Parabolic Resonance: A Route to Hamiltonian Spatio-Temporal Chaos Eli Shlizerman and Vered Rom-Kedar...
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Transcript of Parabolic Resonance: A Route to Hamiltonian Spatio-Temporal Chaos Eli Shlizerman and Vered Rom-Kedar...
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
t xx-i 0
• Change variables to oscillatory frame
• To obtain the autonomous NLS
The perturbed NLS equation
focusingdispersion
20i(Ω t+θ )Ψ(x,t)=B(x,t)e
2 2t xx-iB B ( B -Ω )B ε
+damping : [Bishop, Ercolani, McLaughlin 80-90’s]
20i( t+θ )εe + i
forcing damping
Periodic NLS(Review)The Problem
ODE Phase Spaceand Bifurcations
PDE Phase SpaceDescription
Spatio-Temporal Chaos
Formulation of Results
2 2t xx-iB B ( B -Ω )B ε
• Boundary • Periodic B(x+L,t) = B(x,t)• Even (ODE) B(-x,t) = B(x,t)
• Parameters• Wavenumber k = 2π/L • Forcing Frequency Ω2
The autonomous NLS equation
Periodic NLS(Review)The Problem
ODE Phase Spaceand Bifurcations
PDE Phase SpaceDescription
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
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 Spaceand Bifurcations
PDE Phase SpaceDescription
Spatio-Temporal Chaos
Formulation of Results
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.
B(X,T)=[c(T)+b(T)coskX+η(X,T)]
22 LI = c(T)+b(T)coskX
x
Periodic NLS(Review)The Problem
ODE Phase Spaceand Bifurcations
PDE Phase SpaceDescription
Spatio-Temporal Chaos
Formulation of Results
Integrals of motion
• Integrable case (ε = 0):
*1 B-B
iH dx
L
21I B dx
L
Infinite number of constants of motion: I,H0, …
HT=H0 + εH1
• Define:
• Perturbed case (ε ≠ 0): The total energy is preserved:
All others are not! I(t) != I0
ε)BΩ-B(BiB- 22
xxt
2 4 220 x
1 1B + B - B
2H dx
L
Periodic NLS(Review)
The ProblemODE Phase Space
and BifurcationsPDE Phase Space
DescriptionSpatio-Temporal
ChaosFormulation of
Results
The plane wave solution
Im(B(0,t))
Re(
B(0
,t))
θ₀
0(0, ) e i tpwB t c
Im(B(0,t))
Re(
B(0
,t))
θ₀
0 0
0)BΩ-B(BiB- 22
xxt
Non Resonant: Resonant:
Periodic NLS(Review)
The ProblemODE Phase Space
and BifurcationsPDE Phase Space
DescriptionSpatio-Temporal
ChaosFormulation 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
Periodic NLS(Review)
The ProblemODE Phase Space
and BifurcationsPDE Phase Space
DescriptionSpatio-Temporal
ChaosFormulation of
Results
The plane wave solution
0(0, ) e i tpwB t c
0)BΩ-B(BiB- 22
xxt
Periodic NLS(Review)
The ProblemODE Phase Space
and BifurcationsPDE Phase Space
DescriptionSpatio-Temporal
ChaosFormulation of
Results
Im(B(0,t))
Re(
B(0
,t))
θ₀
Bh
Bpw
Im(B(0,t))
Re(
B(0
,t))
θ₀
Bh
Bpw
Heteroclinic Orbits!
