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Parabolic Resonance: A Route to Hamiltonian Spatio-Temporal Chaos

Parabolic Resonance: A Route to Hamiltonian Spatio-Temporal Chaos. Eli Shlizerman and Vered Rom-Kedar Weizmann Institute of Science. Publications:. [1] ES & VRK , Hierarchy of bifurcations in the truncated and forced NLS model, CHAOS-05.

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Parabolic Resonance: A Route to Hamiltonian Spatio-Temporal Chaos

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  1. Parabolic Resonance: A Route to Hamiltonian Spatio-Temporal Chaos Eli Shlizerman and Vered Rom-Kedar Weizmann Institute of Science Publications: [1] ES & VRK, Hierarchy of bifurcations in the truncated and forced NLS model,CHAOS-05 [2]ES & VRK,Three types of chaos in the forced nonlinear Schrödinger equation,PRL-06 [3]ES & VRK,Parabolic Resonance: A route to intermittent spatio-temporal chaos,SUBMITTED [4]ES & VRK,Geometric analysis and perturbed dynamics of bif. in the periodic NLS,PREPRINT Stability and Instability in Mechanical Systems, Barcelona, 2008 http://www.wisdom.weizmann.ac.il/~elis/

  2. The perturbed NLS equation The Problem Periodic NLS (Review) ODE Phase Space and Bifurcations PDE Phase Space Description Spatio-Temporal Chaos Formulation of Results forcing damping dispersion focusing • Change variables to oscillatory frame • To obtain the autonomous NLS +damping : [Bishop, Ercolani, McLaughlin 80-90’s]

  3. The autonomous NLS equation The Problem Periodic NLS (Review) ODE Phase Space and Bifurcations PDE Phase Space Description Spatio-Temporal Chaos Formulation of Results • Boundary • Periodic B(x+L,t) = B(x,t) • Even (ODE) B(-x,t) = B(x,t) • Parameters • Wavenumberk = 2π/L • Forcing FrequencyΩ2

  4. The problem Classify instabilities near the plane wave in the NLS equation Route to Spatio-Temporal Chaos The Problem Periodic NLS (Review) ODE Phase Space and Bifurcations PDE Phase Space Description Spatio-Temporal Chaos Formulation of Results Regular Solution in time: almost periodic in space: coherent Temporal Chaos in time: chaotic in space: coherent Spatio-Temporal Chaos in time: chaotic in space: decoherent

  5. Main Results Decompose the solutions to first two modes and a remainder: And define: ODE: The two-degrees of freedom parabolic resonance mechanism leads to an increase of I2(T) even if we start with small, nearly flat initial data and with small ε. PDE: Once I2(T) is ramped up the solution of the forced NLS becomes spatially decoherent and intermittent - We know how to control I2(T) hence we can control the solutions decoherence. The Problem Periodic NLS (Review) ODE Phase Space and Bifurcations PDE Phase Space Description Spatio-Temporal Chaos Formulation of Results

  6. Integrals of motion • Define: • Integrable case (ε = 0): Periodic NLS (Review) ODE Phase Space and Bifurcations The Problem PDE Phase Space Description Spatio-Temporal Chaos Formulation of Results Infinite number of constants of motion: I,H0, … • Perturbed case (ε≠ 0): • The total energy is preserved: • All others are not! I(t) != I0 HT=H0 + εH1

  7. Re(B(0,t)) Re(B(0,t)) θ₀ θ₀ Im(B(0,t)) Im(B(0,t)) The plane wave solution Periodic NLS (Review) ODE Phase Space and Bifurcations The Problem PDE Phase Space Description Spatio-Temporal Chaos Formulation of Results Non Resonant: Resonant:

  8. Linear Unstable Modes (LUM) • The plane wave is unstable for 0 < k2 < 2|c|2 • Since the boundary conditions are periodic k is discretized: kj = 2πj/L for j = 0,1,2… (j - number of LUMs) • Then the condition for instability becomes the discretized condition j2 (2π/L)2/2 < |c|2 < (j+1)2 (2π/L)2/2 • The solution has j Linear Unstable Modes (LUM). As we increase the amplitude the number of LUMs grows. Ipw = |c|2, IjLUM = j2k2/2 Periodic NLS (Review) ODE Phase Space and Bifurcations The Problem PDE Phase Space Description Spatio-Temporal Chaos Formulation of Results

  9. The plane wave solution Heteroclinic Orbits! Periodic NLS (Review) ODE Phase Space and Bifurcations The Problem PDE Phase Space Description Spatio-Temporal Chaos Formulation of Results Bh Bh Re(B(0,t)) Re(B(0,t)) θ₀ θ₀ Bpw Bpw Im(B(0,t)) Im(B(0,t))

