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
http://www.wisdom.weizmann.ac.il/~elis/
Stability and Instability in Mechanical
Systems, Barcelona, 2008
The Problem
Periodic NLS
(Review)
ODE Phase Space
and Bifurcations
PDE Phase Space
Description
Spatio-Temporal
Chaos
Formulation of
Results
The perturbed NLS equation
-i t   xx     0εeforcing
2
dispersion
i(2 t+θ0 )
+ i
damping
focusing
• Change variables to oscillatory frame
Ψ(x,t)=B(x,t)e
i(Ω2 t+θ0 )
• To obtain the autonomous NLS
-iBt  Bxx  ( B -Ω )B  ε
2
2
+damping : [Bishop, Ercolani, McLaughlin 80-90’s]
The Problem
Periodic NLS
(Review)
ODE Phase Space
and Bifurcations
PDE Phase Space
Description
Spatio-Temporal
Chaos
The autonomous NLS equation
-iBt  Bxx  ( B -Ω )B  ε
2
• Boundary
• Periodic
• Even (ODE)
2
B(x+L,t) = B(x,t)
B(-x,t) = B(x,t)
• Parameters
• Wavenumber
k = 2π/L
• Forcing Frequency Ω2
Formulation of
Results
The Problem
Periodic NLS
(Review)
ODE Phase Space
and Bifurcations
PDE Phase Space
Description
Spatio-Temporal
Chaos
Formulation of
Results
The problem
• Classify instabilities near the plane wave in the NLS equation
• Route to Spatio-Temporal Chaos
Regular Solution
Temporal Chaos
Spatio-Temporal Chaos
in time: almost periodic
in time: chaotic
in time: chaotic
in space: coherent
in space: coherent
in space: decoherent
The Problem
Periodic NLS
(Review)
ODE Phase Space
and Bifurcations
PDE Phase Space
Description
Spatio-Temporal
Chaos
Formulation of
Results
Main Results
Decompose the solutions to first two modes and a remainder:
B(X,T)=[c(T)+b(T)coskX+η(X,T)]
And define:
I2 = c(T)+b(T)coskX
L2 x
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
Integrals of motion
Spatio-Temporal
Chaos
- iB t  Bxx  ( B - Ω 2 )B  ε
2
• Define:
2
1
I   B dx
L
1 
1 4
2
2
2
H 0    Bx + B -  B dx
L 
2



i
H1   B-B* dx
L
• Integrable case (ε = 0):
Infinite number of constants of motion: I,H0,
…
• Perturbed case (ε ≠ 0):
The total energy is preserved:
All others are not! I(t) != I0
Formulation of
Results
HT=H0 + εH1
The Problem
Periodic NLS
(Review)
ODE Phase Space
and Bifurcations
PDE Phase Space
Description
The plane wave solution
B pw (0, t )  c e
Formulation of
Results
- iB t  Bxx  ( B - Ω 2 )B  0
2
i t 0 
Resonant:
 0
Re(B(0,t))
0
Re(B(0,t))
Non Resonant:
Spatio-Temporal
Chaos
θ₀
θ₀
Im(B(0,t))
Im(B(0,t))
The Problem
Periodic NLS
(Review)
ODE Phase Space
and Bifurcations
PDE Phase Space
Description
Spatio-Temporal
Chaos
Formulation of
Results
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
The Problem
Periodic NLS
(Review)
ODE Phase Space
and Bifurcations
PDE Phase Space
Description
The plane wave solution
B pw (0, t )  c e
Spatio-Temporal
Chaos
Formulation of
Results
- iB t  Bxx  ( B - Ω 2 )B  0
2
i t 0 
Heteroclinic Orbits!
Bh
Re(B(0,t))
Re(B(0,t))
Bh
θ₀
θ₀
Bpw
Im(B(0,t))
Bpw
Im(B(0,t))
The Problem
Periodic NLS
(Review)
ODE Phase Space
and Bifurcations
PDE Phase Space
Description
Modal equations
Spatio-Temporal
Chaos
Formulation of
Results
- iB t  Bxx  ( B - Ω 2 )B  0
2
• Consider two mode Fourier truncation
B(x , t) = c(t) + b (t) cos (kx)
• Substitute into the unperturbed eq.:



1 4 1 2 2 3 4 1 2 2 2 1 2 2 1 2 2 2 2
H0= | c |  | b | | c |  | b | - Ω + k | b | -  | c |  b c + b c
8
2
16
2
2
8
I0
1 2
2
= (| c |  | b | )
2
H1= i
ε
2
(c - c* )
[Bishop, McLaughlin, Ercolani, Forest, Overmann ]

