Knowledge of classical periodic orbits and their bifurca-tions is of great importance in the study of nonintegrableHamiltonian systems. Periodic orbits also play an essential role in investigations of the classical-quantum corre-spondence for such systems. In this paper, we providedetailed study of the periodic orbits in a model Hamil-tonian describing molecular vibrations. Symmetries aris-ing from invariance of the equations of motion undertime-reversal are fully exploited, as is a new permuta-tional representation of the periodic orbits. In this way, aglobal understanding of possible bifurcation sequences ispossible.
I. INTRODUCTION
The importance of the study of periodic orbits~pos! innonintegrable classical Hamiltonian systems has been apciated since the time of Poincare´.1–3 Knowledge of po loca-tions and stabilities provides an understanding of phspace structure and transport.3 Quantities such as escaprates and correlation functions can be expressed in termpo properties.4
Much interest attaches to the process of po creationdestruction with changes in system parameter;5 the study ofpo bifurcations can provide essential insights into the rofrom simple ~integrable! dynamics to complicated, chaotidynamics as nonintegrability is introduced.
The study of classical pos has received added impfrom recent work in semiclassical po theory on the classicquantum correspondence in classically nonintegra
a!Electronic mail: [email protected]!Present address: Department fo Chemistry, Stanford University, Stan
systems.6,7 The essential result here is the Gutzwiller traformula,6 which is a semiclassical expression for the osciltory part of the density of states of a quantum systemterms of the periods, actions, stability parameters aMaslov indices of the classical pos of the corresponding csical system.6 A large amount of work has been devotedexploring the relation between quantum properties and csical pos.4 For both atomic8,9 and molecular10 systems, it ispossible to extract classical po periods and actions direfrom quantum spectra. For example, Delos and coworkhave applied Hamiltonian bifurcation theory to analyzebifurcation patterns derived from spectra of the H-atom inmagnetic field.11
Study of classical pos is important in the interpretatiof molecular vibrational spectra.10,12–14 The pos and theirassociated resonance zones15,16 provide the means to assighighly-excited states in regimes where traditional specscopic methods, based on a harmonic oscillator~integrable!limit picture, fail.17
In this paper we study the periodic orbits of a classinonintegrable Hamiltonian describing two coupled anhmonic ~Morse! oscillators. Our system is a model for molecular effective vibrational Hamiltonians used tospectra,18 and consists of a zeroth-order integrable partgether with a number~up to two, here! of resonant couplingterms. The motivation for our work is the need to understathe organization of classical pos in our system, in order toable to analyze the quantum spectrum via semiclassicatheory.19 Special emphasis is placed on the classical pofurcation sequences. However, the present investigationthe classical bifurcations stands alone as a systematic attto understand the transition from integrability to nonintegbility in a fairly typical molecular vibrational Hamiltonian.
Bifurcations of fundamental classical pos have bestudied in detail by Kellman and coworkers for the integra
820 Chaos, Vol. 9, No. 4, 1999 M. Tsuchiya and G. S. Ezra
case where only a single~1:120 or 2:121! coupling term ispresent. Moreover, semiclassical po theory has been apto these integrable systems.22,23Although the integrable limitmanifests rich structure in the dependence of phase sstructure on system parameter,20,21 it is necessary to understand what happens as integrability is broken by the preseof an additional coupling term.24
Efficient and systematic determination of pos in boarea-preserving maps and continuous time Hamiltoniantems requires the exploitation of symmetries associatedthe invariance of equations of motion under involution oerations such as time-reversal.25–27The search for pos in the2D surface of section is then reduced to an analysis ofintersections of thesymmetry linewith its iterates under thereturn map,25,26,28as intersection points must lie on pos.25 Akey problem then becomes one of understanding the orgzation of ~symmetric! pos along the symmetry line.
A new approach to this problem has recently beenveloped by Tsuchiya and Jaffe´ ~TJ! in their study of periodicand homoclinic bifurcations in the area-preserving He´nonmap.28 A permutational representationis used to describethe intersection configuration of a symmetry line and itserate, and thereby enumerate pos and their periodicitiesgiven order of iteration. Using the permutational represention to understand po organization at each order of iteraenables one to construct the whole set of~symmetric! pos.Local topological constraints on the way in which orbits apear and disappear then allow the formulation ofpruningrules, which in turn enable theglobal bifurcation structurefor symmetric pos to be elucidated. This analysis isscribed in more detail below, while the mathematical foudation for the permutational representation is providedRef. 28. For the case of the area-preserving He´non map, forwhich the strong coupling limit is fully chaotic~hyperbolic!,it is possible to develop a full understanding of po bifurction patterns.28
As described in more detail in Tsuchiya’s thesis,28 thefull power of the permutational representation approachillustrated by the ability to generate the permutational repsentation foranyorder of iteration of the~hyperbolic! HenonmapT from knowledge of the intersection of the symmetline Si with its first iterateTSi ( i 51,2), using a set of indi-ces~cf. Sec. IV C of the present paper!. Once the permutation representation is determined, global bifurcation analyfollows via the pruning rules. Furthermore, TJ have alsoplied the permutational analysis to homoclinic orbit bifurctions on the stable and unstable manifolds of the primunstable fixed point.~Full accounts of this work are currentlin preparation.28! However, since the coupled Morse systeconsidered here is a mixed system, where no indices ethe permutational representations are obtained fully numcally for SiùTmSi , mÞ2n; for m52n, the permutationalrepresentation can always be determined.
In the present paper we apply this approach tocoupled Morse Hamiltonian. We extend the theory by uswinding numbers to characterize pos; this extension is hful for formulating pruning rules for symmetric pos icontinuous-time Hamiltonian systems, as opposed to apreserving maps.28 Several distinct pruning sequences m
ed
ce
ce
s-th-
e
ni-
-
-r a-n
-
--n
-
is-
is-
-y
st,ri-
egp-
a-
be possible for a given po sequence. In addition to determing these sequences, our global analysis of the coupMorse system reveals the mechanism responsible for thepearance of ubiquitous tangent bifurcations in the noningrable case. Moreover, by analyzing po pairing on two symetry lines, we can obtain information on po stabilitiehence changes in local phase space structure. Bifurcationa coupled Morse system have also been studied by Weand Child.29
This paper is organized as follows: Section II introducthe model vibrational Hamiltonian and reviews the topictime-reversal symmetry and symmetry lines. Section IIItroduces the concept of the permutation representation, wSec. IV applies the method to the integrable limit~singleresonance coupling! at low and high energies. Periodic orbpruning patterns are discussed in Sec. V, and Sec. VI exines global bifurcation patterns of symmetric pos in bothintegrable and nonintegrable limits. Bifurcations occurrion the second symmetry line are discussed in Sec. VII. Cclusions are given in Sec. VIII. Some technical points aaddressed in the Appendix.
This paper analysis serves as a foundation for studythe classical-quantum correspondence in our system;work will be presented elsewhere.19
Our emphasis here is on a global understanding of~sym-metric! po bifurcation structure; that is, we wish to undestand how the results of local Hamiltonian bifurcatioanalysis5 ‘‘fit together’’ in the large. In recent related workDelos and coworkers have used normal form theory to demine the fundamental dynamical origin of certain corretions between different po bifurcations that had been nonumerically.30 Such correlations are also the subject of tpresent global analysis using the permutation representaOne key finding of our work here is the importance of certacodimension-2 bifurcations in mediating the transition frointegrability to nonintegrability.31 The influence ofcodimension-2 bifurcations in semiclassical approximatioto the density of states has recently been examinedSchomerus.32
II. CLASSICAL MECHANICS: SYMMETRY ANDPERIODIC ORBITS
A. Classical Hamiltonian
The model vibrational Hamiltonian we study describtwo anharmonic~Morse! oscillators coupled by two resonanterms, a 1:1 resonant termH 1:1 and a 1:2 resonant termH 1:2. In terms of canonical action-angle variable(I 1 ,I 2 ,u1 ,u2), the Hamiltonian is
H5H01H 1:11H 1:2, ~1!
whereH0 is the zeroth-order uncoupled Morse Hamiltonia
and 2p<(u1 ,u2)<p. We shall takev151.0, v250.8,a150.03 anda250.02.
The classical phase space structure ofH depends on thevalues of the parametersE, b11 and b12. For b1250, theHamiltonianH is integrable, and periodic orbit~po! bifurca-tions and classical phase space structure have been esively studied in this limit.20 In the present work, we focusattention on the onset of nonintegrability~chaos! as the twoparametersE andb12 change.
Semiclassical analysis of the level spectrum of the qutum version of Hamiltonian~1! results in plots of po periodt 22 or action S19 versus energyE. Interpretation of theseresults therefore requires investigation of families of clascal pos and their bifurcations in the classical system~1! as afunction of the two parametersE andb12.
B. Time reversal symmetry and symmetry lines
Our study of po bifurcations requires a systematic aproach to the location of pos, as opposed to a random sein the surface of section. It is well known that periodic orbin both area-preserving maps25 and continuous timesystems26 can be found efficiently by locating the intersetions of symmetry lines and their iterates. The symmelines are sets of points invariant under involutiooperations.25 An important example of an involution is thoperation of time reversal.
For a Hamiltonian of the usual kinetic energy plus ptential form,
H5p2
21V~q!, ~4!
Hamilton’s equations of motion (q,p)5(]H/]p,2]H/]q)are invariant under the time reversal operation
t→2t, ~5a!
p→2p, ~5b!
q→q. ~5c!
The surfaceM5$(p,q)up50,q50% is therefore a ‘‘mirrorplane’’: orbits initiated in this plane will follow the sametrajectory in q-space in both forward and backwards timAn orbit that returns toM after starting onM is thereforeperiodic. Such pos intersectM at two different points; it isnot possible to have three or more distinct intersectpoints.
The model Hamiltonian~1! does not have the ‘‘kineticplus potential’’ form. The equations of motion (I ,u)5(]H/]u,]H/]I ) are nevertheless invariant under the folowing time-reversal operation:
u j~2t !52u j~ t ! or 2u j~ t !62p, j 51,2 ~6a!
