Energy Exchanges and Localizationin Forced Nonlinear Oscillators and Strongly

Degenerate Systems

Yuli StarosvetskyFaculty of Mechanical Engineering,

Technion Israel Institute of Technology,Technion City, Haifa

2nd Conference on Localized Excitationsin Nonlinear Complex Systems

Sevilla, July 10, 2012

Collaborators:Prof. L.I. ManevitchN.N. Semenov Institute of Chemical Physics ,

Russian Academy of Sciences, Moscow, 119991, Russia

Mr. Tzahi Ben-MeirFaculty of Mechanical Engineering,Technion Israel Institute of Technology, Haifa, 32000, Israel

Forced Duffing Oscillator

tsFuudtdu

dtud 1sin282 32

2

Complex variables :

uidtduui

dtdu

ee itit *

We introduce a multi-scale expansion: 0 1( , ) nn

n

tF sin~

Slow Evolution:0 1

0

1

20 03 | | .isi iFe

L. I. Manevitch, A. S. Kovaleva, and D. S. Shepelev, Physica D 240, 1 (2011).

Polar representation: = ae i for a > 0 and the transformed

equations of motion:

In the absence of dissipation (γ = 0), there exists an integral of motion

sin3

,cos

3

1

1

Faasdda

Fadda

where Δ = - (s1- π/2)

sin223 42 aFH aas

Stationary states:.0,0

11

dddda

Limiting phase trajectories: (LPT) H=0

Forced Duffing Oscillator

5

Nonlinear oscillator without dissipation before and after the first topological transition

-3 -2 -1 1 2 3 4

0.5

1

1.5

2

a

α = 0.093 Phase trajectories for s=0.4, F=0.13, α = 0.094

-2 -1 1 2 3 4

0.5

1

1.5

2

a

10 20 30 40 50 60 701

0.25

0.5

0.75

1

1.25

1.5

1.75

2a

α = 0.093 α = 0.09410 20 30 40 50 60 70

t1

0.2

0.4

0.6

0.8

1a

2

3

1, 812Fs

cr

Forced Duffing Oscillator

Phase trajectories for s=0.4, F=0.13, α = 0.187

5 10 15 20 251

0.2

0.4

0.6

0.8

1

1.2

1.4

a

Forced Duffing OscillatorSecond Topological Transition …

3

,2 2

481crsF

Concept of Limiting Phase Trajectories – Two Distinct Types of LPTs

31

1 2

281crsF

Transition from one type of LPT to another one (Manevitch et. al.)

Thus:

1cr Quasi-Linear Oscillations

1cr Non-Linear Oscillations

Forced Duffing Oscillator

8

11

11 )(,

2sinarcsin2)(

dde

)(arccos)(),()( 1111 eFkka

Time dependences of the amplitude a (1) and the phase Δ (1)

Non-smooth basis :

Nonsmooth basis functions

a) (1); b) e (1)

a) b)

where the parameter k is determined by the intensity of the excitation

Forced Duffing Oscillator

042

322

35221

2

assaadad

max0 aa

The equation and the phase plane of LPT Forced Duffing Oscillator

Model FormulationSystem under investigation comprises dimensionless, weakly nonlinear

oscillator subject to a two term harmonic excitation in the neighborhood

of 1:1 resonance:

23

1 1 2 22 2 8 2 sin 1 sin 1d u duu u F s t F s tdt dt

1 0 Asymptotical Adoptions :

1 2 1 2, , , , , (1)F F s s O

Y. Starosvetsky, L.I. Manevitch, PHYSICAL REVIEW E 83, 046211

Theoretical Study – Undamped CaseConcept of Limiting Phase Trajectories – Analytical Study

Performing additional change of variables:

0 0 1 1= exp( )is

201 0 0 0 0 1 2 2 2 2

1

3 exp ,is i iF iF i

Integral of motion is easily found:

4 2 * *2 0 1 0 1 0 0 2 0 0

3 exp( ) exp( )2

H i is iF iF i i

‘Super – Slow Time Scale’

H Adiabatic Invariant

The main goal: Study the relationship between the parameters controlling the global system dynamics1 1 2, , , ,s F F

