Persistence of resonant torus doubling bifurcation under polynomial perturbations

Abstract

Resonant torus doubling bifurcation occurs in discrete maps of three or more dimensions. To date, only three discrete maps have been known to exhibit such bifurcations namely Shilnikov map, generalised Hénon map, and quadratic map. In the present study, we explore if the resonant torus doubling bifurcation is persistent under perturbations. In a simplest case, we apply polynomial perturbations of various degrees. Interestingly, different maps even though all have quadratic nonlinearity showcase different existence intervals in the perturbation parameter space. A novel algorithm is used to detect the doubling bifurcation in two-dimensional parameter space. Furthermore, we explore different kinds of codimension-one bifurcation occurring near the non-existence of the doubling bifurcation. It is found in all three maps that near such bifurcation, (a) instead of eigenvalues crossing $-1$ for a period-doubling bifurcations, rather develop complex conjugate eigenvalues and undergo Neimark–Sacker bifurcation, (b) they form a resonant torus bubble which remains robust and no other bifurcations takes place. Moreover, we also explore the persistence of quasiperiodic torus doubling bifurcation in a similar way. For both resonant torus doubling bifurcation and quasiperiodic torus doubling bifurcation, it is found that Shilnikov map and generalised Hénon map showcase persistence in wide intervals of perturbations while the quadratic map is not as persistent and the resonant torus ceases to exist with even smaller range of perturbations.