I ~2t !5I ~ t !, ~6b!
where the vector (u,I )5(u1 ,u2 , I 1 , I 2). The mirror planeM is then the surface
M5$~u,I !uu j50 or 6p, I j50, j 51,2%, ~7!
whereu j5p is identified withu j52p.
ten-
n-
i-
p-rchs
-ryn
-
.
n
l-
For our model vibrational Hamiltonian~1!, the intersec-tion of the mirror planeM with the surface of constant energy E is topologically a circle. The four combinations oangles (u1 ,u2), Q1[(0,0), Q2[(p,0), Q3[(p,p), andQ4[(0,p), are then associated with the first, second, thand fourth quadrants of the symmetry line, respectively.
Both to facilitate numerical integration of the equatioof motion and to avoid dealing with discontinuities in anglar coordinatesu, we define cartesian-like coordinatesq andconjugate momentap via the canonical transformatio(I i ,u i)→(pi ,qi), i 51,2:
qi5A2I i /v i cosu i , ~8a!
pi5A2I iv i sinu i . ~8b!
From Eq.~6!, we see that the time reversal operation in~p,q!space is precisely that given in Eq.~5!, where on the mirrorplaneM, p(t50)50 and q(t50)50. At fixed energyE,the analogue of the equipotential boundary is determinedthe equation
H~q,p50!5E. ~9!
The mirror boundary forE56.98 is projected intoq-space inFig. 1.
The set of trajectories initiated on the circle definedEq. ~9! will eventually intersect the Poincare´ surface of sec-tion ~sos!. With an appropriate choice of sectioning condtion, the resulting symmetry line has a simple functionform on the sos. We define four sos by the following contions:
S1[$u250,u2>0%, ~10a!
S2[$u25p,u2>0%, ~10b!
S3[$u150,u1>0%, ~10c!
FIG. 1. Mirror boundary in (q1 ,q2) space. Points on the boundary lie at thintersection of the mirror planeM with the surface of constant energy,E.Four quadrantsQi are shown, together with several important periodic obits. Trajectories started onM trace out the same path inq-space in bothforward and reverse time. EnergyE56.98, b1250.0, b1150.01.
ad
t
rttr
heho-
eir
n-s ae,
asncymi-
c-e
trictwoup
e-con-
of
tion-
are
-
nts
822 Chaos, Vol. 9, No. 4, 1999 M. Tsuchiya and G. S. Ezra
S4[$u15p,u1>0%. ~10d!
The relation between symmetry linesS, sosS and quadrantsQ is shown in Table I. For example, the symmetry lineS1 onsosS1 is the line p150. The two branchesu150 and u1
5p correspond to the union of the first and second qurants,Q1øQ2 . The symmetry lineS28 on S2 is the linep1
50, and corresponds toQ3øQ4 . The second symmetry lineS2 onS1 is the image of the setS28 ; it is not a straight line onthe sosS1 , but has a more complicated form~see Fig. 2!.The symmetry of pos that lie onS2 in S1 is therefore not asobvious as that for symmetric pos onS1 , where the relevansymmetry operation is:p1→2p1 andq1→q1 . This compli-cation for S2 stems from the fact that the origin of time (t50) for S2 is on the sosS3 , not S1 .
In addition to periodic orbits of finite period that staand end on symmetry lines, there can also exist symmehomoclinic orbits, biasymptotic to the same~unstable! pos as
FIG. 2. Symmetry lines on the surface of sectionS1 , whereS1 is defined
by the conditionS1[$u250, u2>0%. Symmetry lineS1 is the linep150,and is invariant under the momentum reversal operationI 15R:pi→2pi .Symmetry lineS2 is the heavy curve passing through period-1 fixed poion S1 , and is the set of points invariant under the involutionI 25TR.EnergyE54.6, b1150.01, b1250.02.
TABLE I. Surfaces of section, symmetry lines and quadrants.
Section Symmetry line Quadrant
S1 HS1[p150S25image of S28
HQ1øQ2
Q3øQ4
S2 HS28[p150S185image of S1
HQ3øQ4
Q1øQ2
S3 HS3[p250S45image of S48
HQ1øQ4
Q2øQ3
S4 HS485p250S385image of S3
HQ2øQ3
Q1øQ4
-
ic
t→6`. The existence of these homoclinic orbits on tsymmetry lines suggests accumulation of pos to themoclinic orbits.28
In the rest of the paper, we investigate pos and thbifurcations on theS1 andS2 sos as the two parametersEand b12 vary. To study the classical-quantum correspodence, it is necessary to obtain po periods or actions afunction of energyE. Tangency problems can however ariswhere, as a parameter~e.g.,E! is changed, a po onS1 ~say!becomes tangent toS1 , and so disappears from the sosthe energy continues to change. In principle, such tangeproblems can be eliminated by using sos defined dynacally by fundamental pos~cf. Sec. VI B 1!. Missing pos alsooccur on the boundary ofS1 where uq1u has its maximumvalue in (p1 , q1) space; switching to a complementary setion such asS3 , which is still intersected by the po, thaction of such orbits can still be obtained.
C. Poincare return map as a product of involutionoperators
In this subsection, we summarize results on symmepos of area-preserving mappings that are products ofinvolution operators.25–28These results are used in settingthe permutational representation~see below!.
For a continuous-time Hamiltonian system with timreversal symmetry, such as the coupled Morse systemsidered in this paper, Greene showed that the Poincare´ returnmapT, T:S°S, can in general be written as the producttwo involutions:26
T5I 2I 1 , with I i251 ~ i 51,2!. ~11!
The mapT is invertible,T215I 1I 2 . Two symmetry linesSi
are defined as invariant sets for each involutionI i on S
Si[$zuI i z5z,zPR2% ~ i 51,2!. ~12!
For the coupled Morse system, one possible factorizais I 15R, I 25TR, whereR is the momentum reversal operatorR:pi→2pi . It is straightforward to check thatTR isan involution, i.e., thatRTR5T21, and that the symmetryline S2 on the sosS1 is invariant underI 2 . The symmetryline S1 on S1 is clearly invariant underR.
The following results can be proved:25–28
~1! Intersections between a symmetry line and its iteratespos. The set of pointsSiùTn2mSi is equivalent underthe mappingTm to TmSiùTnSi .
~2! The maximum period of pos is 2n for intersections be-tween the symmetry lineSi and itsnth iterateTnSi , sothat the po set forSiùTnSi consists of pos with periodicities that are factors of 2n.
~3! See, e.g., Ref. 28:
~a! If a fixed pointz(0) of a period-2n periodic orbitlies on the symmetry lineS1(S2), then thenth
iterate of the fixed point,z(n) also lies on theS1(S2) symmetry line.
~b! If a fixed pointz(0) of a period-(2n11) po lies onthe S1(S2) symmetry line, then the (n11)th iter-ate of the fixed pointz(n11) lies onS2(S1).
~c! If none of the fixed points of a period-n po lies oneither symmetry line, then there exists a secoperiod-n po that is an image of the first orbit under either involution operation.
~d! No other cases occur.
III. PERMUTATIONAL REPRESENTATION OF POS
Periodic orbits are obtained by iteration of the symmelines. The difficulty with this~and other approaches! is thatthe number of symmetric pos of given period isa prioriunknown. In this paper, we apply the method of Tsuchand Jaffe´ ~TJ!28 to both enumerate and predict the orderpos on time-reversal symmetry lines. The result of tanalysis can then be used to provide a global understanof po bifurcations.
The key idea of the TJ analysis is that a labeling ofintersections of the symmetry line and its iterate~see Fig. 3!is equivalent to a permutation. For the surface of sectshown in Fig. 3, for example, the permutationP(S1ùTS1) is
P~S1ùTS1!5S 1 2 7 6 5 4 3 8
1 2 3 4 5 6 7 8D . ~13!
Note that relabelling of points gives rise to a permutatthat is the conjugate of the original; nevertheless, the unlying po sequence is invariant. In the usual column represtation of the permutation, each column is an intersection,hence corresponds to a point on a po. From TheoremSec. II C, it follows that~for T a diffeomorphism!, the per-mutational representationP(SiùTn2mSi) is invariant underthe mapT:
P~Tm~SiùTn2mSi !!5P~TmSiùTnSi !
5P~SiùTn2mSi !. ~14!
FIG. 3. Intersection of the symmetry lineS1 and its iterateTS1 on thesurface of sectionS1 . (E56.0, b1250.02). The eight intersection points iS1ùTS1 lie on pos of period 1 or 2; the permutationP(S1ùTS1) ~Eq. 13!describes the configuration of fixed points.
d
y
afsng
e
n
r-n-dof
This means that study of the intersections ofSi with its iter-atesTnSi , n>1, provides complete information on symmeric pos.
A. Structure of permutations and associatedperiodicities
From Theorem 3 in Sec. II C, permutations representintersections of iterated symmetry lines consist solelycycles of length one~unit column! ( i
i) or two ~pair of col-umns! (k
j )( jk) ~cf. Ref. 28!. This result is easily understood b
considering the time-reversal mirrorM. Orbits starting fromM in, say, Q1øQ2 , either return directly to the originapoint after one period, or intersectM at a different point inQ1øQ2 before returning to the original point~cf. Fig. 1!, sothat there are either one~unit column! or two intersectionpoints ~pair of columns! on the correspondingS1 symmetryline. Periodicities of pos associated withP(SiùTnSi) aredetermined by the requirement that
TnS ii D5S i
i D ~15!
and
T2nS jkD5TnS k
j D5S jkD , ~16!
so that the period corresponding to the unit column is onethe factors ofn, whereas the period of the po associated wa pair of columns is a factor of 2n, other thann itself or 1.For instance, periodicities of pos for the permutation~13! are
`~P~S1ùTS1!!5~11,12,21,22,13,22,21,14!, ~17!
whereml denotes thel -th distinct po of periodm on the sosconsidered. Note that, if a period-2 po appears onSiùTSi asa pair of columns, then it will be represented by a unit cumn in P(SiùT2Si); pos for SiùTSi form a subset of thepos forSiùT2Si .
