Torus Break-down and Bifurcations in Coupled Oscillators

To discover qualitative changes of solutions of differential equations, one has to study their bifurcations. We start with the well- known bifurcations of equilibria leading to periodic solutions, followed by bifurcations of periodic solutions (Neimark-Sacker and Hopf-Hopf) in a dissipative setting leading to quasi-periodic motion corresponding with tori. In their turn these families of quasi-periodic solutions may bifurcate to produce strange attractors. A number of examples illustrate the theory.