Post on 06-Mar-2018
T. UzerGeorgia Institute of Technology
Collaboration: C. Chandre (Univ. of Marseille)S. Wiggins (Bristol)
KITP Atto 06 KITP Atto 06 –– September 8, 2006September 8, 2006
TimeTime--frequency analysis of chaotic systemsfrequency analysis of chaotic systems
220
2
2 iii
ii
bendCSCO
qpE
EEEE
ωμ
+=
++=normal mode analysisnormal mode analysis
Reference:Reference: D. Carter,D. Carter, et al, J. Chem. Phys,et al, J. Chem. Phys, 77,77, 4208, (1982)4208, (1982)
Intramolecular relaxation ratesIntramolecular relaxation rates
ps 17.01≈=
λτ λ
predicted relaxation rate:predicted relaxation rate:assumption of uniformity of phase space assumption of uniformity of phase space statistical theories ( RRKM theory ) :statistical theories ( RRKM theory ) :
observed relaxation rate:observed relaxation rate: ps 10>τ
CSE COEαE
planar carbonyl sulfide (OCS)planar carbonyl sulfide (OCS)
1R 2R
OC
S
α
m Ii
i ii
H T P P P R VP RVR RAR RR3 3
1 2 1 21
1 2 31
( )( , , , , ( ,, (,) ))α= =
= α + +∑ ∏
quartic polynomialMorsekinetic energy Sorbie-Murrell potential
1
23
2
13
21
32
2
22
1
12213
22
221
1sinsincos
22cos
22 R
PP
R
PP
RRRRPPPPPT
αμ−
αμ−⎟⎟⎠
⎞⎜⎜⎝
⎛ αμ−
μ+
μ+αμ+
μ+
μ= αα
α
( )2)( 0
1)( iii RRiim eDRV −β−−=
( ))~(tanh1)( iiiiI RRRV −γ−=
∑ ∑ ∑ ∑++++=i ji kji lkji
lkjiijklkjiijkjiijii RRRRcRRRcRRcRcRRRP, ,, ,,,
321 1),,(
α−+== cos2 212
22
13 RRRROSR
References:Carter, Brumer, J. Chem. Phys. 77 (1982), 4208Foord, Smith, Whiffen, Mol. Phys. 29 (1975), 1685Bunker, J. Chem. Phys. 37 (1962), 393
>> 3 modes: OC stretch, CS stretch, OCS bend
Who cares ?celestial mechanics, atomics physicschemical physics, particle acceleratorsplasma physics, …
Analysis of trajectories of Hamiltonian systemsAnalysis of trajectories of Hamiltonian systems
{ }t T
Ht
Ht
t t[0, ]
dd
( ), ( )dd
∈
∂⎧ = −⎪ ∂⎪ ⇒⎨ ∂⎪ =+⎪ ∂⎩
pq
qp
p q
H t( , ; )p q1( )q t
t
> chaotic (?)
Questions:>> regular or chaotic ?>> extent of chaoticity: strong or weak chaos>> intermittency: resonance trappings, transitions…>> type of chaos: slow chaos, stable chaos,…
{ }i i i=1, ,Nt t( ), ( )p q
…
d(0)
d t( )
t
d tt d1 ( )
lim ln(0)→∞
λ =0 regular
0 chaotic
λ =λ >
>> Lyapunov exponent
PoincarPoincaréé section section (2 degrees of freedom)
quasiperiodic
periodic
chaotic
p
x
( )H p x t p x x t21( , , ) cos cos( )
2= − ε + −
> only 2 degrees of freedom> qualitative information
individual trajectories phase space structures
Frequency AnalysisFrequency Analysis
>> Fourier analysis
>> Frequency Analysis
Idea: decomposition of a trajectory known on a finiteinterval [0,T] into the basis
0 T
x(t;
x 0)
ω
)(ωSP
2FFT
Tπ
Δω =
{ }i te ω
ω∈
( )2
0)()(
1∫ χ=ωφ ω−T ti dttetz
T
4
1T
Δω∝T
tt
π−=χ
2cos1)( : windowHanning
1(t) : windowwithout =χ 2
1T
Δω∝
