Quasi-Periodic Bifurcations and Chaos

A natural phenomenon in applications is the interaction of quasi-periodic solutions of dynamical systems in a dissipative setting. We study the interactions of two of such ODE systems based on the construction of a nonlinear oscillator with thermostatic (energy) control. This leads to the emergence of complexity, torus doubling, and chaos. We find canards; 1-, 2-, and 3-tori; chaos, and hyperchaos. Detailed analysis is possible in the case of small oscillations and small interactions. Large-scale phenomena are studied by the construction of charts of parameter space using Lyapunov exponents.