1. Parabolic Resonance: A Route to Hamiltonian Spatio-Temporal Chaos Eli Shlizerman and Vered Rom-Kedar Weizmann Institute of Science Stability and Instability in Mechanical Systems, Barcelona, 2008 [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 Publications: http://www.wisdom.weizmann.ac.il/~elis/ [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

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

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

4. The problem Classify instabilities near the plane wave in the NLS equation Route to Spatio-Temporal Chaos 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 Periodic NLS (Review) The Problem ODE Phase Space and Bifurcations PDE Phase Space Description Spatio-Temporal Chaos Formulation of Results

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 I 2 (T) even if we start with small, nearly flat initial data and with small ε. PDE: Once I 2 (T) is ramped up the solution of the forced NLS becomes spatially decoherent and intermittent - We know how to control I 2 (T) hence we can control the solutions decoherence. Periodic NLS (Review) The Problem ODE Phase Space and Bifurcations PDE Phase Space Description Spatio-Temporal Chaos Formulation of Results

6. Integrals of motion Integrable case (ε = 0): Infinite number of constants of motion: I,H 0, … H T =H 0 + εH 1 Define: Perturbed case (ε ≠ 0): The total energy is preserved: All others are not! I(t) != I0 Periodic NLS (Review) The Problem ODE Phase Space and Bifurcations PDE Phase Space Description Spatio-Temporal Chaos Formulation of Results

7. The plane wave solution Im(B(0,t)) Re(B(0,t)) θ₀θ₀ Im(B(0,t)) Re(B(0,t)) θ₀θ₀ Non Resonant:Resonant: Periodic NLS (Review) The Problem ODE Phase Space and Bifurcations PDE Phase Space Description Spatio-Temporal Chaos Formulation of Results

8. Linear Unstable Modes (LUM) The plane wave is unstable for 0 < k 2 < 2|c| 2 Since the boundary conditions are periodic k is discretized: k j = 2πj/L for j = 0,1,2… (j - number of LUMs) Then the condition for instability becomes the discretized condition j 2 (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. I pw = |c| 2, I jLUM = j 2 k 2 /2 Periodic NLS (Review) The Problem ODE Phase Space and Bifurcations PDE Phase Space Description Spatio-Temporal Chaos Formulation of Results

9. The plane wave solution Periodic NLS (Review) The Problem ODE Phase Space and Bifurcations PDE Phase Space Description Spatio-Temporal Chaos Formulation of Results Im(B(0,t)) Re(B(0,t)) θ₀θ₀ BhBh B pw Im(B(0,t)) Re(B(0,t)) θ₀θ₀ BhBh B pw Heteroclinic Orbits!

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

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

12. Preliminary step - Local Stability Fixed PointStableUnstable x=0y=0I > 0I > ½ k 2 x=±x 2 y=0I > ½k 2 - x =0y=±y 3 I > 2k 2 - x =±x 4 y=±y 4 -I > 2k 2 [Kovacic & Wiggins 92’] B(X, t) = [|c| + (x+iy) coskX ] e iγ Periodic NLS (Review) Spatio-Temporal Chaos PDE Phase Space Description Formulation of Results ODE Phase Space and Bifurcations The Problem validity region

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

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 Periodic NLS (Review) Spatio-Temporal Chaos PDE Phase Space Description Formulation of Results ODE Phase Space and Bifurcations The Problem

15. Level 1: Singularity Surfaces Construction of the EMBD - (Energy Momentum Bifurcation Diagram) Fixed PointH(x f, y f, I; k=const, Ω=const) x=0y=0H1H1 x=±x 2 y=0H2H2 x =0y=±y 3 H3H3 x =±x 4 y=±y 4 H4H4 [Litvak-Hinenzon & RK - 03’] Periodic NLS (Review) Spatio-Temporal Chaos PDE Phase Space Description Formulation of Results ODE Phase Space and Bifurcations The Problem