( ) ( )ωφ=ωφω ωmax thatsuch : )(for frequency Main 11tz
Frequency Map AnalysisFrequency Map Analysis (J. Laskar)
>> Fourier analysis
>> Frequency Analysis Idea: decomposition of a trajectory known on a finiteinterval [0,T] into the basis { }i te ω
ω∈
T i tSP x x t x e t t
T0 00
1( , ) ( ; ) ( )d− ωω = χ∫
Sx P x1 0 0( ) maximizes ( ; )ω i
periodic
quasiperiodic
chaotic
>> dynamical information, time varying frequencyx0
0 T
x(t;
x 0)
ω
T1
δω∼
T 4
1δω∼
The windowed Fourier transform of f(t) is given by
( ) ( ) ( )∫∞
∞−
−−= dteutgtfuSf tiξξ,The Gabor transform is obtained by choosing the Gaussian window
( ) ( ) 4/122/ /22 πσσtetg −=
The spectrogram is given by
( ) ( ) 2,, ξξ uSfufPS =where
( ) ( ) ( ) dtetgutfeuSf titi ∫∞+
∞−
−− += ξξξ,
TimeTime--frequency resolutionfrequency resolutionThe time spread around a point (u,ξ) in the time-frequency plane is defined as
( ) ( ) ( )( ) 2/12
,
2, dttgutu utime ∫∞+
∞−−= ξξσ
where
( ) ( ) ti
u eutgtg ξ
ξ
−−=,
Is independent of u and ξ. For a Gaussian windowit is equal to
( ) ( ) 4/122/ /22 πσσtetg −=.2/σ
The frequency spread around a point (u,ξ) is defined as
( ) ( ) ( )2/1
2
,
2 ˆ21, ⎟
⎠⎞
⎜⎝⎛ −= ∫
∞+
∞−ωωξω
πξσ ξ dgu ufreq
Does not depend on u and ξ since ( ) ( ) ( ) .ˆˆ ,
ωξ
ξ ξωω −−= iu
u eggFor a Gaussian window, it is equal to . We notice that the product of the timespread with the frequency spread at a given point (u,ξ) in the time-frequency plane is constant and larger than ½ (Heisenberg uncertainty). It is minimum for a Gaussianwindow.
.2/1 σ
Local Frequency AnalysisLocal Frequency Analysis
Reference: Martens, Ezra, J. Chem. Phys. (1987)
Observation: chaotic motion over long time scale appears regular over shorter time scales
>> intermittency
1:3
( ) '1; ( ') ( ') '
t T i t
tt z t e t dt
T
+ − ωφ ω = χ∫ 1( )t→ω
T
2
1-74 dB Blackman-Harris window :
TΔω∝
1:5
Continuous Wavelet TransformContinuous Wavelet Transform
dts
uttf
ssuWf ∫
∞+
∞−⎟⎠⎞
⎜⎝⎛ −
Ψ= *)(1
),(
Re( )ψ
t
i t tt e e2 2/2( ) η − σψ =
1 with <⎟⎠⎞
⎜⎝⎛ −
Ψ ss
ut1 with >⎟
⎠⎞
⎜⎝⎛ −
Ψ ss
ut
2),(
1),( :Scalogram suWf
ssuPW =
ScaleScale / / frequency representationfrequency representation ::22 )(),( ,)(for η−ωσ−ω == s
Wti esufPetf s
η=ω
Fast algorithmFast algorithm :: )(ˆ)(ˆ),(ˆ * ξΨξ=ξ sfssfW ⇒ 1 FFT + 1 IFFT
T1
: resolution frequency ∝ωΔ
low frequency high frequency
MorletMorlet--Grossman waveletGrossman wavelet
Continuous wavelet transformContinuous wavelet transform
Time-scale representation of trajectory
( ) ( )∫∞
∞−⎟⎠⎞
⎜⎝⎛ −= dt
suttfsuWf s
*1, ψ
Morlet-Grossman wavelet
( ) ( ) 4/122/ /22 πσψ ση tti eet −=Center frequency of wavelet
( )( ) ( )[ ]sutst
d
su
su
/
ˆ2/1
,
2
,21
−=
=−
∞
∞−∫ψψ
ωωψωη π
sηξ =
Normalized scalogram
( )2
,1, suWfss
ufPW =⎟⎠⎞
⎜⎝⎛ =ηξ
freq
uen
cy
time
TimeTime--frequency Analysisfrequency Analysis