B. Generating higher order permutations fromP„Si"TSi…: mixed systems
In order to enumerate symmetric pos, it is importantknow how each order of permutation is generated frP(SiùTSi) and to understand how each order of iteratiorganizes pos on the symmetry lines. For example, knoedge of the permutational representation forSiùT6Si deter-mines the relative ordering of period-3 and 4 pos onsymmetry line, which cannot be determined by knowledgethe intersectionsSiùT3Si or SiùT2Si alone. The basic prin-ciple and fundamental difficulty involved in generatinhigher orders of permutation from the basic intersectP(SiùTSi) in mixed systems are briefly discussed in tAppendix.
isa
tate
os
w
altl
d
h
l
o
ns
mr
hatl
d
rus.he
in
eri-a-t-ctn
retecan
d
824 Chaos, Vol. 9, No. 4, 1999 M. Tsuchiya and G. S. Ezra
IV. PERMUTATIONAL REPRESENTATION ANDPERIODIC ORBITS IN THE INTEGRABLE LIMIT:b1250
The phase space structure and po bifurcation analysthe two Morse oscillator system coupled by a 1:1 resoninteraction is well understood.20,22 The angle C25 1
2(u1
1u2) is ignorable, so that the conjugate variableJ25I 1
1I 2 is a constant of the motion. The existence of a consof the motion in addition to the energy means that the sysis integrable. We consider the integrable limit withb1250,and fixedb1150.01. In this integrable system, almost all poccur in 1-parameter families on rational 2-tori~i.e., a torusfor which the fundamental frequency ratiov1 /v25m/n!.Thus, po ‘‘bifurcation’’ refers to the emergence of a nefamily of pos~rational torus! in the vicinity of an isolated poas the energy or coupling parameter changes. It shouldnoted that all rational tori intersect theS1 andS2 symmetrylines, so that the whole set of~marginally stable! pos existson both symmetry lines in the integrable limit. As we shsee, understanding pos and their bifurcations in the ingrable limit (b1250) provides the foundation for globaanalysis of po bifurcation for nonzerob12. In this section,we show how permutational representations can be useunderstand po organization on the sosS1 .
A. Bifurcation sequences for S1"TS1 and S2"TS2
Consider for example the caseb1150.01, b1250 andE52.0. For these parameter values, both symmetry linesS1
and S2 are intersected by their iterates at two points. Tpermutational representations are
P~S1ùTS1!5S 1 2
1 2D ~18!
and
P~S2ùTS2!5S 1 2
1 2D . ~19!
The corresponding periods are
`~P~S1ùTS1!!5~11,12! ~20!
and
`~P~S2ùTS2!!5~11,12!. ~21!
The pair of fundamental~period 1! pos onS1 is identicalwith the pair of fundamental pos onS2 . We refer to the orbit11 as thecentral periodic orbit~cpo!; it is a generalized locamode.20 The orbit 12 is called theprimary resonant mode~prm!, and is a generalized normal or resonant mode.20 Bothorbits are stable, so that bifurcations of pos can occurthese orbits.
As the energyE increases, the pattern of intersectioP(S1ùTS1) changes as follows
~23!where a family of period-2 rational tori first emerges frothe local mode 11 , (21,11,21), followed by a saddle-cente~tangent! bifurcation, (13,14), where 13 and 12 are, respec-tively, stable and unstable pos~Fig. 4!. The po 13 is anothergeneralized local mode.
On S2 , the bifurcation sequence is
~11,12!→~22,11,22,12!→~13,14,22,11,22,12!, ~24!
where another resonant period-2 bifurcation (22,11,22) oc-curs. In the Appendix, it is shown thatS1 and S2 have thesame period-1, but not period-2, pos in common. Note tthe two period-2 orbits 21 and 22 lie on the same rationatorus. More generally, the appearance of a period-n orbit(n.1) (n1,11 ,n1) on theS1 symmetry line is accompanieby the appearance of a complementary po (n2,11 ,n2) on S2 ,where the complementary po lies on the same rational toIn the integrable limit, both pos are marginally stable; in tnonintegrable case (b12Þ0), one of the pos will be stableand the other unstable~see the discussion of pairing of posSec. VII!. For the integrable case, analysis of pos onS1 alonesuffices to determine po periods and actions.
So far we have examined bifurcation sequences numcally. Numerical construction of the bifurcation tree as a prameter such asE increases, although in principle straighforward, is unsatisfactory insofar as it is difficult to predianalytically which bifurcations will occur as a perturbatioparameter increases~using, for example, normal formanalysis31!. In contrast, if a strongly chaotic limit exists folarge values of the coupling parameter, for which a complset of unstable pos can easily be enumerated, then oneformulatepruning rulesgoverning the merging of orbits an
FIG. 4. Period-1 and period-2 pos onS1 in the integrable limit. The diagramshows the relative location of pos on the symmetry lineS1 as a function ofenergyE, from low E ~bottom of the figure! to high E ~top!. The couplingparameterb1250.
the appearance of stable pos as the coupling parametreduced. TJ have applied this approach to the He´non map.28
Such a ‘‘reverse’’ approach can be applied to~23!, althoughin the coupled Morse oscillator system the strong coupllimit is by no means fully chaotic~hyperbolic!. Once pruningpatterns are established, possible po pruning sequencebe enumerated. Complications can arise if bifurcations ocin the reverse direction. For example, Schomerus32 foundthat certain normal forms lead to po bifurcations followedinverse bifurcations as a single parameter is varied; we hfound such ‘‘nonmonotonic’’ behavior in the system exained here~see below!. The TJ approach establishes theglo-bal structure of the po bifurcation diagram by revealing tpo bifurcation or pruning patterns implicit in the topologythe intersections of the symmetry line and its iterates.
In any particular problem, additional information is usally required to decide between several possible allowpruning sequences determined by the TJ approach. Forample, in the reverse~pruning! direction in Eq. ~23!, wecannot determine the order in which the saddle-center bication (13,14) and the resonant period-2 bifurcatio(22,11,22) occur. In the next subsection we introduce twinding number assignment for pos as a ‘‘pruning gramar’’ that enables some possibilities to be eliminated.
B. Winding number assignments of pos
The permutational representation of pos specifies theriodicity m together with an additional indexl , wherem issimply the number of distinct intersections of the po with tchosen sos~S1 , say!. For the case of a continuous timsystem, as opposed to an area-preserving map,28 it is impor-tant to have additional information on the nature of theThewinding numberfor the po,m/n say, specifies a periodicity n in a complementary sos~S3 , say! in addition tom. Inthe integrable limit, the winding number is simply the ratof fundamental frequencies on a given rational torus.
Close to the bifurcation point, two pos involved intangent~saddle-center! bifurcation must have the same number of intersections with any sos. Conversely, two pos wdifferent winding numbers cannot merge in a tangent bifcation; this constitutes a strict pruning rule. An analogorestriction also holds for other kinds of bifurcation e.g.,land chain, in nonintegrable systems.
For example, the sequence of winding numbers forbifurcation sequence~23! is
~~1/1!cpo,~1/1!prm!→~2/3,~1/1!cpo,2/3,~1/1!prm!
→~1/1,1/1,2/3,~1/1!cpo,2/3,~1/1!prm!.
~25!
Note that the sequence of numerators in~25! matches theperiod sequence~23! determined by permutation analysis othe sosS1 . The denominators, on the other hand, specperiodicities on sosS3 , so that the winding numbers provida link between permutational representations onS1 andS3 .
is
g
canur
ve-
dx-
r-
-
e-
.
h-s-
e
y
C. Permutational representations of pos for highenergy
We now show how the order of pos on time-reverssymmetry lines can be enumerated and predicted. At the rtively high energyE56.98 ~the dissociation energy of thweakest bond is 8.0!, there are 14 intersections forS1ùTS1
on theS1 sos. The permutational representation is
P~S1ùTS1!5S 1 2 13 4 11 6 9 8 7
1 2 3 4 5 6 7 8 9
10 5 12 3 14
10 11 12 13 14D . ~26!
An important empirical observation is that the sum of adcent numbers in the first row of the permutation~26! fromthe second column alternates between the values 15 andThe permutation~26! is said to haveindices^15,17&.28 Indi-ces, when they exist, provide a compact specification ofstructure of the intersection configuration.28 ~Note that thereexists a set of indices for each order of permutation inhyperbolic Henon map. These indices can be used to genate permutations corresponding to arbitrary orders of itetion, P(SiùTnSi), (n.1), from P(SiùTSi).
28! The permu-tation representation~26! also shows that there is symmetaround the cpo, (8
8), which is manifested in the followingsequence of periods@determined by the permutation~26!#@cf. ~23!#:
where the subscript distinguishes between orbits havingsame periodicity. The winding number sequence~28! showsthat the winding numbers (n/m) on the left side of the cpobetween 1/14 and (1/11)cpo decrease monotonically as thcpo is approached, and then increase monotonically toright ~up to (1/12)prm!. On theS1 (u250) sos, the symmetryline S1 is the line p150, which consists of two branchesu150, corresponding toq1>0, andu15p, corresponding toq1<0. The cpo is an orbit of minimal actionI 1.0 and maxi-mal actionI 2 ; the actionI 1 therefore increases as we movway from the cpo on either side, and the behavior ofwinding numbers follows from the negative mode anharmnicities (a i.0). Moreover, as we shall show, the orderinof winding numbers can be understood using a Fareyconstruction.33
For the local mode 1/13 , winding numbers (n/m) on theleft side of the cpo between 1/13 and (1/11)cpo decreasemonotonically towards the cpo. On theS3 (u150) sos, thelocal mode 1/13 is at the center with minimal actionI 2 and
.,
se
14e
u-e-eur-ibleeine
rsm--r
3,oc-
826 Chaos, Vol. 9, No. 4, 1999 M. Tsuchiya and G. S. Ezra
maximal actionI 1 . The winding numbers are flipped, i.em/n, and monotonically decrease towards 1/13 .
At the same energy there are 26 intersections at theond iteration,S1ùT2S1 , with indices^27,29&. The permuta-tional representation is
P~S1ùT2S1!5S 1 26 3 24 5 22 7 20 9 18
1 2 3 4 5 6 7 8 9 10
11 16 13 14 15 12 17 10
11 12 13 14 15 16 17 18
19 8 21 6 23 4 25 2
19 20 21 22 23 24 25 26D , ~29!
The 14 pos of~27! have period 1 or 2, and so appear incycles of length 1 inP(S1ùT2S1). The period sequenc`(P(S1ùT2S1)) in ~29! is then determined uniquely:
There are 38 intersections forS1ùT3S1 with indices^39,41&. The permutation is given by
P~S1ùT3S1!
