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knotted_tori_oct03

knotted_tori_oct03 - R 2 × S 1 ×{• and R 2 ×{• × S...

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Knotted tori and topological bifurcations in quasiperiodically driven oscillators We examine the solutions to a damped, quasiperiodic (QP) Mathieu equation with cubic nonlinearities. The system is suspended in a four-dimensional phase space R 2 × T 2 in which there exist attracting, knotted 2-tori. We develop a topological approach to the study of such tori in which a 2-torus is characterized by two closed braids that exist in two Poincaré sections R
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Unformatted text preview: R 2 × S 1 × {•} and R 2 × {•} × S 1 . Changes of a single parameter lead to a global bifurcation through which the attracting torus loses stability and sheds a torus of different knot type–a topological torus bifurcation (TTB). We show that a pair of TTBs is responsible for sharp, large-amplitude resonances. We develop a variant of the method of multiple scales to study knotted tori and TTBs near this resonance. 1...
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