Wavelet Analysis and Arnold Web Picture for Detecting Energy Transfer inHamiltonian Dynamical System

Akira Shojiguchi∗Physics Department, Nara Womens University Kita-Uoya-Nishimachi Nara, 630-8506, Japan

Akinori Baba and Chun-Biu LiNonlinear Scienece Laboratory, Department of Earth and Planetary Sciences,

Faculty of Science, Kobe University, JST/CREST, Nada, Kobe 657-8501 Japan

Tamiki KomatsuzakiNonlinear Scienece Laboratory, Department of Earth and Planetary Sciences,

Faculty of Science, Kobe University, JST/CREST, Nada, Kobe 657-8501 Japan andDepartment of Theoretical Studied, Institute for Molecular Science, Myodaiji, Okazaki 444-8585, Japan

Mikito TodaPhysics Department, Nara Womens University Kita-Uoya-Nishimachi Nara, 630-8506, Japan

(Dated: November 23, 2005)

In this paper, we introduce a technique to investigate the inhomogeneity of the phase space andmonitor energy transfers among its partial degrees of freedoms in nonergodic systems; this techniqueemploys wavelet analysis and a picture of the Arnold web. First, we examine several simple energytransfer processes, e.g., motion on a resonance line, between resonance lines, and around a resonancejunction in a simple three degree-of-freedom (DOF) system. Then, we show how the elementalprocesses of the intramolecular vibrational energy redistribution (IVR) are detected by our tools.In particular, we explain that the structure of the higher-order resonance of the system can bedetected by wavelet analysis and motion in the action space. Next, we analyze the reaction processin a simple Hamiltonian system with 3 DOF and a double-well potential, i.e., a system with atransition state of the center-saddle-center type, and we detect energy transfers in the reactionprocess. We demonstrate that our tools can divide the entire trajectory into several simple motions.

I. INTRODUCTION

From the conventional viewpoint, the concept of strongchaos is used to support the foundation of statistical me-chanics. An equilibrium system, which can be success-fully described by statistical mechanics, has a homoge-neous phase space. An important result of statisticalmechanical approaches is that the statistical transitiontheory has been successfully used to explain many re-action phenomena [1]. This theory is based on the as-sumption of the time-scale separation between the timescale of thermalization in the reactant well and that ofreaction, i.e., the local ergodicity assumption. This as-sumption means that this theory does not require anyinformation on the phase-space structure of the reactantwell and all members in the reactant well with the sameenergy have an equal chance to react.

However, recent experimental results detected an errorin the statistical transition theory [1, 2] and questionedthe ergodicity assumption [3]. In particular, Yamanouchiand others observed intramolecular vibrational redistri-bution (IVR) of acetylene and found hierarchical struc-tures in the spectra [3]. These structures indicate thatbottlenecks exist between fast motions and slow ones,

∗Electronic address: shojiguchi@ki-rin.phys.nara-wu.ac.jp

which result in a breakdown of ergodicity. This resultalso reveals that in nonergodic reaction dynamics it isimportant to observe the IVR process.

The motivation for this study is to understand how thereaction coordinate effectively gains dynamical energyfrom other degrees of freedom (i.e., bath coordinates) inchemical reactions, thereby avoiding the thermalizationof the redistributed energy. In such a reaction system,the phase-space structure should not be homogeneous,i.e., the system is never ergodic. To deal with noner-godic dynamics, we require a technique to characterizethe inhomogeneity of the phase space. Weak chaotic sys-tems can be studied for this purpose. These systems havenonuniform phase-space structures that are nonuniformmixtures of tori and chaotic sea [4]. The nonuniformitymeans that there are regions with special properties thatdo not appear in other spaces. The question that arises ishow can we characterize the inhomogeneity of the phasespace of nonergodic systems? The key concept that an-swers this question is nonlinear resonance. The frequen-cies in a nonlinear system vary with the action variables[4], and infinite energy transfer between some degrees offreedom does not occur as it does in a linear resonantsystem. Due to the action-value dependence of nonlinearfrequencies, the resonance conditions generate a networkin the action space (each condition generates a line orplane). This network is called the Arnold web [4].