5S 1 2 37 4 35 6 33 8 31 10 29 12
1 2 3 4 5 6 7 8 9 10 11 12
27 14 25 16 23 18 21 20 19 22 17
13 14 15 16 17 18 19 20 21 22 23
c-
24 15 26 13 28 11 30 9
24 25 26 27 28 29 30 31
32 7 34 5 36 3 38
32 33 34 35 36 37 38D . ~32!
Periodicities for the 1-cycles~unit column! are either 1 or 3,and for the 2-cycles either 2 or 6. Knowledge of the permtation ~32! is not sufficient to determine the location of priod 1 versus 3~or 2 versus 6! pos in the sequence. Onpossible way to assign periodicities is to analyze the bifcation sequences associated with each possconfiguration.28 Here, in the integrable limit, we use thFarey tree organization of the winding numbers to determthe periods.
In a region of the symmetry line where winding numbeincrease or decrease monotonically, pos with winding nubers (j /k) and (l /m) have lying between them a po of winding number (j 1l )/(k1m), where the new winding numbeis obtained by Farey addition~add numerators anddenominators!.33
The symmetry of the cycle structure in~32! shows thatthe 1-cycle (20
20) is the cpo (1/1)cpo. Consider winding num-bers of the first four columns in~32!. Due to the monotonicincrease of winding numbers towards 1/13 , the period-3 po3/2, represented by a 1-cycle inP(S1ùT3S1), will appearbetween 1/13 and 4/31 in ~31!. Farey addition of windingnumbers between 3/21 and 1/14 leads to the po sequence 4/5/4, 6/5,.... The unit column (2
2) therefore corresponds tperiod-3, (3
37) and (44) are period six and period one, respe
tively. Similarly, period-3 pos between (44) and (20
20) can belocated by Farey addition of winding numbers from~28! and~31!:
For higher order permutationsP(S1ùTiS1), (i>4),2-cycles ~pairs of columns! appear near (1/1)prm. For ex-ample, in P(S1ùT4S1), which has 52 intersections,period-8 po with winding number 8/8 appears in the vicinof the prm (1/1)prm
P~S1ùT4S1!5S 49 48 47
..., ,...
47 48 49D , ~36!
where (1/1)prm corresponds to (4848) and the period-8 to the
2-cycle (4749
4947). From ~36!, one can see also that the perm
tation no longer has good indices^53,55& due to the occur-rence of the third pruning center (1/1)prm.
It is interesting to note that frequency ratios~windingnumbers! are constant in the 1:1 resonance region~i.e., allhave winding numbersm/m51!. Constant frequency ratioin resonance zones were found by Laskar using localquency analysis of near-integrable systems.34
By continued Farey addition, winding number sequencan be constructed for any order of iteration, where windnumbers in the 1:1 resonant region in the vicinity of (1/1)prm
are constant.
V. PRUNING PATTERNS OF POS
In a mixed system, regular and irregular regions areterleaved in phase space in an extremely intricate mann3
As a result, high-order iterates of the symmetry line can hvery complicated shapes, and it is difficult to determinebifurcation patterns as system parameters change~from thehigh energy limit, for example! by considering all possibledeformations of the iterated symmetry line. It is howevpossible to construct possible global bifurcation diagramsfinding pruning ~bifurcation! patterns at the pruning~bifur-cation! points for the pos on the symmetry lines. Once tpossible pruning patterns are determined, it is straightward to construct pruning~bifurcation! sequences. This isaccomplished by starting with the order of the pos onsymmetry lines for some energy and finding where the pring patterns appear in the periodic sequence. Those pothe patterns are then pruned, and the process continuednext level. In this way, one can see how retreat from the fuchaotic limit occurs~for systems such as the He´non map28!,and in the process construct global bifurcation diagrams
For the system of coupled Morse oscillators under stuhere, we find two relevant pruning patterns~see Fig. 5!. Thefirst ~called type I ! involves three sequential intersectiopoints (ni ,m,nj ), such as an island-chain type bifurcatio(k5n/mPN) or a period-doubling type bifurcation (n/m52). In a typeI pattern, two points (ni ,nj ) are pruned at thepruning center,m. This pattern can be understood as folows: if local analysis~such as a normal form5! indicates thatthe local functional behavior of the iterate follows~generi-cally! a cubic polynomial, then two of the three root(ni ,nj ) become degenerate or complex conjugate at thefurcation point. Note that, ifiÞ j , that is, if the two points(ni ,nj ) are different pos, a pair of identical pruning even(ni ,m,nj ) occurs at exactly the same parameter value. Iimportant to note that, for theiÞ j case, the pruning centerm
-
-
sg
-r.e
o
ry
er-
e-inthey
y
i-
is
must be part of a 2-cycle in the permutation in order to haa pair of identical pruning events; otherwise the seque(ni ,m,nj ) is not a pruning sequence. Furthermore, in ttype I pattern, the period-m po is necessarily stable just aftepruning. Stability of the orbit is indicated by superscripts,ms.
The second pruning pattern~type II ! involves two adja-cent intersection points (ni ,nj ). Here, the two points (ni ,nj )are pruned at the same time. IfiÞ j , a pair of identical prun-ing events occurs at the same parameter value. The typIIbifurcation is atangent bifurcation. Tangent bifurcation canoccur whenever two pos of the same period appear asadjacent columns, ( i
, ji 11j 11,
¯). The local functional behav-ior is ~generically! quadratic, so that two real solutions bcome complex conjugate after pruning/prior to bifurcatioAccording to the Poincare´–Birkhoff theorem,3 a stable andunstable po pair is pruned in the typeII case.
Note that the pruning sequence of the so-calledtouch-and-go bifurcation5,11 (n1 ,m,n2), ~k5n/mPN for k53l !,(n1 ,m,n2)→(n1 ,n2 ,m) or (m,n1 ,n2)→(m), is consideredto be a special case of the typeI pruning. In the rest of thispaper, we show how the origin of ubiquitous tangent bifucations seen upon turning on the nonintegrability paramb12 can be understood in terms of typeI and II pruningpatterns.
VI. GLOBAL BIFURCATION ANALYSIS
The existence ofpruning degeneracy28 means that sev-eral different possible sequences of bifurcations are possstarting from a given periodic sequence. In order to narrthe range of possibilities, we introduce winding numbersignments for pos. For the nonintegrable case, this assment is based on the number of intersections with sosfined by the two fundamental pos~see below!. Byeliminating possible tangency problems~see below!, we en-sure that typeI and II pruning ~bifurcation! can only occurfor pos with the same winding number.
FIG. 5. Two basic pruning patterns for the coupled Morse system.diagram shows two essentially different ways in which the pattern of insections of a symmetry lineSi and its iterate can undergo qualitative chanwith variation in system parameter.~a! Type I pruning. This case corre-sponds to either an island-chain or a period-doubling bifurcation.~b! Type IIpruning. This case corresponds to tangent bifurcation.
828 Chaos, Vol. 9, No. 4, 1999 M. Tsuchiya and G. S. Ezra
A. Bifurcations for the integrable system1. Pruning sequence for P(S 1"TS1)
For the winding number sequence~28! associated with the first iterate ofS1 , the full pruning sequence ending with~25!is
where pos to be pruned are underlined. Pruning degeneracy between (1/13,1/14) and (2/31 ,(1/11)cpo,2/31) occurs in the abovepruning sequence~37! ~see below!.
2. Three pruning centers in the integrable limitThe pruning sequence for the winding numbers~31! associated with the second iterate ofS1 , T2S1 , shows that there are
two ‘‘pruning centers.’’ One of the pruning sequences is shown below.
where one center is the cpo and the other center is the3orbit, called thesecond pruning center~spc!. In order to havethe tangent bifurcation of pos ((1/1)3
spc,1/14) shown in~37!for the winding number sequence~35!, the two 4/31 pos, onebetween ((1/1)spc,1/1) and the other to the right of the resnant mode (1/1)prm, have to be pruned at the spc. The pmary resonant mode (1/1)prm therefore acts as a ‘‘stopperfor pruning these three orbits at the cpo.
The above pruning sequence~38! exhibits pruning
1degeneracy.28 Additional information is required to determine the order of prunings at the two centers. Thus, we cnot predict when the orbit 4/31 is pruned at the spc relative tpo prunings at the cpo. Nevertheless, the order of pruningeach center can be determined uniquely.
We can now construct the pruning sequence forunion of the winding numbers~35! up to the third iterate,T3S1 . We reorder the winding numbers~35! to highlight thepruning order of pos at the two pruning centers:
The stable primary resonant mode (1/1)prm acts as a thirdpruning center for 2-cycles centered at the prm; these apfor higher order iterations,P(S1ùTiS1), i>4 @see Eq.~36!#.
As shown above, there are three pruning groups: thegroup is related to the spc, (1/1)spc, the second group to thcpo, (1/1)cpo, and the third to the prm, (1/1)prm. The numberof pos created around those centers at each order of iteris in general different on each sos. In the rest of the papefocus on the three pruning groups independently.
B. Nonintegrable system „b12Þ0…: tangency and two-parameter „codimension-2 … bifurcations
Once the second coupling parameterb12 becomes non-zero, all rational tori break up into stable and unstablevia Poincare´-Birkhoff island-chain type bifurcation.~The is-lands associated with high-order resonances may besmall, however.24! The coupled Morse system now exhibilocally chaotic motion. The polyad number,35 J25I 11I 2 ,ceases to be an invariant, and regular and chaotic comnents coexist in an extremely complicated manner in phspace.
Classical po energy-period (E,t) plots for integrable andnonintegrable cases are shown in Fig. 6. The (E,t) plot forthe integrable limit exhibits a rich structure. Such structurewell understood;13 each trace in the (E,t) plot correspondseither to a rational 2-torus~1-parameter family of pos! or anisolated po,13 and repetitions thereof. In the integrable lim@Fig. 6~a!#, the tangent bifurcation creating the po pair 13 and14 is an important feature of the plot. Rational tori appearbifurcating off isolated~stable! po traces. The correspondin(E,t) plot for the system with nonintegrable perturbati@Fig. 6~b!# is very similar in overall structure to that of thintegrable case, up to the maximum times considered.tailed comparison shows that onset of nonintegrability issociated with the appearance of ubiquitous tangent bifutions derived from the bifurcation structure for the integralimit @~39! and ~40!#. As explained below, these tangent bfurcations emerge from the integrable limit by a mechaniinvolving a codimension-2 bifurcation, i.e., an essentia2-parameter scenario. Codimension-2 bifurcations haveceived relatively little attention~for recent work, see Refs. 3and 32!.