  10. Modal equations Consider two mode Fourier truncation B(x , t) = c(t) + b (t) cos (kx) Substitute into the unperturbed eq.: ODE Phase Space and Bifurcations The Problem Periodic NLS (Review) PDE Phase Space Description Spatio-Temporal Chaos Formulation of Results [Bishop, McLaughlin, Ercolani, Forest, Overmann ]

  11. General Action-Angle Coordinates For b≠0 , consider the transformation: Then the system is transformed to: We can study the structure of ODE Phase Space and Bifurcations The Problem Periodic NLS (Review) PDE Phase Space Description Spatio-Temporal Chaos Formulation of Results [Kovacic]

  12. Preliminary step - Local Stability ODE Phase Space and Bifurcations The Problem Periodic NLS (Review) PDE Phase Space Description Spatio-Temporal Chaos Formulation of Results B(X , t) = [|c| + (x+iy) coskX ] eiγ validity region [Kovacic & Wiggins 92’]

  13. y x ODE Phase Space and Bifurcations The Problem Periodic NLS (Review) PDE Phase Space Description Spatio-Temporal Chaos Formulation of Results PDE-ODE Analogy ODE Bpw=Plane wave +Bsol=Soliton (X=0) -Bsol=Soliton (X=L/2) PDE +Bh=Homoclinic Solution -Bh=Homoclinic Solution

  14. Hierarchy of Bifurcations Level 1 Single energy surface -EMBD, Fomenko Level 2 Energy bifurcation values -Changes in EMBD Level 3 Parameter dependence of the energy bifurcation values -k, Ω ODE Phase Space and Bifurcations The Problem Periodic NLS (Review) PDE Phase Space Description Spatio-Temporal Chaos Formulation of Results

  15. Level 1: Singularity Surfaces Construction of the EMBD - (Energy Momentum Bifurcation Diagram) ODE Phase Space and Bifurcations The Problem Periodic NLS (Review) PDE Phase Space Description Spatio-Temporal Chaos Formulation of Results [Litvak-Hinenzon & RK - 03’]

  16. EMBD ODE Phase Space and Bifurcations The Problem Periodic NLS (Review) PDE Phase Space Description Spatio-Temporal Chaos Formulation of Results Iso-energy surfaces H4 H1 H3 H2 Parameters k and are fixed. Dashed – Unstable, Solid – Stable

  17. Level 2: Bifurcations in the EMBD ODE Phase Space and Bifurcations The Problem Periodic NLS (Review) PDE Phase Space Description Spatio-Temporal Chaos Formulation of Results Each iso-energy surface can be represented by a Fomenko graph 5* 6 4 Energy bifurcation value

  18. Possible Energy Bifurcations Folds - Resonances I H ODE Phase Space and Bifurcations The Problem Periodic NLS (Review) PDE Phase Space Description Spatio-Temporal Chaos Formulation of Results • Branching surfaces – Parabolic Circles • Crossings – Global Bifurcation [ Full classification: Radnovic + RK, RDC, Moser 80 issue, 08’ ]

  19. Level 3: Changing parameters, energy bifurcation values can coincide Parabolic Resonance: IR=IPk2=2Ω2 ODE Phase Space and Bifurcations The Problem Periodic NLS (Review) PDE Phase Space Description Spatio-Temporal Chaos Formulation of Results • Example: Parabolic Resonance for (x=0,y=0) • Resonance IR= Ω2 • hrpw = -½ Ω4 • Parabolic Circle Ip= ½ k2 • hppw = ½ k2(¼ k2-Ω2)

  20. Perturbed solutions classification ODE Phase Space and Bifurcations The Problem Periodic NLS (Review) PDE Phase Space Description Spatio-Temporal Chaos Formulation of Results Integrable - a point Perturbed – e slab in H0 • Away from sing. curve: • Regular / KAM type ? • Near sing. curve: • Standard phenomena (Homoclinic chaos, Elliptic circles) e e e e • Near energy bif. val.: • Special dyn phenomena (HR,PR,ER,GB-R …) e √e e

  21. I H0 I H0 I H0 ODE Phase Space and Bifurcations The Problem Periodic NLS (Review) PDE Phase Space Description Spatio-Temporal Chaos Formulation of Results Numerical simulations

  22. I H0 I H0 I H0 ODE Phase Space and Bifurcations The Problem Periodic NLS (Review) PDE Phase Space Description Spatio-Temporal Chaos Formulation of Results Numerical simulations – Projection to EMBD

  23. Bifurcations in the PDE Looking for the standing waves of the NLS The eigenvalue problem is received (Duffing system) Periodic b.c. select a discretized family of solutions! Phase space of the Duffing eq. ODE Phase Space and Bifurcations Periodic NLS (Review) PDE Phase Space Description Spatio-Temporal Chaos Formulation of Results The Problem Denote: solution