The Problem
Periodic NLS
(Review)
ODE Phase Space
and Bifurcations
PDE Phase Space
Description
Spatio-Temporal
Chaos
Formulation of
Results
General Action-Angle Coordinates
• For b≠0 , consider the transformation:
c  |c| e
i
b  ( x  iy) e

1
I 
| c |2  x 2  y 2
2
iγ

• Then the system is transformed to:
H( x, y, I ,  )  H0 (x, y, I )+ H1 ( x, y, I ,  )
• We can study the structure of
H0 (x, y, I )
[Kovacic]
The Problem
Periodic NLS
(Review)
ODE Phase Space
and Bifurcations
PDE Phase Space
Description
Spatio-Temporal
Chaos
Formulation of
Results
Preliminary step - Local Stability
B(X , t) = [|c| + (x+iy) coskX ] eiγ
Fixed Point
x=0
y=0
x=±x2
y=0
Stable Unstable
I>0
I > ½ k2
I > ½k2
-
x =0 y=±y3 I > 2k2
x =±x4 y=±y4
validity region
I > 2k2
[Kovacic & Wiggins 92’]
The Problem
Periodic NLS
(Review)
ODE Phase Space
and Bifurcations
PDE Phase Space
Description
Spatio-Temporal
Chaos
Formulation of
Results
PDE-ODE Analogy
ODE
y
x
-Bsol=Soliton (X=L/2)
Bpw=Plane wave +Bsol=Soliton (X=0)
PDE
-Bh=Homoclinic Solution
+Bh=Homoclinic Solution
The Problem
Periodic NLS
(Review)
ODE Phase Space
and Bifurcations
PDE Phase Space
Description
Spatio-Temporal
Chaos
Hierarchy of Bifurcations
Formulation of
Results
H0 (x,y,I)
• 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
The Problem
Periodic NLS
(Review)
ODE Phase Space
and Bifurcations
PDE Phase Space
Description
Spatio-Temporal
Chaos
Formulation of
Results
Level 1: Singularity Surfaces
Construction of the EMBD (Energy Momentum Bifurcation Diagram)
Fixed Point
x=0
y=0
x=±x2
y=0
x =0
y=±y3
x =±x4
y=±y4
H(xf , yf , I; k=const, Ω=const)
H1
H2
H3
H4
[Litvak-Hinenzon & RK - 03’]
The Problem
Periodic NLS
(Review)
ODE Phase Space
and Bifurcations
PDE Phase Space
Description
Spatio-Temporal
Chaos
Formulation of
Results
EMBD
Iso-energy surfaces
H4
H3
H1
H2
Parameters k and are fixed.
Dashed – Unstable, Solid – Stable
The Problem
Periodic NLS
(Review)
ODE Phase Space
and Bifurcations
PDE Phase Space
Description
Spatio-Temporal
Chaos
Level 2: Bifurcations in the EMBD
Each iso-energy surface can be
represented by a Fomenko graph
4
5*
Energy bifurcation
value
6
Formulation of
Results
The Problem
Periodic NLS
(Review)
ODE Phase Space
and Bifurcations
PDE Phase Space
Description
Spatio-Temporal
Chaos
Formulation of
Results
Possible Energy Bifurcations
• Folds
Branching
Crossings
- Resonances
–surfaces
Global Bifurcation
– Parabolic Circles
H

Iθp1 0H 3  0
H
I
I
H
[ Full classification: Radnovic +
RK, RDC, Moser 80 issue, 08’ ]
The Problem
Periodic NLS
(Review)
ODE Phase Space
and Bifurcations
PDE Phase Space
Description
Spatio-Temporal
Chaos
Level 3: Changing parameters, energy
bifurcation values can coincide
• 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)
Parabolic Resonance: IR=IP k2=2Ω2
Formulation of
Results
Periodic NLS
(Review)
The Problem
ODE Phase Space
and Bifurcations
PDE Phase Space
Description
Spatio-Temporal
Chaos
Formulation of
Results
Perturbed solutions classification
Integrable - a point
Perturbed –  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 …)
The Problem
Periodic NLS
(Review)
ODE Phase Space
and Bifurcations
PDE Phase Space
Description
Spatio-Temporal
Chaos
Formulation of
Results
Numerical simulations
I
I
H0
H0
I
H0
The Problem
Periodic NLS
(Review)
ODE Phase Space
and Bifurcations
PDE Phase Space
Description
Spatio-Temporal
Chaos
Formulation of
Results
Numerical simulations – Projection to EMBD
I
I
H0
H0
I
H0
The Problem
ODE Phase Space
and Bifurcations
Periodic NLS
(Review)
PDE Phase Space
Description
Bifurcations in the PDE
Spatio-Temporal
Chaos
Formulation of
Results
- iB t  Bxx  ( B - Ω 2 )B  0
2
Looking for the standing waves of the NLS
B  Ψ E (x)e iEt
Ψ E (x)  R
The eigenvalue problem is received
(Duffing system)
HE Ψ E  ( xx  Ψ E - Ω 2 )Ψ E  EΨ E
Periodic b.c. select a
discretized family of
solutions!
2
Phase space of the Duffing eq.
Denote: U  ΨE , V   x ΨE
solution U  a dn(a x,  ) ,
U x  V
1
2

