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

This paper studies a natural mechanism, called a homoclinic-doubling cascad e, for the disappearance of period-doubling cascades in vector fields. Simp ly put, an entire period-doubling cascade collides with a saddle-type equil ibrium. Homoclinic-doubling cascades are known to have self-similar structu re. in contrast to the well-known Feigenbaum constant, the scaling constant s for homoclinic-doubling depend on the eigenvalues of the saddle equilibri um. Specifically, we present here for the first time a detailed study of ho moclinic-doubling cascades in a smooth vector field, namely a three-dimensi onal polynomial model proposed by Sandstede. A numerical algorithm is prese nted for computing homoclinic-doubling cascades in general vector fields, w hich makes use of the program AUTO/HoMCoNT. This allows us to compute two t ypes of homoclinic-doubling cascades. one where the primary homoclinic orbi t undergoes an inclination flip bifurcation and one where it undergoes an o rbit flip bifurcation. Our results bring out the scaling constants in good agreement with analytical estimates obtained from one-dimensional maps. (C) 2000 Elsevier Science B.V. All rights reserved.