium. Specifically, we present here for the first time a detailed study of homoclinic-doubling cascades in a smooth vector field, namely a three-dimensional polynomial model proposed by Sandstede. A numerical algorithm is presented for computing homoclinic-doubling cascades in general vector fields, which makes use of the program AUTO/HomCont. This allows us to compute two types of homoclinic-doubling cascades, one where the primary homoclinic orbit undergoes an inclination flip bifurcation and one where it undergoes an orbit flip bifurcation. Our results bring out the scaling constants in good agreement with analytical estimates obtained from one-dimensional maps.