2
3
b1cn(b 2 x,  )
V

(
E


)
U
U
 x
The Problem
Periodic NLS
(Review)
ODE Phase Space
and Bifurcations
PDE Phase Space
Description
Spatio-Temporal
Chaos
Formulation of
Results
Bifurcation Diagrams for the PDE
We get a nonlinear bifurcation diagram for the different
stationary solutions ΨE ( x) :
EMBD – I(ΨE ( x)) vs. H(Ψ E ( x))
Standard – I(ΨE ( x)) vs. E
The Problem
Periodic NLS
(Review)
ODE Phase Space
and Bifurcations
PDE Phase Space
Description
Spatio-Temporal
Chaos
Formulation of
Results
Classification of initial conditions in the PDE
Unperturbed
Perturbed KAM like
Perturbed Chaotic
The Problem
Periodic NLS
(Review)
ODE Phase Space
and Bifurcations
PDE Phase Space
Description
Spatio-Temporal
Chaos
Formulation of
Results
Previous: Spatial decoherence
- iB t  Bxx  ( B - Ω 2 )B  εeiθ0  iB
2
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
θ₀
Spatio-temporal chaos
[D. McLaughlin, Cai, Shatah]
The Problem
Periodic NLS
(Review)
ODE Phase Space
and Bifurcations
PDE Phase Space
Description
Spatio-Temporal
Chaos
Formulation of
Results
New: Hamiltonian Spatio-temporal Chaos
- iB t  Bxx  ( B - Ω 2 )B  εeiθ0
2
• All parameters are fixed:
 The initial data B0(x) is almost flat,
asymmetric for all solutions - δ=10-5.
B0(x)
|B|
δ
Bpw(x)
 The initial data is near a unperturbed stable
plane wave I(B0) < ½k2 (0 LUM).
 Perturbation is small, ε= 0.05.
x
Ω2=0.1
• Ω2 is varied:
Ω2=0.225
Ω2=1
The Problem
Periodic NLS
(Review)
ODE Phase Space
and Bifurcations
PDE Phase Space
Description
Spatio-Temporal
Chaos
Formulation of
Results
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’]
The Problem
ODE Phase Space
and Bifurcations
Periodic NLS
(Review)
PDE Phase Space
Description
Spatio-Temporal
Chaos
Formulation of
Results
The Correlation function
t T / 2 L / 2
C T ( B, y , t ) 

*
B
(
x
,
s
)
B
( x  y, s )dxds

t T / 2  L / 2
t T / 2 L / 2
  B ( x, s )
2
dxds
Properties:
• Normalized, for y=0, CT(B,0,t)=1
• T is the window size
• For Spatial decoherence,
the Correlation function decays.
Re(CT(B,y,T/2))
t T / 2  L / 2
1
Coherent
|x/L|
De-correlated
The Problem
Periodic NLS
(Review)
ODE Phase Space
and Bifurcations
PDE Phase Space
Description
Spatio-Temporal
Chaos
Formulation of
Results
Intermittent Spatio-Temporal Chaos
• While the Correlation function over the whole time decays
the windowed Correlation function is intermittent
HR
ER
PR
The Problem
Periodic NLS
(Review)
ODE Phase Space
and Bifurcations
PDE Phase Space
Description
Spatio-Temporal
Chaos
Formulation of
Results
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
The Problem
ODE Phase Space
and Bifurcations
Periodic NLS
(Review)
PDE Phase Space
Description
Spatio-Temporal
Chaos
Formulation of
Results
Conjecture / Formulation of Results
• For any given parameter k, there exist εmin = εmin(k) such
that for all ε > εmin there 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
εmin(k)
Ωpar
Ω
The Problem
Periodic NLS
(Review)
Spatio-Temporal
Chaos
PDE Phase Space
Description
Formulation of
Results
Conjecture / Formulation of Results
• Here we demonstrated that such decoherence can be
achieved with rather small ε values (so εmin(0.9) ~ 0.05).
• Coherence for long time scales may be gained by either
decreasing ε or by selecting Ω2 away from the O(√ε)-interval.
The Problem
Periodic NLS
(Review)
ODE Phase Space
and Bifurcations
PDE Phase Space
Description
Spatio-Temporal
Chaos
Formulation of
Results
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.
Thank you!
http://www.wisdom.weizmann.ac.il/~elis/