Topological Bifurcations of Attracting 2-Tori of Quasiperiodically Driven Oscillators

AbstractWe examine the solutions to a damped, quasiperiodic (QP) Mathieu equation with cubic nonlinearities. The system is suspended in a four-dimensional phase space ℝ2 × T2 in which there exist attracting, knotted 2-tori called torus braids. We develop a topological classification scheme in which a torus braid is characterized by closed braids that exist in two Poincare sections, ℝ2 \times S1 × {·} and ℝ2 × {·} \times S1. Based on the classification scheme, we develop numerical invariants that describe the linkedness of attractors and provide information about the global dynamics. Numerical simulations show that changes of a single parameter lead to a global bifurcation through which the attracting torus loses stability and locally "doubles," shedding a torus of different equivalence class. We call this a topological torus bifurcation of the doubling variety (TTBD). We provide a topological analysis of the doubling produced by TTBDs and we examine the qualitative dynamical changes that result. We also examine the effect of TTBDs on the spectrum of Lyapunov exponents and the time series power spectrum.