  24. Bifurcation Diagrams for the PDE We get a nonlinear bifurcation diagram for the different stationary solutions : Standard – vs. EMBD – vs. ODE Phase Space and Bifurcations Periodic NLS (Review) PDE Phase Space Description Spatio-Temporal Chaos Formulation of Results The Problem

  25. Classification of initial conditions in the PDE Unperturbed Perturbed KAM like ODE Phase Space and Bifurcations Periodic NLS (Review) PDE Phase Space Description Spatio-Temporal Chaos Formulation of Results The Problem Perturbed Chaotic

  26. θ₀ Previous: Spatial decoherence • For asymmetric initial data with strong forcing and damping (so there is a unique attractor) • Behavior is determined by the #LUM at the resonant PW: • Ordered behavior for 0 LUM • Temporal Chaos for 1 LUMs • Spatial Decoherence for 2 LUMs and above ODE Phase Space and Bifurcations Periodic NLS (Review) PDE Phase Space Description Spatio-Temporal Chaos Formulation of Results The Problem Temporal chaos Spatio-temporal chaos [D. McLaughlin, Cai, Shatah]

  27. New: Hamiltonian Spatio-temporal Chaos • All parameters are fixed: • The initial data B0(x) is almost flat, asymmetric for all solutions - δ=10-5. • The initial data is near a unperturbed stable plane waveI(B0) < ½k2 (0 LUM). • Perturbation is small,ε= 0.05. • Ω2 is varied: B0(x) δ |B| Bpw(x) x Ω2=1 Ω2=0.1 Ω2=0.225 ODE Phase Space and Bifurcations Periodic NLS (Review) PDE Phase Space Description Spatio-Temporal Chaos Formulation of Results The Problem

  28. Spatio-Temporal Chaos Characterization A solution B(x,t) can be defined to exhibit spatio-temporal chaos when: B(x,t) is temporally chaotic. The waves are statistically independent in space. When the waves are statistically independent, the averaged in time for T as large as possible, T → ∞, the spatial Correlation function decays at x = |L/2|. But not vice-versa. ODE Phase Space and Bifurcations Periodic NLS (Review) PDE Phase Space Description Spatio-Temporal Chaos Formulation of Results The Problem [Zaleski 89’,Cross & Hohenberg93’,Mclaughlin,Cai,Shatah 99’]

  29. The Correlation function • Properties: • Normalized, for y=0, CT(B,0,t)=1 • T is the window size • For Spatial decoherence, • the Correlation function decays. 1 ODE Phase Space and Bifurcations Periodic NLS (Review) PDE Phase Space Description Spatio-Temporal Chaos Formulation of Results The Problem Re(CT(B,y,T/2)) Coherent De-correlated |x/L|

  30. Intermittent Spatio-Temporal Chaos While the Correlation function over the whole time decays the windowed Correlation function is intermittent HR ER ODE Phase Space and Bifurcations Periodic NLS (Review) PDE Phase Space Description Spatio-Temporal Chaos Formulation of Results The Problem PR

  31. Choosing Initial Conditions Projecting the perturbed solution on the EMBD: Parabolic Resonant like solution ODE Phase Space and Bifurcations Periodic NLS (Review) PDE Phase Space Description Spatio-Temporal Chaos Formulation of Results The Problem • Decoherence can be characterized from the projection • “Composition” to the standing waves can be identified

  32. Conjecture / Formulation of Results • For any given parameter k, there exist εmin= εmin(k) such that for all ε > εminthere exists an order one interval of initial phases γ(0) and an O(√ε)-interval of Ω2 values centered at Ω2par that drive an arbitrarily small amplitude solution to a spatial decoherent state. ε STC √ε ODE Phase Space and Bifurcations Periodic NLS (Review) PDE Phase Space Description Spatio-Temporal Chaos Formulation of Results The Problem εmin(k) Ωpar Ω

  33. TheProblem Conjecture / Formulation of Results Periodic NLS (Review) Spatio-Temporal Chaos PDE Phase Space Description Formulation of Results • Here we demonstrated that such decoherence can be achieved with rather small εvalues (so εmin(0.9) ~ 0.05). • Coherence for long time scales may be gained by either decreasing εor by selecting Ω2 away from the O(√ε)-interval.

  34. Summary • We analyzed the ODE with Hierarchy of bifurcations and received a classification of solutions. • Analogously to the analysis of the two mode model we constructed an EMBD for the PDE and showed similar classification. • We showed the PR mechanism in the ODE-PDE. Initial data near an unperturbed linearly stable plane wave can evolve into intermittent spatio-temporal regime. • We concluded with a conjecture that for given parameter k there exists an ε that drives the system to spatio-temporal chaos. ODE Phase Space and Bifurcations Periodic NLS (Review) PDE Phase Space Description Spatio-Temporal Chaos Formulation of Results The Problem

  35. Thank you! http://www.wisdom.weizmann.ac.il/~elis/

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