In a nonlinear system, the members of energy trans-

2

fers change corresponding to a transition from one res-onance line (plane) to another in the action space. Anergodic system has a dense Arnold web; this assures thatany orbit can visit any location in the phase space. Onthe other hand, nonergodic systems have a sparse Arnoldweb. Thus, we can characterize the inhomogeneity of thephase space by means of the Arnold web and detect theIVR processes as motions in this web[5]. Moreover, theArnold web can be used to describe the doorway of thechemical reaction, which indicates that the phase spaceis not uniform and that there is a specific pathway con-necting to the transition state in the space in the reactantwell.

To observe the IVR process, wavelet analysis has beenperformed[6–8] ] and it has succeeded in characterizingthe properties of weak and strong chaos[8]. In this paper,we present a technique to investigate the inhomogeneousphase-space structures of Hamiltonian systems by incor-porating the Arnold web picture with wavelet analysis.First, we analyze the most elemental processes of IVRfrom the Arnold web picture and determine their char-acteristics. Next, we construct a simple reaction systemand examine a reactive orbit with our tools by applyingknowledge of the elemental processes. The goal of thisstudy is to propose a technique to characterize the inho-mogeneity of the phase space, e.g., the reactive doorway,which leads to the controllability of the chemical reac-tion by light, i.e., the control of the reaction by selec-tively preparing an initial state in the reactive doorwayby optical excitation.

In section 2, we briefly explain the roles of nonlinearresonance in energy transfers or IVR and introduce theconcept of the Arnold web. In section 3, we analyzeelementary processes of energy transfers, which are de-scribed in terms of nonlinear resonance, by using waveletanalysis and extract the characteristics of these processes.In section 4, we construct a simple reaction system withan inhomogeneous phase-space structure and investigatethe IVR process in reactive trajectories. Section 5 is de-voted to the conclusion and discussion. In the appendix,we provide a brief explanation on the properties of thewavelet analysis.

II. NONLINEAR RESONANCE AND ARNOLDWEB

A. Nonlinear resonance

The canonical perturbation theory provides a way toremove the angle variables from the Hamiltonian andconstruct a new Hamiltonian in terms of the constants ofmotion, i.e., the action variables. However, if the systemis essentially nonintegrable, there exists a point in theaction space at which the perturbation series diverges. Anonlinear resonance appears at the divergent point. Us-ing the perturbation procedure, we can make a Hamilto-

J1

J3 J2

Resonance line

resonant

motion

FIG. 1: The resonant motion in the action space with reso-nance lines. The motion occurs transversally to the resonanceline.

nian in the following form:

H = H0(J) +H1(J ,θ) (1)

Therefore, the resonance phenomena indicate the impos-sibility of removing the angle variables from the Hamil-tonian (1). Nonlinear resonance occurs as a result of amatch in the resonance condition among more than twofrequencies as follows:

∑

i

niωi(J) = 0, (nis are integers), (2)

where, ωi(J) = ∂H0(J )∂Ji

. However, since the frequenciesof nonlinear oscillators depend on the action variablesas described above, the energy transfer due to the reso-nance generally occurs at a set of action variables thatsatisfy the resonance condition (2). The resonance pro-cess is explained as follows. If the resonance conditionis satisfied, energy transfer occurs; however, due to theenergy transfer, the frequencies gradually change withthe action variables. As a result of this change, the en-ergy transfer is suppressed until it stops. Then, since theresonant point is stable, reverse energy transfer occurs.Thus, the resonant motion is described as a slow oscil-lation around the resonance condition (2) in terms of anangle variable: θ̂ =

∑i niθi. IIn fact, the Hamiltonian

near the resonance can be approximately described bythe Hamiltonian of a pendulum [4]. The resonant mo-tion has a finite width that corresponds to the amplitudeof the pendulum; this width indicates a region in whichthe resonance has influence.

The resonance condition (2) generates a set of pointsin the action space, especially in a 3-DOF system. Theintersection between this set of points and the constant-energy surface forms a line in the action space; therefore,this set of resonant points is called the resonance line.In Fig. 1, we show the motion of a resonant orbit in theaction space with the corresponding resonance line. Sincethe interaction part in the Hamiltonian (1) (1.1) typicallygenerates many resonance conditions, the Hamiltoniansystem generally has many resonance lines that intersect

3

FIG. 2: The resonant widths of two crossing resonance linesand the resonance overlap region between them.

each other and form a network in the action space. Thenetwork of resonance lines is called the Arnold web[4].

