knotted_tori_oct03 - R 2 S 1 {} and R 2 {} S 1 . Changes of...

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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
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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 dierent knot typea 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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This note was uploaded on 01/04/2012 for the course CDS 280 taught by Professor Marsden,j during the Fall '08 term at Caltech.

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