Topological Bifurcations of Attracting 2-Tori of Quasiperiodically Driven Oscillators
2005
We 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 schemein which a
torus
braidis characterized by closed
braidsthat 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
torusloses stability and locally "doubles," shedding a
torusof different
equivalence class. We call this a topological
torusbifurcation 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.
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