Assigning vibrational polyads using relative equilibria: application to ozone.

Abstract

We demonstrate how relative equilibria of a vibrating molecule, which are families of principal periodic orbits otherwise known as nonlinear normal modes, can be used to describe the global polyad structure of vibrational energy levels. The classical action integral n(E) computed along these orbits at different energies E corresponds to the polyad quantum number n so that the energy En of different relative equilibria describes the splitting of n-polyads. Further information on the internal polyad structure can be driven from the stability analysis of relative equilibria. We use the ozone molecule as a concrete example where n-polyads or "hyperpolyads" should be distinguished from the well-known polyads of the 1:1 stretching mode resonance; the stretching polyads are structural elements of hyperpolyads. We give dynamical interpretation of the relation between relative equilibria and n-polyads based on the normal form reduction in the limit of small vibrations near the equilibrium.

Full text

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Spectrochimica Acta Part A 61 (2004) 2867–2885
Assigning vibrational polyads using relative equilibria:
application to ozone
I.N. Kozina,b,1, D.A. Sadovski´ ıc,∗, B.I. Zhilinski´ ıc
aMathematics Institute, University of Warwick, Coventry CV4 7AL, UK
bDepartment of Chemistry, University of Aberdeen, Meston Walk, Aberdeen AB24 3UE, UK
cUniversit´ e du Littoral, UMR 8101 du CNRS, 59140 Dunkerque, France
Received 6 July 2004; received in revised form 30 September 2004; accepted 28 October 2004
Abstract
Wedemonstratehowrelativeequilibriaofavibratingmolecule,whicharefamiliesofprincipalperiodicorbitsotherwiseknownasnonlinear
normal modes, can be used to describe the global polyad structure of vibrational energy levels. The classical action integral n(E) computed
along these orbits at different energies E corresponds to the polyad quantum number n so that the energy E(n) of different relative equilibria
describes the splitting of n-polyads. Further information on the internal polyad structure can be driven from the stability analysis of relative
equilibria. We use the ozone molecule as a concrete example where n-polyads or “hyperpolyads” should be distinguished from the well-
known polyads of the 1:1 stretching mode resonance; the stretching polyads are structural elements of hyperpolyads. We give dynamical
interpretation of the relation between relative equilibria and n-polyads based on the normal form reduction in the limit of small vibrations
near the equilibrium.
© 2004 Elsevier B.V. All rights reserved.
PACS: 33.15.Mt; 33.20.Vq
Keywords: Relative equilibrium; Normal form; Periodic orbit; Vibrational polyad; Dynamical symmetry; Local mode
1. Introduction
Inthisarticleweintendtodescribetheglobalpolyadstruc-
tureofthevibrationallevelsofozone.Weconsidertheclassi-
cal mechanical analogue of the vibrational molecular system
and use the results of the qualitative analysis of this classical
analogue in order to characterize the polyad structure of the
original quantum system. After specifying the dynamical or
polyadsymmetryofoursystem,wefinditsrelativeequilibria
(RE), which provide the framework of the qualitative study.
Definition of relative equilibrium. Consider a Hamilto-
nian dynamical system with phase space P and Lie symme-
∗Corresponding author. Tel.: +33 328658263; fax: +33 328658263.
E-mail addresses: i.kozin@dl.ac.uk (I.N. Kozin);
sadovski@univ-littoral.fr (D.A. Sadovskii); zhilin@univ-littoral.fr
(B.I. Zhilinskii).
1Presentaddress:DepartmentofComputationalScienceandEngineering,
CCLRCDaresburyLaboratory,KeckwickLane,WarringtonWA44AD,UK.
try group G. Let gt⊆ G (with parameter t ∈ R) be a one-
parameter continuous subgroup of G. A phase curve of our
system which coincides with the group orbit of the action of
gton P is called relative equilibrium (RE), see Appendix 5C
of [1a] and Chapter 3.3 of [1b]. Reduction of the symmetry
group gtmaps such phase curves to equilibria of the reduced
system.
WedescribetheREofozonefirstas“short”periodicorbits
and use their action-energy diagram to describe the polyad
structure. Subsequently, we normalize the classical system
and uncover the explicit relation of RE to the polyad integral
ofthenormalform.Finally,wequantizethenormalformand
compare the results to the ab initio energy levels.
1.1. Rotational relative equilibria
The most obvious molecular example of RE comes up in
the study of free molecule rotation. In this case, since the
1386-1425/$ – see front matter © 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.saa.2004.10.039

