Bifurcations of quasi-periodic dynamics: torus breakdown

To study the dynamics of quasi-periodic bifurcations, we consider a system of two nonlinearly coupled oscillators using averaging, continuation and numerical bifurcation techniques. This relatively simple system displays considerable complexity. Assuming the internal resonance to be 1:2, we find a 2π-periodic solution which undergoes a supercritical Neimark– Sacker bifurcation, yielding a stable torus. Choosing a route in parameter space, we show by numerical bifurcation techniques how the torus gets destroyed by dynamical and topological changes in the involved manifolds (Krauskopf and Osinga in J Comput Phys 146:404–419, 1998). The 1:6 resonance turns out to be prominent in parameter space, and we detected a cascade of period doublings within the corresponding resonance tongue yielding a strange attractor. The phenomena agree with the Ruelle–Takens (Commun. Math. Phys. 20:167–192, 1971, Commun. Math. Phys. 23:343–344, 1971) scenario leading to strange attractors. Other periodic regimes are present in this system, and there is interesting evidence that two different regimes interact with each other, yielding yet another type of strange attractor. In this context, certain π-periodic solutions emerge that are studied by continuation following the Poincar´e–Lindstedt method using Mathieu functions; when the implicit function theorem breaks down, the analysis is supplemented by numerical bifurcation techniques.