Modal equations
• Consider two mode Fourier truncation B(x , t) = c(t) + b (t) cos (kx)
• Substitute into the unperturbed eq.:
2222222224224 cb+c b8
1 |c|
2
1-|b|k+Ω
2
1-|b|
16
3|c||b|
2
1|c|
8
1 0H =
0I = )|b||c(|2
1 22 *εi (c - c )
2 1H =
[Bishop, McLaughlin, Ercolani, Forest, Overmann ]
0)BΩ-B(BiB- 22
xxt
Periodic NLS(Review)
Spatio-Temporal Chaos
PDE Phase SpaceDescription
Formulation of Results
ODE Phase Spaceand Bifurcations
The Problem
General Action-Angle Coordinates
• For b≠0 , consider the transformation:
• Then the system is transformed to:
• We can study the structure of
| | ic c e iγ ( ) e b x iy 2 2 21 | |
2 I c x y
0 1H( , , , ) H ( , , )+ H ( , , , )x y I x y I x y I
0H ( , , )x y I
[Kovacic]
Periodic NLS(Review)
Spatio-Temporal Chaos
PDE Phase SpaceDescription
Formulation of Results
ODE Phase Spaceand Bifurcations
The Problem
Preliminary step - Local Stability
Fixed Point Stable Unstable
x=0 y=0 I > 0 I > ½ k2
x=±x2 y=0 I > ½k2 -
x =0 y=±y3 I > 2k2 -
x =±x4 y=±y4 - I > 2k2
[Kovacic & Wiggins 92’]
B(X , t) = [|c| + (x+iy) coskX ] eiγ
Periodic NLS(Review)
Spatio-Temporal Chaos
PDE Phase SpaceDescription
Formulation of Results
ODE Phase Spaceand Bifurcations
The Problem
validity region
Bpw=Plane wave +Bsol=Soliton (X=0)
+Bh=Homoclinic Solution
-Bsol=Soliton (X=L/2)
-Bh=Homoclinic Solution
x
y
PDE-ODE Analogy
Periodic NLS(Review)
Spatio-Temporal Chaos
PDE Phase SpaceDescription
Formulation of Results
ODE Phase Spaceand Bifurcations
The Problem
ODE
PDE
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 SpaceDescription
Formulation of Results
ODE Phase Spaceand Bifurcations
The Problem
0H (x,y,I)
Level 1: Singularity Surfaces
Construction of the EMBD -(Energy Momentum Bifurcation Diagram)
Fixed Point H(xf , yf , I; k=const, Ω=const)
x=0 y=0 H1
x=±x2 y=0 H2
x =0 y=±y3 H3
x =±x4 y=±y4 H4
[Litvak-Hinenzon & RK - 03’]
Periodic NLS(Review)
Spatio-Temporal Chaos
PDE Phase SpaceDescription
Formulation of Results
ODE Phase Spaceand Bifurcations
The Problem
EMBD
Parameters k and are fixed.
Dashed – Unstable, Solid – Stable
Periodic NLS(Review)
Spatio-Temporal Chaos
PDE Phase SpaceDescription
Formulation of Results
ODE Phase Spaceand Bifurcations
The Problem
H2
H1
H4
H3
Iso-energy surfaces
Level 2: Bifurcations in the EMBD
Periodic NLS(Review)
Spatio-Temporal Chaos
PDE Phase SpaceDescription
Formulation of Results
ODE Phase Spaceand Bifurcations
The Problem
4 6
Each iso-energy surface can be represented by a Fomenko graph
5*
Energy bifurcationvalue
Possible Energy Bifurcations
• Branching surfaces – Parabolic Circles• Crossings – Global Bifurcation• Folds - Resonances
H
I
0I
H0θ
pI 31 HH
Periodic NLS(Review)
Spatio-Temporal Chaos
PDE Phase SpaceDescription
Formulation of Results
ODE Phase Spaceand Bifurcations
The Problem
[ Full classification: Radnovic + RK, RDC, Moser 80 issue, 08’ ]
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
Periodic NLS(Review)
Spatio-Temporal Chaos
PDE Phase SpaceDescription
Formulation of Results
ODE Phase Spaceand Bifurcations
The Problem
Perturbed solutions classification
Periodic NLS(Review)
Spatio-Temporal Chaos
PDE Phase SpaceDescription
Formulation of Results
ODE Phase Spaceand Bifurcations
The Problem
?