Although the simple oscillating motion near the reso-nance line implies that the system has an approximatedconstant of motion, i.e., the action variable assigned tothe pendulum near the resonance, it leads to an ex-tremely slow change in the action variable and the trajec-tory trapped by the resonance eventually moves outsidethe resonance region. This change in the action vari-able generates a motion along the resonance line, whichis called the Arnold diffusion. However, the motion alongthe resonance line is less important than that across theresonance line in the energy transfer between different de-grees of freedom[5] because of the time-scale separationbetween these motions.

Generally, a new frequency generated by a resonancebetween two frequencies can also resonate with the origi-nal frequencies or frequencies of other degrees of freedom;this leads to the generation of another new frequencycalled secondary resonance. Although this process cancontinue endlessly and there appear higher-order reso-nances without limit, their strengths and widths expo-nentially decrease with the order.

B. resonance overlap

In Fig. 2, we show individual resonant motions on twocrossing resonance lines from several initial conditions.. Around the crossing point (called the resonance junc-tion), we find an overlap between the resonant motionon one resonance line and that on the opposite line. Asexplained in the previous subsection, the coexistence ofmore than two interacting resonances generates higher-order resonances. Although a dividing surface (KAMtorus) generally exists between two resonances, if the tworesonances are close enough to each other, the KAM torusdisappears and a stochastic (chaotic) region develops be-tween them. This is the standard scenario by which chaos

develops in Hamiltonian systems. The overlap region in-dicated in Fig. 2 corresponds to the stochastic region;this region is termed as the resonance overlap region.Therefore, if we place an initial point on a resonance linein the overlap region, it leads to a transition between theresonance lines due to the chaos that develops in this re-gion. This transition indicates a change of the membersparticipating in the interaction, and the energy transfercorresponding to the transition is an elemental process ofIVR in Hamiltonian systems.

Thus, since the resonance overlap region is the originof chaotic motion and it is the stage where the elemen-tal process of IVR occurs, the structure of resonancesdescribed by the Arnold web can accurately character-ize the dynamical inhomogeneity of the system, i.e., wecan state that the location where the resonance lines aredensely distributed has high mobility and, in contrast,the location with sparse density has low mobility.

III. ARNOLD WEB AND WAVELET ANALYSISTO DETECT IVR

A. A minimal model describing IVR

To analyze the motion of the transition between reso-nance lines in the resonance overlap region, we introducea minimal model with only two primary resonance lines.It should be noted that the most generic structure con-sists of two crossing resonance lines that contribute tothe IVR process. Our model is described by the follow-ing Hamiltonian:

H=H0 +H1, (3)

H0=3∑

i=1

(p2

i

2+ω2

i q2i

2+ bq4i

),

H1=a1q21q

22 + a2q

21q

23 .

The unperturbed Hamiltonian H0 consists of nonlinearoscillators of the Hamiltonian p2

2 + ω2q2

2 + bq4, which isintegrable with the following constant of motion:

J(h)=12π

∮dqp(q)

=A

3k2

[(2k2 − 1)E(k) + (1− k2)K(k)

], (4)

where h = p2

2 + ω2q2

2 + bq4, k2 = α2

α2+β2 ,

α2 = −ω2+√

ω4+16bh4b , β2 = ω2+

√ω4+16bh4b , A =

2α2

π

√2b(α2 + β2), and with the elliptic functions

E(k)=∫ π/2

0

dφ

√1− k2 sin2 φ

K(k)=∫ π/2

0

dφ1√

1− k2 sin2 φ

4

J1

J3 J2

Resonance linetransition

through

resonance

overlap

ω 1

= ω 2

ω 1

= ω 3

FIG. 3: Transition from one resonance (ω̄1 = ω̄2) line toanother ( ω̄1 = ω̄3) is shown in the action space. The initialpoint is placed on the resonance line corresponding to thecondition ω̄1 = ω̄2.