As already mentioned, the system of coupled Morsecillators under study is not strictly hyperbolic in the limit ohigh energy or large coupling parameter. We shall analthe evolution of po sequences from the integrable lim
ar
st
ione
s
ry
o-se
s
y
e--
a-
e-
s-
et
(b11Þ0, b1250) as the coupling parameterb12 increases atfixed energy (E56.98). In this way, the route to tangenbifurcations is revealed.
1. Winding number assignments in the nonintegrablecase
The winding number assignment depends on the sosused. One problem with using a pair of complementarysuch asS1 and S3 , as done in the integrable case, is tpossibility that a po might become tangent to one of the sleading to a discontinuous change in winding number~as afunction of E, for example!. We can however use the twfundamental pos discussed above, 1cpo and 1prm, to definetwo new dynamically determined sos, denotedJ1 andJ2 ,and so avoid the tangency problem.36 Tangency of any po tothe fundamental pos is forbidden~unless this orbit is in-volved in a bifurcation with the fundamental pos!, and so allpos must intersect both sosJ i defined by trajectory crossings of the fundamental pos in configuration space. Newordinates (j i ,h i), i 51,2, are defined by the perpendiculdistance (h i) from the fundamental orbiti and the position(j i) of intersections on the fundamental po, where we tathe origin j i50 to lie on the time-reversal mirrorM. ThesosJ i are then defined byh i50 andj i>0. A winding num-ber n/m is obtained from the number of intersections withe fundamental pos withj i>0, wheren andm are the num-ber of intersections with 1cpo and 1prm, respectively; that is,the number of points on the two sosJ i . It is important to
FIG. 6. Classical (E,t) plots for coupled Morse oscillators,b1150.01.Traces are associated with either isolated pos or rational tori, and repetit~a! b1250, integrable limit. The tangent bifurcation creating the po p(13 ,14) is an important feature of this plot.~b! b1250.022, nonintegrablecase. This plot differs from the integrable case by the presence of madditional tangent bifurcations involving high-order pos.
eth
eis
tht
ane
-le
e
itsi-
thetheypee
bi-r-ingmark
-e
bits
830 Chaos, Vol. 9, No. 4, 1999 M. Tsuchiya and G. S. Ezra
note that the winding number of pos bifurcating from thfundamental pos corresponds to the periodicity of pos ondynamical sosJ i , (i 51,2), and is an invariant quantitycharacterizing the po. The numerator for such pos agrwith the period determined by the permutational analysFor pos that do not bifurcate from the fundamental pos,numerator of the winding number is in general not equalthe periodicity onS1 . In the integrable limit, both windingnumber assignments become equivalent. In the nonintegrcase, however, winding numbers no longer vary monotocally along the symmetry line,34 and the Farey tree structuris disrupted.
2. Abrupt change in the position of the cpo
For b1250.0003 andE56.98, the permutational representation forS1ùTS1 is the same as that for the integrabsystem~26!. The winding number sequence is however
The relative position of the~1/1! cpo on the symmetry linechanges abruptly, so that the winding number sequenc
e
es.eo
blei-
is
different from that of~28!, and the po sequence now exhibsymmetry about the po (1/27)c, which appears at the prevous location of the cpo. The sosS1 (E55.0) for both theintegrable (b1250) and the nonintegrable case (b12
50.0003) is shown in Fig. 7.The winding number sequence clearly shows that
pruning sequence must involve tangent bifurcations of1/2 orbits, as all of the 1/2 orbits cannot be created via tI bifurcations. Sinceb12 is small, a drastic change in thpruning sequence from that of the integrable system,~37!, isnot expected. The first pruning group is just the tangentfurcation (1/13,1/14). For these parameter values, no bifucation appears in the third group at the first iteration. Prunsequences are shown below for the second group. We rethat the pruning analysis for the sequence of periods~27!shows all possible pruning~or bifurcation! sequences, including that for~41!, that are inherent in the topology of thconfiguration of the intersections.
There are two possible prunings for the period-1/2 orin the second group:
where two possible prunings for the tangent bifurcationthe 1/2 orbits are shown inside the curly brackets: ‘‘left tagency’’ ( . . . ,1/26 ,(1/27)c,1/28 , . . . ) or ‘‘right tangency’’(...,1/26 ,(1/27)c,1/28 ,...).
Bifurcation diagrams for~42! and~43! are shown in Fig.8. Numerically, the pruning sequence~42! with one tangentbifurcation ~1/2,1/2! occurs as energyE varies with b12
50.0003, while the sequence~43! with right tangency oc-curs for higher values ofb12 ~see Fig. 8,b1250.015!, so thatthe po 18 is unstable (1/28
u) and the po 16 is stable (1/26s). As
a general rule, we find that, as the coupling parameterb12
increases, so does the number of tangent bifurcations.pair of pos involved in a tangent bifurcation, one is staand the other is unstable. For instance, the po 1/26
s is stable,so that the existence of a tangent bifurcation (1/25,1/26
s)
FIG. 7. Surface of sectionS1 for the coupled Morse system, whereS1 is
defined by the conditionS1[$u250, u2>0%. Period-1 pos on the symmetry line p150 are shown as filled circles. EnergyE55.0. ~a! b1250.0. ~b!b1250.0003. Tangent bifurcation of pos 15 and 17. Note the reordering ofthe central pos on the symmetry line as the nonintegrable perturbatioturned on.
f-
ae
means that the po 1/25 must be unstable, 1/25u . The period-1
po stabilities are therefore
~1/13s,1/14
u,1/25u,1/26
s ,~1/27s!c,1/28
u ,~1/11s!cpo,~1/12
s!prm!.~44!
Note that a change in stability of period-1 pos from unstato stable is not observed in the coupled Morse systemenergies belowE57.0.
The mechanism of onset of tangency for period-1 posdisconnection of the two period-1 orbits from the~inte-grable! bifurcation tree,~37! ~see Fig. 9; cf. also the work oWeston and Child29!. As we shall see below, onset of tangency for higher-order pos follows a different route.
It is important to note that the period-2 orbits in thsequence~41! cannot undergo tangent bifurcations. Thperiod-1 orbits are sandwiched between the period-2 pand so act as stoppers for tangent bifurcations of the periopos. New period-2 orbits~or more period-1 orbits! need toappear in the sequence in order to see tangencies ofperiod-2 orbits.This is the first example where the structuof the period sequence determines which pruning sequenpossible.We show below that the mechanism by which tagent bifurcations of period-n orbits (n>2) appear is differ-ent from that for period-1 orbits. Moreover, analysis of tintersectionS1ùT2S1 ~b1250.02 andE56.98! reveals thatthe onset of tangency of period-2 pos is correlated withtangency of period-4 orbits~see below!.
is
FIG. 8. Schematic bifurcation analysis forS1ùTS1 . The relative orderingof period-1 and period-2 pos on the symmetry lineS1 is shown as a functionof energy for fixed parametersb1150.01 andb12 . Both sequences shownare found numerically.~a! b1250.0003@cf. Eq. ~42!#. Note that the reorder-ing of the three middle period-1 pos on the symmetry line seen in Fig. 7~b!occurs as a consequence of the ‘‘disconnection’’ of the pos 15 , 11 and 18
~cf. Fig. 4!. ~b! b1250.015@cf. Eq. ~43!#. Another disconnection involvingpos 16 , 11 and 17 is seen~cf. Fig. 4!.
ec-
.
832 Chaos, Vol. 9, No. 4, 1999 M. Tsuchiya and G. S. Ezra
FIG. 9. Emergence of tangent bifurcations of period-1 pos by ‘‘disconntion.’’ The value of the coordinateq1 for period-1 pos on the symmetry lineS1 is shown as function of energyE. ~a! b1250.0. Integrable case, cf. Fig4. ~b! b1250.0003. This is the disconnection illustrated in Fig. 8~a!. ~c!b1250.015, cf. Fig. 8~b!.
nfo
-
,
he
3. Tangency for period-2 bifurcations: collision ofperiod-2 bifurcations
As b12 is increased at fixedE, the number of pos in-creases. The number of pos forS1ùTS1 with b1250.02 andE56.98 increases from 14@see~26!# to 16. An increase inthe number of intersections is associated with the creatioa new period-2 orbit. The permutational representationS1ùTS1 is
P~S1ùTS1!5S 1 2 15 4 11 6 9 8 7
1 2 3 4 5 6 7 8 9
10 5 14 13 12 3 16
10 11 12 13 14 15 16D , ~45!
and the corresponding periodic sequence@cf. ~27! and ~44!#is
ofr
`~S1ùTS1!5~13s,14
u,21,15u,22,16
s,23 ,~17!c,23,18u,22,
24 ,~11s!cpo,24,21 ,~12
s!prm!, ~46!
where the period-24 orbit is new. The winding number sequence is
~1/13s,1/14
u,2/31,1/25,2/42,1/26,2/43 ,~1/27!c,2/43,1/28,
2/42,2/34 ,~1/11s!cpo,2/34,2/31 ,~1/12
s!prm!. ~47!
The first pruning group is (1/13,1/14). For the second groupexcept for the new period-2/34 po, the rest of the pruningsequence for~47! follows ~43! with the right tangency, sothat the following two pruning patterns are possible for twinding numbers 2/3 orbits~see Fig. 10!:
Pruning degeneracy occurs for pruning pattern~48! be-tween (2/43 ,(1/2)c,2/43) and (2/34 ,(1/1)cpo,2/34); that is,we cannot determine the order of pruning of the 2/34 and2/43 orbits. However, in order to have a pair of tangentfurcations~2/3,2/3! in the pruning sequence, there can bepruning degeneracy in~49! between (2/43 ,(1/2)c,2/43) and(2/34 ,(1/1)cpo,2/34). As b12 increases at fixed energy, wexpect the pruning sequence of typeI to occur first, followedby the typeII sequence. Numerically, the pruning sequen~48! occurs forb1250.02, while the sequence~49! occurs atb1250.022.~The transition between the two types of behaior occurs atb1250.2118, cf. Fig. 11.!
4. Analysis of codimension-2 collisions for higher-order pos
The above pruning degeneracy between typeI and IIprunings for period-2 pos can be seen easily by considethe permutational representation for the po configurat(2i ,2j ,1
cpo,2j ,2i) ~see Fig. 12!:
P~SiùTSi !5S 5 4 3 2 1
1 2 3 4 5D . ~50!