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 …)
Numerical simulations
H0
I
H0
I
H0
I
Periodic NLS(Review)
Spatio-Temporal Chaos
PDE Phase SpaceDescription
Formulation of Results
ODE Phase Spaceand Bifurcations
The Problem
Numerical simulations – Projection to EMBD
H0
I
H0
I
H0
I
Periodic NLS(Review)
Spatio-Temporal Chaos
PDE Phase SpaceDescription
Formulation of Results
ODE Phase Spaceand Bifurcations
The Problem
Bifurcations in the PDE
iEtE (x)eΨB
EE22
ExxE EΨ)ΨΩ-Ψ ( ΨEH
Looking for the standing waves of the NLS
The eigenvalue problem is received
(Duffing system)
Phase space of the Duffing eq.
R (x)ΨE
32
x
x
U- U)E( V
V U
ExE Ψ V,Ψ U Denote:
)x,cn(bb
, )x,dn(aaU
21
21
solution
0)BΩ-B(BiB- 22
xxt
Periodic b.c. select a discretized family of solutions!
Periodic NLS(Review)
Spatio-Temporal Chaos
PDE Phase SpaceDescription
Formulation of Results
ODE Phase Spaceand BifurcationsThe Problem
Bifurcation Diagrams for the PDE
We get a nonlinear bifurcation diagram for the different
stationary solutions : )(ΨE x
Standard – vs.
))(I(ΨE x EEMBD – vs. ))(I(ΨE x ))(H(ΨE x
Periodic NLS(Review)
Spatio-Temporal Chaos
PDE Phase SpaceDescription
Formulation of Results
ODE Phase Spaceand BifurcationsThe Problem
Unperturbed
Perturbed KAM like
Perturbed Chaotic
Classification of initial conditions in the PDE
Periodic NLS(Review)
Spatio-Temporal Chaos
PDE Phase SpaceDescription
Formulation of Results
ODE Phase Spaceand BifurcationsThe Problem
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 chaos Spatio-temporal chaos
Biεe)BΩ-B(BiB- 0iθ22
xxt
Previous: Spatial decoherence
θ₀
Periodic NLS(Review)
Spatio-Temporal Chaos
PDE Phase SpaceDescription
Formulation of Results
ODE Phase Spaceand BifurcationsThe Problem
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 wave I(B0) < ½k2 (0 LUM).
Perturbation is small, ε= 0.05.
• Ω2 is varied:
0iθ22
xxt εe)BΩ-B(BiB-
Ω2=0.1 Ω2=0.225 Ω2=1
x
B0(x)
Bpw(x)|B| δ
Periodic NLS(Review)
Spatio-Temporal Chaos
PDE Phase SpaceDescription
Formulation of Results
ODE Phase Spaceand BifurcationsThe Problem
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 SpaceDescription
Formulation of Results
ODE Phase Spaceand BifurcationsThe Problem
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.
2/
2/
2/
2/
2
2/
2/
*2/
2/T
),(
),(),(
),,(C Tt
Tt
L
L
Tt
Tt
L
L
dxdssxB
dxdssyxBsxB
tyB
|x/L|
1
Coherent
De-correlated
Re(
CT(B
,y,T
/2))
Periodic NLS(Review)
Spatio-Temporal Chaos
PDE Phase SpaceDescription
Formulation of Results
ODE Phase Spaceand BifurcationsThe Problem
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 SpaceDescription
Formulation of Results
ODE Phase Spaceand BifurcationsThe Problem
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 SpaceDescription
Formulation of Results
ODE Phase Spaceand BifurcationsThe Problem
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 Ω2
par that drive an arbitrarily small amplitude solution to a spatial decoherent state.
Ω
ε
Ωpar
√ε
εmin(k)
STC
Periodic NLS(Review)
Spatio-Temporal Chaos
PDE Phase SpaceDescription
Formulation of Results
ODE Phase Spaceand BifurcationsThe Problem
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.
Periodic NLS(Review)
Spatio-Temporal Chaos
PDE Phase SpaceDescription
Formulation of Results
The Problem
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 SpaceDescription
Formulation of Results
ODE Phase Spaceand BifurcationsThe Problem
Thank you!
http://www.wisdom.weizmann.ac.il/~elis/