Over the Rainbow (Arlen/Harburg – Keith Jarrett)
For a chaotic trajectory, a ridge is a curve or a segment of curve in the time-frequencyWhich is at each time u a local maximum of the normalized scalogram, i.e.,
Each ridge has a weight that is the value of the normalized scalogram on this ridge(it varies continuously in time). We will refer to this value as the amplitude of the ridge.We call the main ridge or main frequency ridge (or a set of ridges) where thenormalized scalogram is a maximum:
( )( )
( )( )
0,,0,2
2
<∂∂=
∂∂
== u
W
u
W
locloc
ufPufPξξξξ
ξξ
ξξ
( )ulocξ
( )umξ
( )( ) ( )ξξξ
,max, ufPuufP WmW =
Instantaneous frequenciesInstantaneous frequencies
ReferenceReference: : C. Chandre and S. Wiggins and T. UzerC. Chandre and S. Wiggins and T. Uzer , , Physica DPhysica D (2003)(2003)
∑ ϕ+ω=k
tik
kkeAtf )()(
( ) ( )[ ] ( )∑>
ωω−ωησ−ϕ+ω−ωω−ωωω
+ω=ω1
2/11
111
21
21
22
cos)(~k
kkkk
kketAt
ω
ωt
t
22 )(),( ,)(for η−ωσ−ω == sW
ti esufPetf
2),(1),( :Scalogram suWfs
suPW =
0),( ,0),()(
2
2
)(
=∂∂
=∂∂
== ulocloc
uPuP Wu
Wωωωω
ωω
ωω)( tωω = uttimeat =
periodic trajectoryperiodic trajectory
quasiperiodic trajectoryquasiperiodic trajectory
sηω =
TimeTime--frequency analysis based on waveletsfrequency analysis based on wavelets
t
ωf t t t t t2( ) cos( ) cos( sin )= α + + β +
ω
t
>> periodic
>> quasiperiodic
ω
t
ω
t
),( ωtPW
tt 2)(1 +α=ω
tt cos)(2 +β=ω
Reference: Chandre, Wiggins, Uzer, Physica D (2003)
ridge extraction
∑ ϕ+ω=k
tik
kkeAtf )()(
( ) ( )[ ] ( )∑>
ωω−ωησ−ϕ+ω−ωω−ωωω
+ω=ω1
2/11
111
21
21
22
cos)(~k
kkkk
kketAt
tietf ω=)(
TimeTime--frequency resolutionfrequency resolution
( ) ( ) ( )( ) 2/12
,
2, dttutsu sutime ∫∞+
∞−−= ψσ
( ) ( )2/1
2
,
2
ˆ, 21 ⎟
⎠
⎞⎜⎝
⎛⎟⎠⎞
⎜⎝⎛ −= ∫
∞+
∞−ωωψηωσ π d
ssu sufreq
Mother wavelet : ( ) ( )tget tiηψ =
Gabor wavelet : ξσησ 2/=time
Frequency spread around s/ηξ =
Gabor wavelet : 2/σηξσ =freq
Heisenberg uncertainty is minimum for the Gabor wavelet
Wavelet decomposition for periodic trajectoryWavelet decomposition for periodic trajectory
Poincare surface of sectionPoincare surface of section timetime--frequency plane (scalogram)frequency plane (scalogram)
timetimex
y
for a periodic trajectory the maximum of scalogram is a constant curve
timetimex
y ξ
ξ
for a chaotic trajectory the maximum of scalogram is time-dependent
Extraction of resonances from wavelet decomposition of a single Extraction of resonances from wavelet decomposition of a single trajectorytrajectory
xx
xx xxxx
yy yy yy
yy
timetime
ξ
[ ] 310 10 ,8 ×=t[ ] 310 4 ,2 ×=t[ ] 310 2 ,1 ×=t
Rydberg atoms in crossed fieldsRydberg atoms in crossed fieldsE t( )
B
z
B BH t E x ωt y ωt L
22 21 1
( , , ) ( cos sin )2 2 8
p x p xx
= − +α+ + +
] [
0
1
0,1
α =α =
α∈
linear polarization
circular polarization (2 d.o.f.)