The permutation~50! is consistent not only with the occurrence of resonant period-2 bifurcations (2j ,1
cpo,2j ) and(2i ,1
cpo,2i), but also with the tangent bifurcation (2i ,2j ) and
(2 j ,2i) ~cf. Sec. V!. In ~50!, the pairs of columns (15
24) and
-o
e
-
gn
(42
51), which are equivalent to po sequences (2i ,2j ) and
(2 j ,2i), define the pruning sequence for the tangent bifurtion. The local behavior of the first iterate ofS1 is shown inFig. 12, where the pruning degeneracy is apparent. For ping of period-2 pos by tangent bifurcation, the two pairspos (2i ,2j ) and (2j ,2i) must be pruned at the same energyE~fixed b12!. For period-3 pos, (3i ,3j ,1,3k,3l), for whichP(SiùT3Si)5(1
122
33
44
55), the two pairs of period-3 pos
(3i ,3j ) and (3k,3l) are in general pruned at differentE.The pruning sequences ‘‘before’’~48! and ‘‘after’’ ~49!
reveal the mechanism for tangency of the period-2 orbFor smallb12, the two period-2 orbits, 2/31 and 2/34 , areboth pruned by typeI bifurcations. Asb12 increases, the twopos collide and separate to form a pair of pos pruned by tII ~tangent! bifurcations~see Fig. 11!. The collision of thetwo period-2 pos is an unlikely event requiring both enerE and coupling parameterb12 to have particular values; thais, the collision is a codimension-2 bifurcation. Suchcodimension-2 bifurcation provides a generic mechanismemergence of tangent bifurcations for period-n (n>2) pos.Detailed examination of po behavior just before collisireveals a more complicated pruning and bifurcation sceninvolving, for period-n (nÞ3), an inverse pitchfork typebifurcation on the cpo, while forn53m, an inverse tangenbifurcation occurs~see Fig. 13!. The above pruning analysishows two limiting regimes~type I and type II ! forcodimension-2 prunings/bifurcations as the coupling para
orils
t
a-
riod-nd
ig.
t
dic
run-
s
tion
n
834 Chaos, Vol. 9, No. 4, 1999 M. Tsuchiya and G. S. Ezra
eterb12 is varied; application of local analysis such as nmal form theory30,32 can presumably describe the full detaof the collision scenario. The pruning patterns discussedSec. V should therefore be taken to describe the net effecmore complicated prunings/bifurcation sequences~Fig. 14!.
5. A possible mechanism for correlation betweentype II bifurcations
The number of intersections forS1ùT2S1 for b12
50.02 andE56.98 increases from 26@see ~29!# to 36,where a period-24 ~two unit columns!, and four period-47210
~four pairs of columns! pos are newly created. The permuttional representation is
P~S1ùT2S1!
5S 1 36 3 34 9 26 27 8 5 32 11
1 2 3 4 5 6 7 8 9 10 11
24 13 22 15 20 17 18 19 16 21
12 13 14 15 16 17 18 19 20 21
14 23 12 31 6 7 28 29 30 25
22 23 24 25 26 27 28 29 30 31
10 33 4 35 2
32 33 34 35 36D , ~51!
FIG. 10. Schematic bifurcation analysis forS1ùTS1 . The relative orderingof period-1 and period-2 pos on the symmetry lineS1 is shown as a functionof energy for fixed parametersb1150.01 andb12 , with b12 values largerthan those of Fig. 8. Both sequences shown are found numerically.~a! b12
50.02. For this case, the period-2 pos 21 and 24 are created by island-chain~type I! bifurcations @cf. Eq. ~48!#. ~b! b1250.022. For this case, theperiod-2 pos 24 and 21 are created by tangent bifurcation@type II , cf. Eq.~49!#.
-
inof
and the corresponding period sequence@cf. ~30!#, where thesequence~46! appears in the unit columns, is
`~P~S1ùT2S1!!5~13s,41,14
u,42,47,48,49,21,47,43,15u,44,22,
45,16s,46,23 ,~17
s!c,23,46,18u,45,22,44,
410,48,49,24 ,~11s!cpo,24,410,43,21,42 ,
~12s!prm,41!. ~52!
The winding number sequence@cf. ~31!# is
~~1/13s!spc,3/31,1/14
u,4/52,4/67,4/68,4/69,2/31,4/67,4/73,
1/25u,4/84 ,2/42,4/85,1/26
s,4/86,2/43 ,~1/27s!c,2/43,
4/86,1/28u,4/85,2/42,4/84,4/710,4/68 ,4/69,2/34 ,
~1/11s!cpo,2/34,4/710,4/73,2/31,4/52 ,~1/12
s!prm,3/31!. ~53!
The pruning sequence in the first group is
~3/31 ,~1/13s!spc,3/31,1/14
u!→~~1/13s!spc,1/14
u!→~B !,~54!
where the 3/31 po on the far right end of~53! now appears onthe left. Note that the winding number for the period-41 po is3/31 rather than 4/31 . The pair ((1/13)spc,1/14) is created~pruned! by a tangent bifurcation~type II !; neither of thesepos bifurcates from the cpo or the prm~see Sec. IV B!. Nothird group bifurcation appears. The presence of the pesequence~46! in ~53! provides a foundation for understanding the bifurcation tree for the period-4 orbits. In the secogroup, there are two pruning subgroups at (1/27)c and(1/11)cpo, respectively. Expanding the sequences~48! and~49! we have the two bifurcation sequences shown in F15. Both sequences shown are found numerically atb12
50.02 andb1250.022, respectively.For b1250.02 ~cf. Fig. 15!, the tangent bifurcation
((1/27)c,1/28) occurs, rather than (1/26 ,(1/27)c) @cf. ~42!and~43!#. The presence of the po (4/67) in ~53! implies thatthe pair of pos (4/68,4/69) has to be involved in a tangenbifurcation. The pair therefore undergoes typeII pruning.The pruning of the pair of pos(4/68,4/69) provides a secondexample of the way in which the structure of the periosequence determines possible pruning of pos.Note also thatthe corresponding columns in~51! for the pair are (6
26727) and
(266
277 ) ~cf. the pruning pattern for typeII in Sec. V!. When
the pair is pruned, however, cannot be predicted due to ping degeneracy between (4/68,4/69) and the prunings at thecenter (1/27)c. At b1250.02, it is found numerically that thepair is pruned right after the period-2 po 2/42 is pruned.Following the final pruning by a pair of tangent bifurcationthere are two possible routes, labeled~a! and ~b!:
~a! For ~48!, the period-2/34 po undergoes typeI prun-ing, followed by two possible prunings~type I andII ! for theorbits (4/73,4/710). For the typeII (b1250.02) case, there ispruning degeneracy between the resonant-2 bifurca(2/34 ,(1/11)cpo,2/34) and tangent bifurcation (4/73,4/710).Numerically, it is found that the tangent bifurcatio(4/73,4/710) occurs first, then (2/34 ,(1/11)cpo,2/34):
FIG. 11. Origin of tangent bifurcations via collisions of period-2 pos.~a!b1250.021. Here, both period-2 pos are created by type-I island-chain bi-furcations @cf. Fig. 10~a!#. ~b! b1250.02117. ~c! b1250.02118. Here,period-2 pos are created by type-II tangent bifurcations@cf. Fig. 10~a!#.
fio
rrateen-
fur-
FIG. 12. Permutational representation and the local behavior of theiterate ofS1 , showing pruning degeneracy for the pruning of period-2 pThe two limiting regimes for codimension-2 bifurcation/pruning are showThe first case~type I! corresponds to island-chain bifurcations; the seco~type-II ! to creation of period-2 pos by tangent bifurcations.
rsts.n.nd
FIG. 13. Origin of tangent bifurcations via collisions of period-3 pos~se-quential touch-and-go bifurcations!. ~a! b1250.009810. For this parametevalue, each stable-unstable pair of period-3 pos is involved in a sepatouch-and-go bifurcation.~b! b1250.009818. For this parameter value, onpair of period-3 pos is not involved in a touch-and-go bifurcation; the ustable po of the other pair is involved in two sequential touch-and-go bications.
836 Chaos, Vol. 9, No. 4, 1999 M. Tsuchiya and G. S. Ezra
IH ~4/52,4/67,2/31,4/67,4/73 ,4/710,~1/11s!cpo,4/710,4/73,2/31,4/52 ,~1/12
s!prm!
↓~4/52,4/67,2/31,4/67 ,4/73 ,~1/11
s!cpo,4/73,2/31,4/52 ,~1/12s!prm!
Jor
II H ~4/52,4/67,2/31,4/67 ,~4/73,4/710!,~2/34 ,~1/11s!cpo,2/34!,~4/710,4/73!,2/31,4/52 ,~1/12
s!prm!
↓~4/52,4/67,2/31,4/67 ,2/34 ,~1/11
s!cpo,2/34,2/31,4/52 ,~1/12s!prm!
J↓
~4/52 ,4/67s,2/31
s ,4/67s ,~1/11
s!cpo,2/31,4/52 ,~1/12s!prm!
↓~4/52 ,2/31
s ,~1/11s!cpo,2/31
s,4/52 ,~1/12s!prm!
↓~4/52 ,~1/11
s!cpo,4/52 ,~1/12s!prm!
↓~~1/11
s!cpo,~1/12s!prm!. ~55!
~b! For ~49! ~occurring atb1250.022!, in order that the period-2/34 po be pruned as typeII , the orbits (4/73,4/710) mustbe pruned as typeII . In other words, if the period-4 po 4/73 or 10 po undergoes typeI bifurcation~the routea,I !, so must theperiod-2 2/31 or 4 po:
The po 4/67 bifurcates from the period-2 po 2/31 , which istherefore stable; the po 2/34 is therefore unstable. Combininresults of po pruning onS1 with an analysis of po pruning onthe S2 symmetry line, the stabilities of more pos candetermined~e.g., the po 4/67 is stable, cf. next section!.