From the unperturbed Hamiltonian, we obtain the non-linear frequencies up to the first order of the action vari-able as follows:

ω̄i(Ji) =∂H0

∂Ji= ωi +

3bω2

i

Ji, (i = 1.2.3)

where we use the action variables introduced in Eq. (4)for calculating the nonlinear frequencies. The interactionHamiltonian generates the following two primary reso-nance conditions:

ω̄1 = ω̄2, ω̄1 = ω̄3. (5)

We use the following system parameters: ω1 = ω2 =ω3 = 1, a1 = a2 = 0.1, b = 0.5.

B. Transition between resonance lines and itswavelet analysis

In Fig. 3, we depict the transition between resonancelines defined by the resonance conditions (5) when weplace the initial point on the resonance line correspond-ing to the first condition of Eq. (5). Now, we apply thewavelet analysis described in the appendix to the transi-tion orbit. In Fig.4, we show the wavelet transform of theaction variable J1 of the orbit shown in Fig.3, the orig-inal signal of J1 , and decompositions of the total orbitshown in Fig. 3 corresponding to the plateaus marked inthe wavelet transform.

We can find several plateaus in the wavelet transformof J1 in Fig. 4 (a)), which indicate periodic motions oc-curring in the time regions marked in the time series (Fig.4 (b)))). We can also observe excitations of a wide rangeof frequencies when a sudden change occurs in the signal(cf. Appendix). We extracted several segments of mo-tions in the action space corresponding to the plateausof the wavelet transform in Fig. 4 (c) and (d). From Fig.4(c) and (d) ), we can infer that the motions are tem-porarily trapped by the two resonance lines. However,

the motion depicted in Fig. 4 (e) ) is not trapped bythe primary resonance (this will be a higher-order reso-nance), and it is unclear as to what resonance traps thismotion. In Fig. 4, we superpose a 2-DOF motion, whichis generated by the Hamiltonian in Eq. (1.3) with a1 = 0and an initial condition on the resonance line. We alsoshow the wavelet transform of J1 of the motion. Fromthis figure, we can deduce that the motion and frequencyof partial motion trapped by a resonance line and 2-DOFmotion are in very good agreement. This indicates thatthe motion trapped by a resonance line is restricted to a2-DOF motion.

To provide further information on the resonance condi-tions, we present the wavelet transform of J1, J2 and J3

and the time evolution of the frequency ratios ω̄2/ω̄1 andω̄3/ω̄1 in Fig. 6. By observing the wavelet transforms ofall degrees of freedom, we can determine which degreesof freedom are interacting at a given point of time. Fromthe time evolution of the frequency ratios, we can obtainthe precise resonance conditions at the plateaus in thewavelet transforms. We determine the resonance condi-tion of the higher-order resonance extracted in Fig. 4 (e)as 2ω̄1 = ω̄2 + ω̄3. Thus, the wavelet analysis provides in-formation on the higher-order resonance, and by feedingthis information to the Arnold web, we can obtain infor-mation on the structure of the higher-order resonance inthe action space.

C. Motion on the resonance junction

The motion around a resonance junction has notbeen analytically explained without the restricted case[9].However, this motion has an important role in the globalmotion in an inhomogeneous phase space. We analyzethis motion with the model described by the Hamilto-nian (1) and try to detect its characteristics with ourtools; the results are described in this subsection. In Fig.7, we show the wavelet transforms of the action variablesof the orbit, which starts just from the resonance junc-tion. In these transforms, we do not find a long plateaubut short fragmentations of excitations. This fact indi-cates that transitions between resonance lines occur veryfrequently in the system. Although Chandre et al. [8]pointed out that the very short fragmentations of excita-tion in the wavelet transforms are signatures of a chaoticsystem, we attribute the origin of these fragmentationsby means of the Arnold web picture to the very frequenttransitions between resonance lines.

