2021/08/17 by Muni, Sishu Shankar, McLachlan, Robert I., Simpson, David J. W. · 1 citation
#Chaotic Dynamics (nlin.CD) #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences
paper · doi:10.48550/arxiv.2108.07476
Global resonance is a mechanism by which a homoclinic tangency of a smooth map can have infinitely many asymptotically stable, single-round periodic solutions. To understand the bifurcation structure one would expect to see near such a tangency, in this paper we study one-parameter perturbations of typical globally resonant homoclinic tangencies. We assume the tangencies are formed by the stable and unstable manifolds of saddle fixed points of two-dimensional maps. We show the perturbations display two infinite sequences of bifurcations, one saddle-node the other period-doubling, between which single-round periodic solutions are asymptotically stable. Generically these scale like |λ|2 k, as k → ∞, where -1 < λ< 1 is the stable eigenvalue associated with the fixed point. If the perturbation is taken tangent to the surface of codimension-one homoclinic tangencies, they instead scale like (|λ|k)/(k). We also show slower scaling laws are possible if the perturbation admits further degeneracies.