Parabolic%20Resonance:%20A%20Route%20to%20Hamiltonian%20Spatio-Temporal%20Chaos

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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/
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The perturbed NLS equation
dispersion
focusing

• Change variables to oscillatory frame
• To obtain the autonomous NLS

damping Bishop, Ercolani, McLaughlin 80-90s
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The autonomous NLS equation

• Boundary
• Periodic B(xL,t) B(x,t)
• Even (ODE) B(-x,t) B(x,t)
• Parameters
• Wavenumber k 2p/L
• Forcing Frequency O2

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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
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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 e. 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.
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Integrals of motion

• Define

• Integrable case (e 0)

Infinite number of constants of motion
I,H0,

• Perturbed case (e ? 0)
• The total energy is preserved
• All others are not! I(t) ! I0

HTH0 eH1
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The plane wave solution
Non Resonant
Resonant
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Linear Unstable Modes (LUM)

• The plane wave is unstable for
• 0 lt k2 lt 2c2
• Since the boundary conditions are periodic k is
  discretized
• kj 2pj/L for j 0,1,2 (j - number
  of LUMs)
• Then the condition for instability becomes the
  discretized condition
• j2 (2p/L)2/2 lt c2 lt (j1)2 (2p/L)2/2
• The solution has j Linear Unstable Modes (LUM).
  As we increase the amplitude the number of LUMs
  grows.
• Ipw c2, IjLUM j2k2/2

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The plane wave solution
Heteroclinic Orbits!
Bh
Bh
Re(B(0,t))
Re(B(0,t))
?0
?0
Bpw
Bpw
Im(B(0,t))
Im(B(0,t))
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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
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General Action-Angle Coordinates

• For b?0 , consider the transformation
• Then the system is transformed to
• We can study the structure of

Kovacic
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Preliminary step - Local Stability
B(X , t) c (xiy) coskX ei?
Fixed Point Fixed Point Stable Unstable
x0 y0 I gt 0 I gt ½ k2
xx2 y0 I gt ½k2 -
x 0 yy3 I gt 2k2 -
x x4 yy4 - I gt 2k2
validity region
Kovacic Wiggins 92
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PDE-ODE Analogy
ODE
BpwPlane wave
BsolSoliton (X0)
-BsolSoliton (XL/2)
PDE
BhHomoclinic Solution
-BhHomoclinic Solution
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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, O

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Level 1 Singularity Surfaces

• Construction of the EMBD -
• (Energy Momentum Bifurcation Diagram)

Fixed Point Fixed Point H(xf , yf , I kconst, Oconst)
x0 y0 H1
xx2 y0 H2
x 0 yy3 H3
x x4 yy4 H4
Litvak-Hinenzon RK - 03
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EMBD
Iso-energy surfaces
H4
H1
H3
H2
Parameters k and are fixed. Dashed Unstable,
Solid Stable
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Level 2 Bifurcations in the EMBD
Each iso-energy surface can be represented by a
Fomenko graph
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Possible Energy Bifurcations

• Branching surfaces Parabolic Circles

• Crossings Global Bifurcation

• Folds - Resonances

Full classification Radnovic RK, RDC, Moser
80 issue, 08
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Level 3 Changing parameters, energy bifurcation
values can coincide

• Example Parabolic Resonance for (x0,y0)
• Resonance IR O2
• hrpw -½ O4
• Parabolic Circle Ip ½ k2
• hppw ½ k2(¼ k2 - O2)

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Perturbed solutions classification
Integrable - a point
Perturbed e slab in H0

• 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 )

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Numerical simulations
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Numerical simulations Projection to EMBD
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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.
Denote
solution
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Bifurcation Diagrams for the PDE
We get a nonlinear bifurcation diagram for the
different stationary solutions
Standard vs.
EMBD vs.
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Classification of initial conditions in the PDE
Unperturbed
Perturbed KAM like
Perturbed Chaotic
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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

Temporal chaos
Spatio-temporal chaos
D. McLaughlin, Cai, Shatah
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New Hamiltonian Spatio-temporal Chaos

• All parameters are fixed
• The initial data B0(x) is almost flat,
  asymmetric for all solutions - d10-5.
• The initial data is near a unperturbed stable
  plane wave I(B0) lt ½k2 (0 LUM).
• Perturbation is small, e 0.05.
• O2 is varied

B0(x)
d
B
Bpw(x)
x
O21
O20.1
O20.225
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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 ?
  8, the spatial Correlation function decays at x
  L/2.
• But not vice-versa.

Zaleski 89,Cross Hohenberg93,Mclaughlin,Cai,S
hatah 99
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The Correlation function

• Properties
• Normalized, for y0, CT(B,0,t)1
• T is the window size
• For Spatial decoherence,
• the Correlation function decays.

1
Re(CT(B,y,T/2))
Coherent
De-correlated
x/L
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Intermittent Spatio-Temporal Chaos

• While the Correlation function over the whole
  time decays the windowed Correlation function is
  intermittent

HR
ER
PR
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Choosing Initial Conditions
Projecting the perturbed solution on the EMBD
Parabolic Resonant like solution

• Decoherence can be characterized from the
  projection
• Composition to the standing waves can be
  identified

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Conjecture / Formulation of Results

• For any given parameter k, there exist emin
  emin(k) such that for all e gt emin there exists
  an order one interval of initial phases ?(0) and
  an O(ve)-interval of O2 values centered at O2par
  that drive an arbitrarily small amplitude
  solution to a spatial decoherent state.

e
STC
ve
emin(k)
Opar
O
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Conjecture / Formulation of Results

• Here we demonstrated that such decoherence can
  be achieved with rather small e values (so
  emin(0.9) 0.05).
• Coherence for long time scales may be gained by
  either decreasing e or by selecting O2 away from
  the O(ve)-interval.

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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 e that drives the
  system to spatio-temporal chaos.

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