IV. ENERGY TRANSFER IN THE REACTIONPROCESS

In this section, we examine a reaction process by ap-plying our tools discussed in the previous section. Witha simple reaction system, we detect the IVR processes ina reactive orbit. We use a 3-DOF reaction system with

5

0.05

0.1

0.15

0.2

0.25

0.3

}

}

} 0

0.05 0.1

0.15 0.2

0.25 0.3

0.35

0.16 0.18

0.2 0.22

0.24 0.26

0.28 0.3

0.32 0.34

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Plateau 1

Plateau 2Plateau 3

0 0.05

0.1 0.15

0.2 0.25

0.3 0.35

0.16 0.18

0.2 0.22

0.24 0.26

0.28 0.3

0.32 0.34

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 0.05

0.1 0.15

0.2 0.25

0.3 0.35

0.16 0.18

0.2 0.22

0.24 0.26

0.28 0.3

0.32 0.34

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

(a)

(b)

(d) (f)(e)

J3

J1

J2

Time evolution of J1

J3

J1

J2 J3

J1

J2

an

gu

lar

fre

qu

en

cy

[

rad

/un

it t

ime

] Wavelet transform of J1

ω 1

= ω 2

ω 1

= ω 3

FIG. 4: (a) is the wavelet transform of the action variable J1 and (b) is the time evolution of J1 of the orbit shown in Fig.3(c), (d), and (e) are the partial motions in the action space corresponding to the plateaus detected by the wavelet analysis.

Blue line: non-reactive

resonant oscillation

w/o interataction with J2

i.e. a1=0

angula

r fr

equen

cy

[

rad/u

nit

tim

e]

Dots: partial motion on Arnold web

{

"line2.dat""line3.dat"

"jjj1d4_2.dat"

0 0.05

0.1 0.15

0.2 0.25

0.3 0.35

0.16 0.18

0.2 0.22

0.24 0.26

0.28 0.3

0.32 0.34

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35 ω 1

= ω 2

ω 1

= ω 3

FIG. 5: The wavelet transform of the action variable J1 andcorresponding motions on the Arnold web of partial motiondescribed in Fig. 4 (e) and 2-DOF motion by setting a1 = 0in the Hamiltonian in Eq.3.

a double-well potential, i.e., a system with a transitionstate of the center-saddle-center type, described by thefollowing Hamiltonian:

H=H0 +H1,

H0 =p21

2− λ2q21

2+ bq41 +

3∑

i=2

(p2

i

2+ω2

i q2i

2+ bq4i

), (6)

H1 =exp[−(q1 − 1)2/σ2

] [a1q

22q

23 + a2(q1 − 1)2(q22 + q23)

].

where q1 is the reaction coordinate and q2 and q3 are thebath coordinates. The bath coordinates have the sameaction variables and nonlinear frequencies as in Eqs. (4)and (5). The action variable of the reaction coordinate

is as follows:

J(h) =√

2λ3π

(α+ β) [E(k)− αβK(k)] ,

where p212 −

λ2q21

2 +bq41 , −λ2

4 < h < 0, α =

√1−

√1 + 4h

λ2 ,

β =

√1 +

√1 + 4h

λ2 , and k = β−αβ+α . The nonlinear fre-

quency up to the first order of the action variable (??)is

ω̄1 =∂H0

∂J1= ω1 − 3

4J1.

We use the system parameters ω1 = 1.02, ω2 =0.94, ω3 = 1.04, a1 = 0.1, a2 = 0.9 and b = 0.5, whereω1 = λ/

√2 is the unperturbed frequency around the well

q1 = 1. This system has the following type of resonance:

nω̄1= ω̄i, (i = 2, 3, n = 1, 2, · · · )nω̄1= ω̄2 + ω̄3, (n = 1, 2, · · · )ω̄2= ω̄3.

However, below the energy barrier, there exist three res-onance lines corresponding to the following conditions:

2ω̄1 = ω̄2, 2ω̄1 = ω̄3, ω̄2 = ω̄3.

This system has three resonance junctions. The structureof the primary resonance of the system is described inFig. 9.

