Energy Exchanges and Localization in Forced Nonlinear Oscillators and Strongly Degenerate Systems
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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
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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
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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).
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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
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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
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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
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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
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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
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042
322
35221
2
assaadad
max0 aa
The equation and the phase plane of LPT Forced Duffing Oscillator
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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
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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
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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
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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
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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
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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
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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’
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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?
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Highly Degenerate System
1. Energy Localization on the first oscillator: 0.17TH The answer is: Yes!
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Highly Degenerate System 1:1 Resonance
2. Strong Energy Exchanges between the oscillators: 0.17TH
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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
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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
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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
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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
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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
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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
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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
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Energy Exchanges and Localization in Highly Degenerate Scalar Models
Weakly coupled chains (k<<1):
Wire
Wire
ix 1ix 1ix
1iy iy 1iy
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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
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Energy Exchanges and Localization in Highly Degenerate Scalar ModelsBelow the threshold: 0.15 0.1826TH
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Energy Exchanges and Localization in Highly Degenerate Scalar Models
0.2 0.1826TH Above the threshold:
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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)
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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)
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