Numerically, typeI bifurcation ~the routea,I ! does notoccur since the period-2 po 2/34 is not created yet at thevalue ofb12 at which the period-4 po 4/710 orbit appears bytype I bifurcation; when the period-2 po 2/34 is created by aresonant period-2 bifurcation, the route (a,II ) occurs. Ouranalysis however clearly shows the possibility of correlatbetween bifurcations of orbits of different periods. Such crelations are expected for largerb12 and higher energy, sincmore newly created pos will appear in the sequence~53!, andbecome involved in tangent bifurcations as the coupling cstantb12 continues to increase.
n-
-FIG. 14. ~a! A complicated bifurcation scheme whose net effect is equilent to a typeI pruning. ~b! Bifurcation scheme involving touch-and-gbifurcation whose net effect is equivalent to typeI pruning.
Finally, from the pruning schemes shown above~55! and~56!, the mechanism of onset of tangency for period-4 poseen to be the same as that of period-2 pos, namely, colliof period-4 bifurcations.
VII. BIFURCATION ANALYSIS ON S2 : PAIRING OFSTABLE AND UNSTABLE PERIODIC ORBITSON SYMMETRY LINES
So far, we have focused on the analysis of pos on theS1
symmetry line. In this section we show how stability of p
FIG. 15. Schematic bifurcation analysis forS1ùT2S1 . The relative orderingof period-1, period-2 and period-4 pos on the symmetry lineS1 is shown asa function of energy for fixed parametersb1150.01 andb12 . Both se-quences shown are found numerically.~a! Bifurcation diagram for the route(a,I ) @see Eq.~55!#. ~b! Bifurcation diagram for the route~b! @occurs nu-merically atb1250.022, see Eq.~56!#.
ison
on the two~S1 andS2! symmetry lines onS1 can be foundby analyzing pruning sequences onS2 . For E56.98 andb1250.02, there are 16 and 32 intersections forS2ùTS2 andS2ùT2S2 . Figure 2 shows thatS2 passes through the samperiod-1 pos asS1 ~see Appendix!, but the relative positionsof the cpo and the period-1 po 1/15 are exchanged. Thischange of relative position is due to the fact that the symmtry line S2 is not simply a straight line onS1 , but a morecomplicated curve~see Sec. II B!. The permutational representation forS2ùTS2 is
P~S2ùTS2!5S 1 2 15 6 5 6 13 8 11
1 2 3 4 5 4 7 8 9
10 9 12 7 14 3 16
10 11 12 13 14 15 16D . ~57!
The period and winding number sequences@cf. ~44! for postabilities# are, respectively,
`~P~S2ùTS2!!5~~13s!spc,14
u,25,26 ,~11s!cpo,26,27,
16s,28,17
c,28,18u,27,15
u,25 ,~12s!prm!, ~58!
and
~~1/13s!spc,1/14
u,2/35,2/36 ,~1/11s!cpo,2/36,2/47,1/26
s,2/48 ,
~1/27s!c,2/48,1/28
u,2/47,1/25u,2/35 ,~1/12
s!prm!. ~59!
The first group is the tangent bifurcation of the spc andstable resonant modes ((1/1)spc,(1/1)u). In the pruning se-quence for the second group@cf. ~48! and ~49!#, the period-25 and 26 pos are pruned as either typeI or type II asfollows:
~2/35 ,~2/36 ,~1/11s!cpo,2/36!,2/47,1/26
s ,2/48 ,~1/27s!c,2/48,1/28
u,2/47,1/25u,2/35 ,~1/12
s!prm!
↓~2/35 ,~2/36 ,~1/11
s!cpo,2/36!,2/47,1/26s ,~~1/27
s!c,1/28u!,2/47,1/25
u,2/35 ,~1/12s!prm!
↓~2/35 ,~2/36 ,~1/11
s!cpo,2/36!,2/47,1/26s ,2/47,1/25
u,2/35 ,~1/12s!prm!
↓~2/35 ,~2/36 ,~1/11
s!cpo,2/36!,~1/26s ,1/25
u!,2/35 ,~1/12s!prm!
↓
I H ~2/35 ,2/36 ,~1/11s!cpo,2/36,2/35 ,~1/12
s!prm!
↓~2/35 ,~1/11
s!cpo,2/35 ,~1/12s!prm!
Jor
II ~~2/35u ,2/36
s!,~1/11s!cpo,~2/36
u ,2/35s!,~1/12
s!prm!
↓~~1/1!cpo,~1/1!prm!, ~60!
can
osirs-
re
e-
the2n
-n
n
d
s.-
-Fig.
a--4
nd
of
bitnals.nce
sal
fem-en-e-
a-am-
838 Chaos, Vol. 9, No. 4, 1999 M. Tsuchiya and G. S. Ezra
For the typeI pruning of period-2 pos 2/36 , pruning degen-eracy between (2/36 ,(1/11
s)cpo,2/36) and (2/48 ,(1/27s)c,2/48)
occurs. However, in order to have a pair of tangent bifurtions (2/35,2/36) in the pruning sequence, there can bepruning degeneracy between (2/36 ,(1/1)cpo,2/36) and(2/48 ,(1/2)c,2/48).
The pos (2/31s,2/35), (2/34
u,2/36), (2/42,2/47), and(2/43,2/48) @cf. ~48! and ~49!# form stable/unstable~Poincare´–Birkhoff! pairs, so that the po 2/35 is unstable(2/35
u) and the po 2/36 is stable (2/36s). These pairings are
consistent with the results of the Appendix on period-2 pOne can determine the stability of pos in the pa(2/42,2/47) and (2/43,2/48) by analyzing possible po prunings for higher order iterations (n.2) or higher energy (E.6.98). For the second iteration, the permutational repsentation is
P~S2ùT2S2!
5S 1 32 3 30 5 28 11 8 9 10 7 26
1 2 3 4 5 6 7 8 9 10 11 12
13 22 15 20 17 18 19 16 21 14
13 14 15 16 17 18 19 20 21 22
25 24 23 12 27 6 29 4 31 2
23 24 25 26 27 28 29 30 31 32D .
~61!
The period of@cf. ~58!# and winding number sequences, rspectively, are
Observe that there is no 4/6 po complementary toperiod-4/67 on S1 in ~63!. The reason is as follows: in thsequence~53!, the po 4/67 bifurcates from the stable period-po 2/31
s , so that no bifurcation can occur on the complemetary unstable period-2 po 2/35
u on S2 . Therefore, if theperiod-4 po 4/6 existed onS2 , it could only have been created by tangent bifurcation. This is impossible, as there isstable period-2 po 2/31 on S2 from which the complementary4/6 po would have to have bifurcated. There is therefore oone possible scenario; that is, the po 4/67 on S1 is created bya period-doublingbifurcation. This conclusion is confirmenumerically ~see Fig. 16!. The po 4/67 is therefore stablebefore bifurcation and unstable after. Finally, ifr 52s for atype I pruning, (r i ,s,r j ) (r ,sPQ), then the bifurcation is
-o
.
-
e
-
o
ly
period-doubling or resonant island-chain@see pos 2/42 in~42! and 2/48 in ~60!#; otherwise the bifurcation pattern iisland-chain, except forr 53s, touch-and-go in our systemNote that the po 4/818 is created by a period-doubling bifurcation.
We have analyzed pruning sequences for~63! before~type I ! and after~type II ! collision of the period-2 po pairs(2/35,2/36) @cf. ~60!#.37 Numerically observed bifurcation sequences corresponding to those of Fig. 15 are shown in17.
Analysis of pruning reveals that the following correltion of bifurcations between the period-2 and the periodpos occurs: In order that the period-2/36 po be pruned as typeII , the orbits (4/713,4/714) must be pruned as typeII ~seeFig. 17!. In other words, if the period-4/713 or 14orbit under-goes typeI bifurcation (a,I ), so must the po 2/35 or 6.
Comparing Figs. 15 and 17, the period-4 pos are fouto be paired as (4/52,4/512), (4/73,4/713), (4/710,4/714),(4/84,4/815), (4/85,4/816), and (4/86,4/817). The stability ofthose period-4 pos must be determined by examinationhigh-order iterates, especially the fourth order iterate.
VIII. CONCLUSION
In this paper we have studied classical periodic or~po! bifurcation sequences in a model molecular vibratioHamiltonian consisting of two coupled Morse oscillatorThe zeroth-order Hamiltonian has a single 1:1 resonacoupling term and is integrable; a second~2:1! coupling termof variable strength introduces nointegrability. Time reversymmetry was exploited to systematically find~symmetric!pos by iteration of symmetry lines.
The permutational representation of Tsuchiya and Jaf´28
was applied to analyze the intersection topology of a symetry line with its iterates. Using the permutational represtation, pruning rules describing possible local bifurcation b
FIG. 16. Creation of a period-4 po onS1 by period doubling. Analysis ofstability of pos on the two symmetry linesS1 andS2 shows that the period-4po 4/67 on symmetry lineS1 must be created by a period-doubling bifurction. This is confirmed numerically in the surface of section shown. Pareters areE55.68, b1150.01, b1250.02.
havior of pos were readily formulated. Moreover, a globunderstanding of allowed symmetric po bifurcation squences together with po stabilities could be obtained.
Examination of classical (E,t) plots for the coupledMorse system showed that breakdown of nonintegraphase space structure is associated with the appearanubiquitous tangent~saddle-center! bifurcations. The mechanism for onset of tangent bifurcations was studied. Fperiod-1 pos a ‘‘disconnection’’ mechanism applies~cf. Ref.29!. For period-2 and higher periods a different mechaniinvolving two parameter~codimension-2! collisions of bifur-cations was found. Such collisions are accompanied bbreakdown of the zeroth-order winding numbers.
