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

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






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


in time: chaotic

in space: decoherent






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






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:





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





ChaosFormulation of

Results

The plane wave solution

0(0, ) e i tpwB t c

0)BΩ-B(BiB- 22

xxt





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






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]






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






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






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






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’]






The Problem

EMBD

Parameters k and are fixed.

Dashed – Unstable, Solid – Stable






The Problem

H2

H1

H4

H3

Iso-energy surfaces

Level 2: Bifurcations in the EMBD






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






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






The Problem

Perturbed solutions classification






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






The Problem

Numerical simulations – Projection to EMBD

H0

I

H0

I

H0

I






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!





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






Unperturbed

Perturbed KAM like

Perturbed Chaotic

Classification of initial conditions in the PDE






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






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






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 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))






Intermittent Spatio-Temporal Chaos

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

HR

ER

PR






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






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






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

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/

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Parabolic Resonance: A Route to Hamiltonian Spatio-Temporal Chaos