A study of resonance tongues near a Chenciner bifurcation using MatcontM

Abstract

MatcontM is a matlab toolbox for numerical analysis of bifurcations of fixed points and periodic orbits of maps. It computes codim 1 bifurcation curves and supports the computation of normal coefficients including branch switching from codim 2 points to secondary curves. Recently, the initialization and computation of connecting orbits was improved. Moreover, a graphical user interface was added enabling interactive control of all these computations. To further support these computations it allows to compute orbits of the map and its iterates and to represent them in 2D, 3D and numeric windows. We demonstrate the use of the toolbox in a study of Arnol’d tongues near a degenerate Neimark-Sacker (Chenciner) bifurcation. Here we illustrate the recent theory of [4] how resonance tongues interact with a quasi-periodic saddle-node bifurcation of invariant curves in maps. Using normal form coefficients we find evidence for one of their cases, but not the other. Actually, we find another unfolding, i.e. a third possibility. We also find a structure that resembles a quasi-periodic cusp bifurcation of invariant curves.