16. EMBD Parameters k and are fixed. Dashed – Unstable, Solid – Stable Periodic NLS (Review) Spatio-Temporal Chaos PDE Phase Space Description Formulation of Results ODE Phase Space and Bifurcations The Problem H2H2 H1H1 H4H4 H3H3 Iso-energy surfaces

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

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

19. Level 3: Changing parameters, energy bifurcation values can coincide Example: Parabolic Resonance for (x=0,y=0) Resonance I R = Ω 2 h rpw = -½ Ω 4 Parabolic Circle I p = ½ k 2 h ppw = ½ k 2 (¼ k 2 - Ω 2 ) Parabolic Resonance: I R =I P k 2 =2Ω 2 Periodic NLS (Review) Spatio-Temporal Chaos PDE Phase Space Description Formulation of Results ODE Phase Space and Bifurcations The Problem

20. Perturbed solutions classification Periodic NLS (Review) Spatio-Temporal Chaos PDE Phase Space Description Formulation of Results ODE Phase Space and Bifurcations The Problem  ? Integrable - a point Perturbed –  slab in H 0 Away from sing. curve: Regular / KAM type   Near sing. curve: Standard phenomena (Homoclinic chaos, Elliptic circles)    √√ Near energy bif. val.: Special dyn phenomena (HR,PR,ER,GB-R …)

21. Numerical simulations H0H0 I H0H0 I H0H0 I Periodic NLS (Review) Spatio-Temporal Chaos PDE Phase Space Description Formulation of Results ODE Phase Space and Bifurcations The Problem

22. Numerical simulations – Projection to EMBD H0H0 I H0H0 I H0H0 I Periodic NLS (Review) Spatio-Temporal Chaos PDE Phase Space Description Formulation of Results ODE Phase Space and Bifurcations The Problem

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

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

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

26. 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 [D. McLaughlin, Cai, Shatah] Temporal chaosSpatio-temporal chaos Previous: Spatial decoherence θ₀θ₀ Periodic NLS (Review) Spatio-Temporal Chaos PDE Phase Space Description Formulation of Results ODE Phase Space and Bifurcations The Problem

27. New: Hamiltonian Spatio-temporal Chaos All parameters are fixed:  The initial data B 0 (x) is almost flat, asymmetric for all solutions - δ=10 -5.  The initial data is near a unperturbed stable plane wave I(B 0 ) < ½k 2 (0 LUM).  Perturbation is small, ε= 0.05. Ω 2 is varied: Ω 2 =0.1 Ω 2 =0.225 Ω 2 =1 x B 0 (x) B pw (x) |B| δ Periodic NLS (Review) Spatio-Temporal Chaos PDE Phase Space Description Formulation of Results ODE Phase Space and Bifurcations 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. [Zaleski 89’,Cross & Hohenberg93’,Mclaughlin,Cai,Shatah 99’] Periodic NLS (Review) Spatio-Temporal Chaos PDE Phase Space Description Formulation of Results ODE Phase Space and Bifurcations The Problem

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

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

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

32. Conjecture / Formulation of Results For any given parameter k, there exist ε min = ε min (k) such that for all ε > ε min there exists an order one interval of initial phases γ(0) and an O(√ε)-interval of Ω 2 values centered at Ω 2 par that drive an arbitrarily small amplitude solution to a spatial decoherent state. Ω ε Ω par √ε√ε ε min (k) STC Periodic NLS (Review) Spatio-Temporal Chaos PDE Phase Space Description Formulation of Results ODE Phase Space and Bifurcations The Problem

33. Conjecture / Formulation of Results Here we demonstrated that such decoherence can be achieved with rather small ε values (so ε min (0.9) ~ 0.05). Coherence for long time scales may be gained by either decreasing ε or by selecting Ω 2 away from the O(√ε)-interval. Periodic NLS (Review) Spatio-Temporal Chaos PDE Phase Space Description Formulation of Results The Problem

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

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