Brian_Spears

Brian_Spears - 1 Topological Bifurcations of Knotted Tori in

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Unformatted text preview: 1 Topological Bifurcations of Knotted Tori in Quasiperiodically Driven Oscillators Brian Spears with Andrew Szeri and Michael Hutchings University of California at Berkeley Caltech CDS Seminar October 24, 2003 2 Introduction z Description of system z Techniques for analysis z Presence of knotted 2-torus attractors z Characterization of the tori z Topological torus bifurcations (TTBs) z TTBs near QP resonance z Approximation of tori by method of multiple scales 3 System Equation z Quadrupolar ion trap used to store charged particles z Model for axial motion z µ, ε, β are small; z α, γ are stability parameters for Mathieu equation z Stability parameters for unstable origin z ¡ f is irrational γ α , > µ ) )( 2 cos 2 cos ( 4 3 = + − − + + + z z t t z z f β ω ε α γ µ & & & 4 Genesis of Attractors z Periodic excitation z (q 1 ,q 2 , θ 1 ) L R 2 x S 1 z 2-periodic limit cycle z (2,1)-torus knot 2 1 q q = & ) )( cos ( 4 3 1 1 1 2 2 q q q q β θ α γ µ + − + − − = & 2 1 = θ & 5 Genesis of Attractors z Quasiperiodic excitation 2 1 q q = & ) )( cos cos ( 4 3 1 1 2 1 2 2 q q q q β θ ε θ α γ µ + − − + − − = & 2 1 = θ & f ω θ 2 2 = & ( ) { } 10 1 2 2 , 10 θ θ θ θ = × ∈ = Σ T R q ( ) { } 20 2 2 2 , 20 θ θ θ θ = × ∈ = Σ T R q z (q 1 ,q 2 , θ 1, θ 2 ) L R 2 x T 2 z Curves codimension 3, surfaces codimension 2 z Define two Poincaré sections z Study the iterated action of each map 6...
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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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Brian_Spears - 1 Topological Bifurcations of Knotted Tori in

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