Theoretical Study – Undamped CaseConcept of Limiting Phase Trajectories – Adiabatic Evolution of LPTs Adiabatic variation of LPTs in super slow time

scaleBifurcation of LPT of the first kind and

transition to the LPT of the second kind

Evolution of LPT of the first kind Evolution of LPT of the second kind

31 1 2

3 2 cos ( ) 2 cos( ( )) 02N s N F N F N

Condition for the bifurcation of LPT

23 2 2

1 1 2

1 2

1

3 4 42cos

8

29

cr cr

cr

cr

N s N F Fa

FF

sN

Theoretical Study – Damped CaseGlobal Dynamics on a Slow Invariant Manifold (SIM)

21 1 2 2

1

3 exp( )is i i F F i

Slow-Flow Model for a Damped Case

0

1 - Slow Time Scale

2 1 2( ) ( ) -Super Slow Time Scale

Proceeding with the multi-scale expansion:

1 21 2

, ,

21 21 1 2 1 2 1 2 1 2 1 2 2

1

,( , ) 3 , , , expis i i F F i

Theoretical Study – Damped CaseGlobal Dynamics on a Slow Invariant Manifold (SIM)

Assuming strong time scales separation , we would like to analyze separately the slow and the super slow evolution of the damped oscillator.Super-Slow Evolution

11 2 2lim , ( )

Setting: 1 2

1

,0

The evolution on a SIM is governed by:

21 1 2 23 exp( )is i i F F i

Simple algebraic manipulations will bring us to the following convenient form (Projection of the SIM to the plane )

22 2 22 2 2

1 1 2 1 2 2 23 2 cos ( )s F F F F Y

,Y

Theoretical Study – Damped CaseGlobal Dynamics on a Slow Invariant Manifold (SIM)

Various regimes of a Duffing Oscillator subject to a bi-harmonic forcing

22 2 22 2 2

1 1 2 1 2 2 23 2 cos ( )s F F F F Y

1 20, 0F F Case 1:Only stationary (fixed) pointson the stable branches of SIMare possible (Simple Periodic Regimes)

Case 2: 1 20, 0F F

Amplitude of excitation doesn’t reach the fold points which results in permanent, weakly

modulated regime

Case 3: 1 20, 0F F

Amplitude of excitation exceeds the fold points which results in strongly modulated

response, with - relaxations

Theoretical Study – Damped CaseGlobal Dynamics on a Slow Invariant Manifold (SIM)

Sufficient conditions for Relaxation Oscillations

22 2 22 2 2

1 1 2 1 2 2 23 2 cos ( )s F F F F Y

2 2

21 2 1 2 1 212 2

1 2 1 2 2

2, 3 , 1,2

2 i i i i

F F F F YY Z Z s Z i

F F F F Y

Fold points

‘Case 2’ ‘Case 3’

Highly Degenerate System

331 1 2 1

332 2 1 2

1 1 2 2(0) 1, (0) (0) (0) 0

x x x x

x x x x

x x x x

Excited oscillator!

Main objectives:• Conditions for energy localization• Conditions for recurrent energy exchanges between the oscillators

Note: System is governed by a single parameter

Is the ideology of LPT valid for a strongly degenerate case?

Highly Degenerate System

1. Energy Localization on the first oscillator: 0.17TH The answer is: Yes!

Highly Degenerate System 1:1 Resonance

2. Strong Energy Exchanges between the oscillators: 0.17TH

Highly Degenerate System Regime of strong localization

331 1 2 1

332 2 1 2

1 1 2 2(0) 1, (0) (0) (0) 0

x x x x

x x x x

x x x x

Assume that energy is permanently localized on the first oscillator:

1 2 1 2( ) ( ) , ( ) (1), ( ) ( )x t x t x t O x t O

Master and Slave:3

1 1 1 1

3 32 2 1 2 2

(1 ) 0, (0) 1, (0) 0

( ) , (0) (0) 0

x x x x

x x x t x x

Highly Degenerate System Solution for a Master Equation

1/21 1 ,1/ 2x t cn t

Rescaling: 1/22 2 , (1 ) , 1x x t

Slave Equation: Forced Cubic Oscillator

33 3 1/22 2 1( ) (1 ) ,1/ 2x x x t cn t

332 2 ;1/ 2 ,

1x x cn

Seek for the strongly localized regime in the form:

2 20 21 20 21( ) ( ) ( ) ( )x x x x x

Highly Degenerate System Multi-Scale Expansion

320

20 0 1

( ) : ,1/ 2

,1/ 2

O x cn

x cn C C

Looking for periodic regimes, we set: 0 1 0C C

Next order approximation:

20 ,1/ 2x cn

1 1

3321 20 21 1( ) : 0, O x x x

Direct expansion:1 1

3 2 2 321 20 20 21 20 21 213 3 0x x x x x x x

Fast Components

Highly Degenerate System Averaging with respect to a fast time scale, yields

1 1

321 21 21

21 20 21

21 20 21

3 0

(0) (0) 0 (0) 1

(0) (0) 0 (0) 0

x x x

x x x

x x x

43 3

200

42 2

200

4

200

,1/ 2 0

,1/ 2 3.3888

,1/ 2 0

K

K

K

x cn d

x cn d

x cn d

1/221 1

13 1 ; ,6 2

x cn m m

Localized solution, yields

1/22 1

1;1/ 2 3 1 ; ,6 2

x cn cn m m

Highly Degenerate System Linear stability analysis, of the localized solution:

332 2

2 2

;1/ 2

, ( )QP

x x cn

x x o

Quasi-Periodic Parametric Excitation

22 23 , 2 , ,1/ 2 ,1/ 2 0cn m cn m cn cn

Stability Analysis - Crude Approximation

2 4

2

3 1 ( ) 0, (1/ )

( ) 2 ( ,1/ 2) ( ,1/ 2)

p O t t O

p cn cn

1. Broer H.W., Puig, J. and Sim´o, C., Resonance tongues and instability pockets in the quasi-periodic Hill-Schr¨odinger equation, Commun. Math. Phys. 241, pp. 467-503, 2003.

2. Zounes, R.S. and Rand, R.H., Transition curves for the quasi-periodic Mathieu equation, SIAM J. Appl. Math. 58, pp. 1094-1115, 1998

Highly Degenerate System Linear stability analysis, of the localized solution:

0, 0 1

( ) ( ), 4 (1/ 2)

p

p p T T K

20 1 ( )O

Applying the method of strained parameters:

20 1 ( )O

Proceeding with the analysis, yields:

4

01

cos2

4

K

p dK

K

Seeking for intersection with the first tongue:2

0 4nK

Highly Degenerate System Linear stability analysis, of the localized solution:

Boundary of the first tongue:

2 4

2 21 0 1

0

13 , cos4 4 2

K

p dK K K

For the original equation:

Conditions for a critical value of

2 20 3 0p

2 20 3

CR2 2

2 21 2

1

3 3 0.18264 (1/ 2) 48 (1 )CRK K

0.17NUM

Energy Exchanges and Localization in Highly Degenerate Scalar Models

Weakly coupled chains (k<<1):

Wire

Wire

ix 1ix 1ix

1iy iy 1iy

Energy Exchanges and Localization in Highly Degenerate Scalar Models

Weakly coupled chains (k<<1):

Excitation of spatially periodic and localized modes on a single chain:

3 3 31 1

3 3 31 1

i i i i i i i

i i i i i i i

x x x x x k x y

y y y y y k y x

1n

( ) ( 1) ( )

( ) ( 1) ( )

nn n

nn n

x t t

y t t

1. Anti-Phase Mode 2 332

2 332

16

ddtddt

k

Energy Exchanges and Localization in Highly Degenerate Scalar ModelsBelow the threshold: 0.15 0.1826TH

Energy Exchanges and Localization in Highly Degenerate Scalar Models

0.2 0.1826TH Above the threshold:

Energy Exchanges and Localization in Highly Degenerate Scalar Models

Weakly coupled chains (k<<1):

Excitation of spatially periodic and localized modes on a single chain:

3 3 31 1

3 3 31 1

i i i i i i i

i i i i i i i

x x x x x k x y

y y y y y k y x

2. Strong localization of compactons

3 31 1i i i i ix x x x x

( ) [0... 0 -1/2 1 -1/2 0 ... 0]nx t cn t

Y.S. Kivshar,’Intrinsic Localized Modes as Solitons with Compact Support’, Phys. Rev. E., Vol. 48, 1 (1993)

Energy Exchanges and Localization in Highly Degenerate Scalar ModelsWeakly coupled chains (k<<1): (Above a Certain Threshold)

Excitation of spatially periodic and localized modes on a single chain:

3 3 31 1

3 3 31 1

i i i i i i i

i i i i i i i

x x x x x k x y

y y y y y k y x

2. Strong localization of compactons

3 31 1i i i i ix x x x x

( ) [0... 0 -1/2 1 -1/2 0 ... 0]nx t cn t

Y.S. Kivshar,’Intrinsic Localized Modes as Solitons with Compact Support’, Phys. Rev. E., Vol. 48, 1 (1993)

THANKS!!!