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 1
Introduction Description of system Techniques for analysis Presence of knotted 2-torus attractors Characterization of the tori Topological torus bifurcations (TTBs) TTBs near QP resonance Approximation of tori by method of multiple scales 2
System Equation Quadrupolar ion trap used to store charged particles Model for axial motion µ, ε, β are small; α, γ are stability parameters for Mathieu equation Stability parameters α, γ for unstable origin is irrational f && z + µ z& + 4( γ + α cos 2t ε cos 2ω f t)( z + βz µ > 0 3 ) = 0 3
Genesis of Attractors Periodic excitation q & 1 = q 2 q& = µ q 4( γ + α cosθ )( q + β 3 2 2 1 1 q1 θ& 1 = 2 (q 1,q 2, θ 1 ) R 2 x S 1 2-periodic limit cycle (2,1)-torus knot ) 4
Genesis of Attractors 5 Quasiperiodic excitation q & 1 = q 2 q& = µ q2 4( γ + α cosθ1 ε cosθ2)( q1 + β & θ 1 = 2 & θ 2 = 2ω Σ 3 2 q1 θ f (q 1,q 2,θ 1, θ 2 ) R 2 x T 2 Curves codimension 3, surfaces codimension 2 Define two Poincaré sections Σ θ ( q, θ ) { 2 2 R T θ = } 10 = θ ( q, θ ) 1 10 { 2 2 R T θ = } 20 = θ Study the iterated action of each map 2 20 )
Attractor Cross Sections Instructive case = 0 ( θ decouples) ε 2 Points cover braids densely in both sections Stable normally hyperbolic invariant manifolds (NHIMs) Cross sections of a knotted torus in R 2 x T 2 Multiple braid components; connected torus 6 θ 10 θ20 Σ closure Σ closure
Attractor Cross Sections ε = 0 θ2 Instructive case ( decouples) Points cover braids densely in both sections Stable normally hyperbolic invariant manifolds (NHIMs) Cross sections of a knotted torus in R 2 x T 2 7
Attractor Cross Sections For ε 0 equations coupled Different braids in both sections Stable, attracting NHIMs Different number of strands How shall we characterize tori? 8
Attractor Cross Sections For ε 0 equations coupled Different braids in both sections Stable, attracting NHIMs Different number of strands How shall we characterize tori? 9
Classifying the Torus Describe the torus by a map F : T 2 C n Theorem: The set of homotopy classes of such maps is in one-to-one correspondence with the set of braid pairs a,b on n strands, modulo equivalence ( ) [ 2 ] n T, C ~ ( a, b) ( a b) ~ ( a, b ) { B B ab = ba} ~, ( ) ( 1 1 a, b = cac, cbc ) n n 10 We distinguish among topologically distinct tori using the braids in the two Poincaré sections
Topological Torus Bifurcation Bifurcation: attractors before and after are not isotopic As ε, torus changes equivalence class Use pair of braids to observe change Topological torus bifurcation (TTB) 11
Topological Torus Bifurcation Parent braids become saddle-type Each parent strand sheds 2 daughter strands Strands form boundary of the unstable 2-manifold of the saddle-type braid Daughter braids are stable TTB of the doubling type, or TTBD 12
Topological Torus Bifurcation Implications for the flow Local: each patch of torus splits into two Global: patches connected to form one torus of different knot type Tori and their invariant manifolds organize the flow 13
Type-I TTBD The number of components in each braid is preserved The unstable 2-manifold is non-orientable The saddle-type knot is Möbius The daughter knot is a cable of the parent Analogue of period doubling in 3-D flows, but here of invariant manifolds 14
15 Type-I TTBD
16 Type-I TTBD
17 Type-I TTBD
18 Type-I TTBD
19 Type-I TTBD
20 Type-I TTBD
Type-II TTBD The number of components in one braid doubles The unstable 2-manifold is orientable The saddle-type knot is hyperbolic The daughter knot is not a cable of the parent The multiple components are covered by a single orbit 21
22 Type-II TTBD
Type-II TTBD 1 2 1 2 23
24 Type-II TTBD
25 Type-II TTBD
Type-II TTBD 1 3 1 2 4 2 1 2 3 4 1 2 3 4 26
27 Type-II TTBD
28 Type-II TTBD
Linkedness of Attractors Linking data computed from crossings Tori are self-linked Tori are linked with one another Linking data is isotopy invariant + - 29
Lyapunov Exponents Calculate all 3 exponents for each map Attractors: 0 = λ1 > λ2 λ3 λ > = λ > Saddles: Near bifurcation 1 0 2 λ3 2 passes through zero for parent Loss of normal hyperbolicity Daughter again stable 30
Power Spectra TTBDs reflected in easily observable dynamics Complicated attractors imply complicated dynamics Power spectra for flow time-history show additional peaks 31
Physical Significance TTBs bound narrow resonance peak Useful for control of ion motion or ejection Two-frequency excitation critical 32
Modified scheme Method of Multiple Scales Gives resonance conditions and solutions ω = p + λ f,res z( t, τ ) = A( τ ) D2n cos(2n + λ) t + B( τ ) D2n sin(2n + λ) t n= Fast oscillation from linear Mathieu equation ~ Fundamental frequency λ = λ( α, γ ) Captures topology of attractors Helps locate saddle-type tori n= 33
Method of Multiple Scales Study the easier slow system TTBs appear as period doubling in the slow time amplitude equations B 34 A
Conclusions Damped, nonlinear, QP Mathieu equation possesses dynamics governed by knotted tori Tori identified by pairs of braids in Poincaré sections Bifurcations (TTBs) occur which double the number of strands change of knot type of torus in R 2 x T 2 TTBs appear exploitable in physical systems 35
Invariant Linking Data [isotopy invariance of linking data; How to compute it linking (self & others) (1) can t unlink tori (2) can distinguish by examining the data] 36