elliptic polarization
time
freq
uen
cy
x
y
freq
uen
cy
time
>> CPCPxxBB
>> EPEPxxBB
TimeTime--frequency decomposition for strongly chaotic trajectoryfrequency decomposition for strongly chaotic trajectory
transition from resonant zonetransition from resonant zone 1:2:1:: 321 =ωωω to strongly chaotic zoneto strongly chaotic zone
TimeTime--frequency decomposition for weakly chaotic trajectoryfrequency decomposition for weakly chaotic trajectory
1:4:2:: 321 =ωωωlonglong--time trapping for t~20 pstime trapping for t~20 ps
planar carbonyl sulfide (OCS)planar carbonyl sulfide (OCS)
1R 2R
OC
S
α
m Ii
i ii
H T P P P R VP RVR RAR RR3 3
1 2 1 21
1 2 31
( )( , , , , ( ,, (,) ))α= =
= α + +∑ ∏
quartic polynomialMorsekinetic energy Sorbie-Murrell potential
1
23
2
13
21
32
2
22
1
12213
22
221
1sinsincos
22cos
22 R
PP
R
PP
RRRRPPPPPT
αμ−
αμ−⎟⎟⎠
⎞⎜⎜⎝
⎛ αμ−
μ+
μ+αμ+
μ+
μ= αα
α
( )2)( 0
1)( iii RRiim eDRV −β−−=
( ))~(tanh1)( iiiiI RRRV −γ−=
∑ ∑ ∑ ∑++++=i ji kji lkji
lkjiijklkjiijkjiijii RRRRcRRRcRRcRcRRRP, ,, ,,,
321 1),,(
α−+== cos2 212
22
13 RRRROSR
References:Carter, Brumer, J. Chem. Phys. 77 (1982), 4208Foord, Smith, Whiffen, Mol. Phys. 29 (1975), 1685Bunker, J. Chem. Phys. 37 (1962), 393
>> 3 modes: OC stretch, CS stretch, OCS bend
a.u.100.0-1cm 90021 at ==+→ ESOCOCS
COR
CSR
1 unimolecular dissociation of OCS
2 intramolecular energy redistribution
OCS potentialOCS potential
equilibrium
Energy redistributionEnergy redistribution
Question: time scale for energy transfer
example: 2D coupled Morse oscillators, Henon-Heiles, cyclobutanone
1 ?τ =
λOHC 64
1τ
λ
1λ
time (ps)
Reference: Davis, Wagner, ACS Symposium (1984)
collinear OCS (2 d.o.f.)collinear OCS (2 d.o.f.)
Reference: Davis, Chem. Phys. Lett. (1984)
ps 10
ps 1.0
ER ≈τ
≈τλ
Frequency analysis applied to planar OCSFrequency analysis applied to planar OCS
>> normal mode frequencies
0 1CO
0 1CS
0 1b
2087 cm
873 cm
520 cm
−
−
−
ω =
ω =
ω =
1T
c=
ω
0CO
0CS
0b
0.016 ps
0.038 ps
0.064 ps
T
T
T
=
=
=
CO
CS
b
87
36
21
n
n
n
=
=
=
1.4 ps
>> resonance CS/bend 3:2 or 2:1
>> resonance CO/bend 4:1 or 7:2
0CO0b
0CS0b
4.01
1.68
ω≈
ω
ω≈
ω
COCSb
CO
CS
b
1 (ps )−ω
>> locking of CS/bend into 3:2 resonance
>> long time of resonance junctions
3: 2
3:1
CS/
bC
O/b
time
CS/
bC
O/b
CO
/CS
O
CS
R1R1R2R2
),,,()(),,,,,( 321
3
1321321 RRRVRVPPPRRRTH INTi
ii ++= ∑
=
)cos(2 ,~2
tanh1),,( )(
2122
213
3
1321
α
γ
RRRRRRRR
RRRRPARV
iii
i
iiINT
−+=−=Δ
⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ Δ
−ΔΔΔ= ∏=
polynomial quartica is )Δ,Δ,(Δ , 321 RRRP
Hamiltonian of the planar carbonyl sulfide (OCS)Hamiltonian of the planar carbonyl sulfide (OCS)
masses reduced theare where,sinsin
cos2
12
1cos12
12
1
13
2
23
1
2132
222
11
221
3
22
2
21
1
iRPP
RPP
RRRRPPPPPT
μμ
αμ
α
μα