Using this model, we analyze an orbit that starts fromthe saddle point, moves toward the well , stays in the well

6

angula

r fr

equen

cy

[

rad/u

nit

tim

e]

angula

r fr

equen

cy

[

rad/u

nit

tim

e]

angula

r fr

equen

cy

[

rad/u

nit

tim

e]

1.2

1.1

1.0

0.9

0.8

ω2/ω

1

ω3/ω

1

FIG. 6: The wavelet transform of J1, J2, and J3 and timeevolution of frequency ratios ω̄2/ω̄1 and ω̄3/ω̄1.

for a considerably long duration, and then escapes fromit. The system has two time scales in the survival proba-bility with respect to the residence time in the well: in theshort term, the probability decreases with a power lawof the residence time, and in the long term, it decreaseswith an exponential of the residence time[10]. The orbitanalyzed here has a large residence time and belongs tothe exponential regime.

In Fig. 8, we show the wavelet transforms of the actionvariables J1, J2 and J3 of the orbit. We find two timeregions in this figure: in the first, the orbit appears tobe trapped by a resonance, and in the second, frequentenergy transfers occur, which implies that the orbit goesaround the resonance junctions. In Fig. 9, we show themotions in the action space corresponding to the two timeregions. We can see that in the first time region, the orbitis trapped by the secondary resonance between the reso-nance lines corresponding to the conditions 2ω̄1 = ω̄2 and2ω̄1 = ω̄3, and in the second region, the orbit encountersa resonant junction and moves around the resonance line2ω̄1 = ω̄2. This fact agrees with the tendency in thewavelet transforms in Fig. 8, i.e., in the second region,the intensity of the wavelet signals of J1 and J2 is strong;in other words, the interaction mainly occurs between thefirst and second coordinates.

J1

J2

J3

J1

J3 J2

ω 1

= ω 2

ω 1

= ω 3

FIG. 7: The wavelet transforms of the action variables J1, J2,and J3 of the orbit, which starts from the resonance junction,and its motion in the action space.

V. CONCLUSION AND DISCUSSIONS

We characterized the IVR processes with the transitionbetween the nonlinear resonances and examined them byemploying wavelet analysis and a picture of the Arnoldweb. To investigate the most elemental processes, we an-alyzed the motion of an orbit trapped by a resonance, thetransition between two resonance lines, and the motionaround a resonance junction. Then, we determined howour tools detect these elemental processes. In particular,we found a characteristic of the motion around a reso-nance junction by wavelet analysis. Next, we applied ourtools to a reaction process with a simple 3-DOF systemand found that we can detect a transition process be-tween a static state (state trapped by a resonance) anda frequently interacting state (state around a resonancejunction) by decomposing the elementary processes intoseveral parts. Moreover, by comparing the wavelet trans-forms of all degrees of freedom, we could determine which

7

ang

ula

r fr

equ

ency

[

rad

/un

it t

ime]

ang

ula

r fr

equ

ency

[

rad

/un

it t

ime]

ang

ula

r fr

equ

ency

[

rad

/un

it t

ime]

J1

J2

J3

FIG. 8: The wavelet transforms of the action variables J1, J2,and J3 of a reactive orbit that starts from the saddle point,moves toward the well q1 = 1 and escape from the well.

(b) T = 8000 -- 12500(a) T = 2000 -- 8000

J1

J3 J2

J1

J3 J2

2ω 1

= ω 2

2ω 1

= ω 3

ω 2

= ω 3

FIG. 9: The partial motion in the action space of time periods(a) t = 2000 to 8000 and (b) t = 8000 to the escaping point(t = 12000).

one participates in the interaction at each time point.Our tools can be applied to a system with higher de-

grees of freedom by trailing orbits in a projected actionspace of lower degree-of-freedom in conjunction with thewavelet analysis. Further, by comparing the wavelettransforms of different degrees of freedom, we can de-duce the effective degrees of freedom that participate inthe motion at each time point.

Two issues should be addressed in the future. First,the IVR processes in a massive system that has a fewhundred degrees of freedom should be investigated; sec-ond, the entire pathway connecting the saddle and wellshould be understood. For the first issue, the embeddingmethod [11] provides a way to count the effective degreesof freedom that participate in the interaction[12]; the re-currence plot[13] of an orbit of a system enables us todecompose the orbit into units of approximately periodic

angula

r fr

equen

cy

[

rad/u

nit

tim

e]

FIG. 10: The wavelet transform of sin ωt with ω =0.1[rad/unit time]. The vertical and horizontal lines repre-sent the frequency and time, respectively.

motions in several time scales[14]. For the second issue,a statistical investigation of the system, i.e., an analysisfrom the ensemble viewpoint, can yield new informationon the inhomogeneity of the phase space[10]. A visual-ization of the structures of the higher-order resonancesis required for investigating shorter time scales and theregion near the saddle.

