Multiple Homoclinic Bifurcations From Orbit-Flip I: Successive Homoclinic Doublings

Abstract

The purpose of this and forthcoming papers is to obtain a better understanding of complicated bifurcations for multiple homoclinic orbits. We shall take one particular type of codimension two homoclinic orbits called orbit-flip and study bifurcations to multiple homoclinic orbits appearing in a tubular neighborhood of the original orbit-flip. The main interest of the present paper lies in the occurrence of successive homoclinic doubling bifurcations under an appropriate condition, which is a part of the entire bifurcation for multiple homoclinic orbits. Since this is a totally global bifurcation, we need the aid of numerical experiments for which we must choose a concrete set of ordinary differential equations that exhibits the desired bifurcation. In this paper we employ a family of continuous piecewise-linear vector fields for such a model equation. In order to explain the cascade of homoclinic doubling bifurcations theoretically, we also derive a two-parameter family of unimodal maps as a singular limit of the Poincaré maps along homoclinic orbits. We locate bifurcation curves for this family of unimodal maps in the two-dimensional parameter space, which basically agree with those for the piecewise-linear vector fields. In particular, we show, using a standard technique from the theory of unimodal maps, that there exists an infinite sequence of doubling bifurcations which corresponds to the sequence of homoclinic doubling bifurcations for the piecewise-linear vector fields described above. Since our unimodal map has a singularity at a boundary point of its domain of definition, the doubling bifurcation is slightly different from that for standard quadratic unimodal maps, for instance the Feigenbaum constant associated with the accumulation of the doubling bifurcations is different from the standard value 4.6692.…

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

“…Our investigations confirm the theory in [6,19] about the existence of a homoclinic-doubling cascade and closely correspond to the results of [5] where a piecewise-linear vector field and a one-dimensional map modelling a general homoclinic-doubling cascade are considered. The novel approach in this paper is that a smooth vector field is used, which can be seen as typical, so that it serves as a model for what homoclinic-doubling cascades look like in applications.…”

“…The simplest consistent way seems to be the one depicted in Figure 6. This was conjectured in [6] and indicated by the numerics on a piecewise-linear vector field in [5]. As we take a closer look, the homoclinic orbit at the curve ➀ ☎ ⑧ which originates from the flip at ✬ ✮ ➆ undergoes a further homoclinic flip bifurcation itself.…”

“…This behaviour was theoretically studied in [5] and [7]. In [5] a numerical investigation is performed of a piecewise-linear vector field, which is complemented and compared with results for an appropriate normal form, given by a one-dimensional unimodal map. The authors compare the scaling of the distances in parameter space between successive ✬ ➧ ✭ -points with the Feigenbaum constant of…”

“…As a second parameter is changed, the period of each ✑ ✓ ✒ -periodic limit cycle approaches infinity and in the limit a homoclinic-doubling cascade is obtained. This was first studied numerically in [5] for a piecewise-linear vector field and then proved to exist in generic families of vector fields in [6]. …”

“…Note that the accumulation rate of the points ✬ is not the same as the usual Feigenbaum number of the period doublings. In fact there are two scaling constants and these depend on the eigenvalues of the saddle points as is explained in Section 4 below; see also [5,7].…”