We have not investigatednonsymmetricpos in this pa-per. However, understanding bifurcations of symmetric pcan provide some information on nonsymmetric pos. Siall fundamental~period-one! pos exist on the symmetrlines, generic bifurcations of pos from fundamental poscur on the symmetry lines. Tangent bifurcations, exceptthose on symmetry lines, will produce nonsymmetric posthe area-preserving He´non map, tangent bifurcations on thsymmetry lines have been shown to accumulate on~symmet-ric! homoclinic orbit bifurcations on symmetry lines.28 Inanalogous fashion, a set of non-symmetric pos could besociated with nonsymmetric homoclinic orbit bifurcations
The present work on the classical mechanics ofcoupled Morse system forms the foundation for an analyof the classical-quantum correspondence. This analysisbe presented elsewhere.19 Our results point to the influencof non-symmetric pos on the quantum (E,t) spectrum.19
With regard to the semiclassical mechanics of the sys
FIG. 17. Schematic bifurcation analysis forS2ùT2S2 . The relative orderingof period-1, period-2 and period-4 pos on the symmetry lineS2 is shown asa function of energy for fixed parametersb1150.01 andb12 . Both se-quences shown are found numerically.~a! Bifurcation diagram for the routein which period-2 pos undergo type-I bifurcations. (b1250.02). ~b! Bifur-cation diagram for the route in which period-2 pos undergo type-II bifurca-tions. ~occurs numerically atb1250.022!.
l-
leof
r
a
se
-r
n
s-
eisill
m
under study, it is interesting to speculate on possible seclassical manifestations of the different possible bifurcatsequences. Different sequences might be characterizedifferent patterns of singularities, and so lead to qualitativdifferent deviations from GOE spectral statistics.38
Finally, we note that the pruning analysis using a permtational representation can also be applied to homoclinicbits as well as pos.28 Just as sequences of pos are knownconverge to homoclinic orbits, so sequences of po tangbifurcations can converge to homoclinic tangencies. Appcation of these ideas to the semiclassical mechanics ofmoclinic tangles39 would be of interest.
ACKNOWLEDGMENT
This research was supported by the National ScieFoundation, under NSF Grant No. CHE-9709575.
APPENDIX
1. Period-1 and period-2 pos on symmetry lines S1and S2
Let zPS1 be a periodic point of period one,Tz5z. Thenz also lies onS2 , zPS2 :
Tz5I 2I 1z5I 2z5z. ~A1!
Let zPS1 be a periodic point of period two onS1 , T2z5z. Then z does not lie onS2 , z¹S2 . This is proved bycontradiction: noting thatT215I 1I 2 , and assumingI 2z5z,we have
z5I 2z5I 2T2z5I 1Tz5T21I 1z5T21z5Tz[z8. ~A2!
However,zÞz8 by hypothesis, so we have a contradictionThe same results clearly hold withS1 and S2 inter-
changed. Moreover, these results are true regardless onature of the dynamics~regular or chaotic!.
2. Generating higher order permutations fromP„Si"TSi…
Without loss of generality, one of the two involutiooperatorsI i can be always taken to be a reflection. Becauperiodicities of symmetric pos cannot change unless thesociated permutation changes, any intersection configuraSiùTnSi , can be deformed so thatSi becomes a straighline. Higher order iterates are generated from the givP(SiùTSi) as follows. The inverse image ofSi is obtainedby reflection alongSi , I i(TSi)5T21Si , so that the intersection between T21Si and TSi , i.e., the permutationP(T21SiùTSi), is obtained uniquely. Invariance of the pemutation~eq. 14! then determinesP(SiùT2Si):
P~T21SiùTSi !5P~SiùT2Si !. ~A3!
Repeating the same procedure forT2Si and continuing forhigher order iterates gives all permutationsP(SiùT2n
Si),(n>0). It is important to note that the permutation for th(2n)th iterate is uniquely determined fromP(SiùTSi) re-gardless of the nature of the dynamics~integrable, chaotic~mixed!, or hyperbolic!.
r-r
ee
r
tatirulsn
em
hy
e, J.
a D
ya
d
s
usly
840 Chaos, Vol. 9, No. 4, 1999 M. Tsuchiya and G. S. Ezra
In mixed systems, difficulties occur in determining pemutationsP(SiùTmSi), m a prime number. Consider, foexample, the permutationP(SiùT3Si). From the above, weknow P(SiùT4Si), which is equivalent toP(T21SiùT3Si).Then knowledge of the permutationP(T21SiùT3Si) and theshape ofT21Si determines the shape ofT3Si . Knowledge ofthe shape of theT3Si is not however sufficient to determinP(SiùT3Si), due to its unknown position relative to thstraight line Si . Additional information is given byP(T21SiùT2Si), where the shapes ofT21Si and T2Si areknown. The fact that the set of pos forSiùT21Si , which isequivalently to that ofSiùTSi , is a subset of pos foSiùT2Si , gives information on the way in whichT21Si andT2Si , henceSi and T3Si , can intersect, but this is still ingeneral not enough to determineP(SiùT3Si).
These difficulties in generating higher order permutions have their origin in the fact that regular and chaocomponents coexist in an extremely complicated mannethe phase space of mixed systems. By contrast, in the fchaotic limit of the He´non map, the number of intersectionis maximal at each iteration, so that all permutatioP(SiùTnSi) can be determined.28
1H. Poincare´, New Methods of Celestial Mechanics~AIP, New York,1993!.
2G. D. Birkhoff, Dynamical Systems~AMS, Providence, 1927!.3R. S. MacKay and J. D. Meiss,Hamiltonian Dynamical Systems~Hilger,Berlin, 1987!.
4See, for example, P. Cvitanovic, Physica D51, 138 ~1991!; 83, 109~1995!.
5K. R. Meyer, Trans. Am. Math. Soc.149, 95 ~1970!; K. R. Meyer and G.R. Hall, Introduction to Dynamical Systems and the N-body Probl~Springer, New York, 1992!.
6M. C. Gutzwiller,Chaos in Classical and Quantum Mechanics~SpringerVerlag, Berlin, 1990!.
7M. S. Child, Semiclassical Mechanics with Molecular Applications~Ox-ford, New York, 1991!; M. Brack and R. K. Bhaduri,Semiclassical Phys-ics ~Addison-Wesley, Reading, MA, 1997!.
8H. Friedrich and D. Wintgen, Phys. Rep.183, 37 ~1989!.9See, for example, G. Tanner, K. Richter, and J-M. Rost, Rev. Mod. P~to appear!, and references therein.
10See, for example, R. Schinke,Photodissociation Dynamics~CambridgeUniversity Press, Cambridge, UK, 1993!, Chap. 8.
-cinly
s
s.
11J. M. Mao and J. B. Delos, Phys. Rev. A45, 1746~1992!.12H. S. Taylor, Acc. Chem. Res.22, 263 ~1989!; S. C. Farantos, Int. Rev.
Phys. Chem.15, 345 ~1996!.13G. S. Ezra, inAdvances in Classical Trajectory Methods, Vol. 3, edited by
W. L. Hase~JAI, New York, 1998!.14P. Gaspard and P. van Ede van der Pals, J. Chem. Phys.110, 5611~1999!.15R. S. MacKay, J. D. Meiss, and I. C. Percival, Physica D28, 1 ~1987!.16M. J. Davis, J. Chem. Phys.107, 4507~1997!.17M. J. Davis, Int. Rev. Phys. Chem.14, 15 ~1995!.18See, for example, R. Marquardt, M. Quack, J. Stohner, and E. Sutcliff
Chem. Soc., Faraday Trans. 286, 1173~1986!.19M. Tsuchiya and G. S. Ezra~work in progress!.20L. Xiao and M. E. Kellman, J. Chem. Phys.90, 6086 ~1989!; Z. Li, L.
Xiao, and M. E. Kellman,ibid. 92, 2251 ~1990!; L. Xiao and M. E.Kellman, ibid. 93, 5805~1990!.
21L. Xiao and M. E. Kellman, J. Chem. Phys.93, 5821~1990!; J. Svitak, Z.Li, J. Rose, and M. E. Kellman,ibid. 102, 4340~1995!.
22D. C. Rouben and G. S. Ezra, J. Chem. Phys.103, 1375 ~1995!; G. S.Ezra, ibid. 104, 26 ~1996!.
23M. Joyeux, Chem. Phys. Lett.247, 454 ~1995!.24A. J. Lichtenberg and M. A. Lieberman,Regular and Stochastic Motion
~Springer, Berlin, 1983!.25R. DeVogelaere, inContributions to the Theory of Nonlinear Oscillations,
Vol. 4, edited by S. Lefschetz~Princeton University Press, NJ, 1958!, p.35.
26J. M. Greene, AIP Conf. Proc.57, 257 ~1979!.27J. M. Greene, R. S. MacKay, F. Vivaldi, and M. J. Feigenbaum, Physic
3, 468 ~1981!.28M. Tsuchiya, Ph.D. thesis, West Virginia University, 1995; M. Tsuchi
and C. Jaffe´, preprint.29T. Weston and M. S. Child, Chem. Phys. Lett.262, 751 ~1996!. See also
S. Cho and M. S. Child, Mol. Phys.81, 447 ~1994!, Ref. 21.30D. A. Sadovskii, J. A. Shaw, and J. B. Delos, Phys. Rev. Lett.75, 2120
~1995!; D. A. Sadovskii and J. B. Delos, Phys. Rev. E54, 2033~1996!.31K. R. Meyer, J. B. Delos, and J.-M. Mao, inConservative Systems an
Quantum Chaos, edited by L. M. Bates and D. L. Rod~American Math-ematical Society, Providence, 1996!.
32H. Schomerus, J. Phys. A31, 4167~1998!.33G. H. Hardy and E. M. Wright,An Introduction to the Theory of Number
~Oxford University Press, New York, 1979!.34J. Laskar, Physica D67, 257 ~1993!; J. Laskar, C. Froeschle´, and A.
Celletti, ibid. 56, 253 ~1992!.35L. E. Fried and G. S. Ezra, J. Chem. Phys.86, 6270~1987!; M. E. Kell-
man, ibid. 93, 6630~1990!.36Such dynamically defined surfaces of section have been used previo
by C. Jaffe~private communication!.37M. Tsuchiya and G. S. Ezra~unpublished!.38M. V. Berry, J. P. Keating, and S. D. Prado, J. Phys. A31, L245 ~1998!.39S. Tomsovic and E. J. Heller, Phys. Rev. E47, 282 ~1993!.
![arXiv:1402.4886v4 [math.CO] 10 Jul 20186.1. Proof of Lemma 3.3 on codimension up to 2 12 6.2. Proof of Lemma 3.4 on codimension 3 14 Acknowledgement 22 Dictionary of Notation 23 References](https://static.documents.pub/doc/80x56/5eda0d29b3745412b570acaa/arxiv14024886v4-mathco-10-jul-2018-61-proof-of-lemma-33-on-codimension-up.jpg)
![Cisco IPv6 Deployment Statics, by Shishio Tsuchiya [APRICOT 2015]](https://static.documents.pub/doc/80x56/55a78f1d1a28ab9a318b46ca/cisco-ipv6-deployment-statics-by-shishio-tsuchiya-apricot-2015.jpg)