Acknowledgments

This work was supported by the Japan Society for thePromotion of Science and a Grant-in-Aid for Science Re-search from the Ministry of Education, Culture, Sports,Science and Technology in Japan.

APPENDIX A: APPENDIX

The wavelet analysis is basically a time-dependent fre-quency analysis (see [15] for detail). We use a continu-ous wavelet transform and present the result visually byshowing the distribution of the frequency amplitudes ingrayscale(online: show intensity blue to red). The con-tinuous wavelet transform of f(t) is written as follows:

(Twavf)(a, b) = |a−1/2|∫dtf(t)ψ

(t− b

a

), (A1)

where we use the Gabor type mother wavelet ψ(x) =1

2√

πσexp

[−ix− x2

σ2

], which constructs complete set and

does not satisfy orthogonality. a and b describe frequencyand time dependence, respectively.

In this section, we describe the basic properties of thewavelet transform by using elemental examples. First,we show the wavelet transform of sinωt in Fig.10. . Thevertical and horizontal lines represent the frequency aand time b, respectively. Since this signal is completelyperiodic with the angular frequency , the wavelet trans-form shows a time-independent horizontal line (exactlyshows a band due to the tradeoff property between thevalidity of the frequency and that of time). In Fig.11, weshow the wavelet transform of a signal whose frequencychanges gradually with time as follows:

(1+tanh(t−30000)) sin(ωt)+(1+tanh(−t+30000) sin 2ωt.

8

ang

ula

r fr

equ

ency

[

rad

/un

it t

ime]

FIG. 11: The wavelet transform of the function (1 + tanh(t−30000)) sin(ωt) + (1 + tanh(−t + 30000) sin 2ωt with ω =0.1[rad/unit time].

angula

r fr

equen

cy

[

rad/u

nit

tim

e]

FIG. 12: The delta-function-like signal sin ω(t−30000)t−30000

(bottom)

and its wavelet transform (top) with ω = 0.1[rad/unit time].

We see that the wavelet transform can detect the fre-quency transition at a given point of time. In Fig.12, weshow how a sudden change in the signal amplitude or aspike signal is described by the wavelet analysis. We canobserve an excitation of frequencies over a wide range atthe peak of the spike. This is on account of the fact thata sharp spike includes a very high number of frequencies.

[1] T. Bear and W. Hase, Unimolecular Reaction Dynamics:Theory and Experiments (New York, Oxford, 1996).

[2] T. Shibata, H. Lai, H. Katayanagi, and T. Suzuki, J.Phys Chem. A102, 3643 (1998).

[3] K. Yamanouchi, N. Ikeda, S. Tsuchiya, D. Joans,J. Lundberg, G. Adamson, and R. Field, J. Chem. Phys.95, 6330 (1991).

[4] A. Lichtenberg and M. Lieberman, Regular and ChaoticDynamics 2nd Ed. (Springer-Verlag, 1991).

[5] J. Laskar, Physica D 67, 257 (1993).[6] C. Martens, M. Davis, and G. Ezra, Chem. Phys. Lett.

142, 519 (1987).[7] A. Askar, A. Cetin, and H. Rabitz, J. Phys. Chem. 67,

257 (1996).

[8] C. Chandre, S. Wiggins, and T. Uzer, Physica D 181,171 (2003).

[9] G. Haller, Chaos Near Resonance (Springer-Verlag,1999).

[10] A. Shojiguchi and M. Toda, (to be published).[11] M. Kennel, R. Brown, and H. Abarbanel, Phy. Rev. A

45, 3403 (1992).[12] A. Shojiguchi, S. Dai, and M. Toda, (to be published).[13] M. Casdagli, Physica D 108, 12 (1997).[14] S. Dai, A. Shojiguchi, and M. Toda, (to be published).[15] I. Daubechies, Ten Lectures on Wavelets (Springer-

Verlag, 1992).