μμα
μμμ
αα
α
−−
⎟⎟⎠
⎞⎜⎜⎝
⎛−++++=
Hamiltonian:Hamiltonian:
bend stretch CS stretch CO
α
3 modes:3 modes:α
SorbieSorbie--Murrell interaction term:Murrell interaction term:
SorbieSorbie--Murrell termMurrell termMorse potentialMorse potentialkinetic termkinetic term
casecollinear for the distances mequilibriu theare ~ ,~ where
,1),,(,,,,,,
321
iiii
lkjilkjiijkl
kjikjiijk
jijiij
iii
RRRs
sssscssscsscscsssP
−=
++++= ∑∑∑∑
OCS potential surface OCS potential surface
a.u.100.0-1cm 90021 at ==+→ ESOCOCS
equilibriumCSR
COR
( )a.u. 1.0
)1(1),,(
)cos(2
31
2)(231
3
1
2)(321
2122
213
0222
0
=+≥
−++=−=∞→=
−+=
−−
=
−−∑DDE
eDDDeDRRRVE
RRRRR
d
RR
i
RRid
iii ββ
α
dE
saddlesaddle
saddlesaddle
ReferenceReference: : C. C. Martens, C. C. Martens, et al, Chem. Phys. Lett.,et al, Chem. Phys. Lett.,142, 142, 519, (1987) 519, (1987)
Phase space bottlenecks to longPhase space bottlenecks to long--time energy relaxationtime energy relaxation
M. J. Davis, M. J. Davis, J. Chem. Phys.,J. Chem. Phys., 83,83, 1016, (1985)1016, (1985)
32.5 ,2/)35(
≤≤+=
γγ
Barriers for diffusion of trajectoriesBarriers for diffusion of trajectories
intramolecular energy relaxation intramolecular energy relaxation are related to phase space structuresare related to phase space structures
33--DOF systemDOF system
3D constant energy surface3D constant energy surface2D invariant tori 2D invariant tori 3D invariant tori3D invariant tori
5D constant energy surface5D constant energy surface
22--DOF systemDOF system
barrier to chaotic transport in 3barrier to chaotic transport in 3--DOF system is 4D subspace:DOF system is 4D subspace: 0)( =ωF
TimeTime--frequency analysis for hydrogen in EP fieldsfrequency analysis for hydrogen in EP fields
Periodic trajectoryPeriodic trajectory
Spectral data versus FLI resultsSpectral data versus FLI results
weakly chaotic trajectoryweakly chaotic trajectory
Spectral data versus FLI resultsSpectral data versus FLI results
strongly chaotic trajectorystrongly chaotic trajectory
TimeTime--frequency planes for 3D hydrogen in crossed fieldsfrequency planes for 3D hydrogen in crossed fields
weakly chaotic trajectoryweakly chaotic trajectory
TimeTime--frequency planes for strongly chaotic trajectoryfrequency planes for strongly chaotic trajectory
dts
uttf
ssuWf ∫
∞+
∞−⎟⎠⎞
⎜⎝⎛ −
Ψ= *)(1
),(
s
η=ω
2),(1),( :Scalogram suWfs
suPW =
TimeTime--frequency decompositionfrequency decomposition
ReferenceReference: : P. Goupillaud and A. Grossmann and J. MorletP. Goupillaud and A. Grossmann and J. Morlet , , GeoexplorationGeoexploration ((19841984 ))
1 , >⎟⎠⎞
⎜⎝⎛ −
Ψ ss
uttt
))(Re( tψ
freq
uen
cy
timeGrossmann Grossmann -- Morlet wavelet transform Morlet wavelet transform
i t tt e e2 2/2( ) η − σψ =
1 , <⎟⎠⎞
⎜⎝⎛ −
Ψ ss
ut