A new route to chaos: Sequences of topological torus bifurcations

We consider a sequence of topological torus bifurcations (TTBs) in a nonlinear, quasiperiodicMathieu equation. The sequence of TTBs and an ensuing transition to chaos are observed by computing the principal Lyapunov exponentover a range of the bifurcation parameter. We also consider the effect of the sequence on the power spectrum before and after the transition to chaos. We then describe the topology of the set of knotted tori that are present before the transition to chaos. Following the transition, solutions evolve on strange attractors that have the topology of fractal braidsin Poincare sections. We examine the topology of fractal braidsand the dynamics of solutions that evolve on them. We end with a brief discussion of the number of TTBs in the cascade that leads to chaos.