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I.N. Kozin et al. / Spectrochimica Acta Part A 61 (2004) 2867–2885
system is invariant with regard to rotations of the laboratory
fixed frame, G = SO(3), gt= SO(2), and RE correspond to
rotation about stationary axes [1a,2]. Due to additional fi-
nite symmetries, the molecule has usually several equivalent
stationary axes. Molecular spectroscopists have recognized
long ago that such axes manifest themselves in the quantum
spectrum as level clusters [3]. Applications [4,5] normally
study the reduced system, for which stationary axes of rota-
tionbecomeequilibria.Foreachnon-zerovalueofthelength
j of the total angular momentum, the reduced phase space is
a 2-sphere S2
j. The reduced Hamiltonian is a function on this
space. It is often called rotational energy surface [6] and is
depicted as a deformed sphere whose maxima, minima, or
saddle points represent RE.
1.2. Vibrational relative equilibria
We can extend our definition of RE by allowing that the
symmetry G is not necessarily strict, but is an approximate
dynamical symmetry. This extension is particularly useful in
the study of molecular vibrations.
1.2.1. Resonances
ThemolecularvibrationalHamiltonianinthecaseofsmall
vibrations about a well-defined molecular equilibrium (the
case of “rigid” molecules) is a power series
H = H(0)+ ?H(1)+ ?2H(2)+ ···
indisplacementsqkandconjugatemomentapk,whoseterms
can be distinguished by the uniform smallness parameter ?.
The quadratic part of H is
H(0)=1
2
K
?
k=1
ωk(q2
k+ p2
k) =
K
?
k=1
ωknk,
where K is the number of vibrational degrees of freedom
and frequencies ω1: ω2: ··· : ωK obey, approximately or
exactly, the resonance condition m1: m2: ··· : mK, where
mkare positive integers. (Note that here nkare classical ac-
tions which quantize as nk→ Nk+1
etc.) The resonance condition defines G. For example, con-
dition 1:1:···:1 results in G = SU(K). The subgroup gtis
givenbytheflowϕtofthevectorfieldXnoftheHamiltonian
2, where Nk= 0,1,2,
n =1
2
K
?
k=1
mk(q2
k+ p2
k).
Comparing H(0)and n, we note that frequencies ωkin the
linearized Hamiltonian H(0)are approximated in n by inte-
gersmk.Furthermore,itisoftenconvenienttorescaleenergy
so, that H(0)ω−1≈ n, where ω is the mean characteristic vi-
brational frequency of the molecule. We also find from the
equations of motion for the system with Hamiltonian n that
ϕtacts as a rotation on the phase space R2K
q,p.
1.2.2. Reduction of dynamical symmetry
In general, the Poisson bracket {H(s),n} does not van-
ish unless s = 0. We assume, however, that the higher or-
ders H(s)are approximately G-invariant. More precisely, as-
suming that {H(s),n} for s > 0 are small, we can normal-
ize H so that all terms H(s)in the transformed Hamilto-
nian H Poisson commute with n. The normal form H is,
therefore, strictly G-invariant. The canonical transformation
L?: H → H is a near unity transformation which becomes
identity when ? → 0 [7]; in particular, H(0)= H(0). Like the
original Hamiltonian H, the normal form H is an ?-series.
The most practical and direct method of normalizing such
series is the Lie series method [8a–d]. In most cases with
K > 1, the series H diverges when taken to unreasonably
high orders (see, for example, Appendix 7 of [1a]). Con-
sequently, we have to truncate H and specify conditions at
which such truncated normal form is useful. Normally we
restrict the energy H, the value of n, and/or the perturbation
scale ?.
By construction, the Hamiltonian function n is an integral
ofmotionofthenormalizedsystem.Wecan,therefore,reduce
this system at each given value of n > 0, so that the value of
n becomes a parameter. The reduced Hamiltonian Hnis the
normal form H expressed as a function on the reduced phase
space Pn.
In the case of the resonance condition 1:1:···:1, the re-
duced phase space is the complex projective space CPK−1.
Dynamical variables of the reduced system are quadratic
polynomials in (q, p) which Poisson commute with n and
generate a Poisson algebra su(K). In this paper we will en-
counter spaces CP2and CP1.
Thereducedphasespaceofthetwo-modesystem(K = 2)
with resonance 1:1 is a 2-sphere S2which is isomorphic to
CP1[9]. This basic case has, of course, been studied in great
detail, notably in application to the H´ enon–Heiles system
[10] and its molecular analogues [11], and 1:1 resonant vi-
brationalsubsystemsofpolyatomicmolecules[12].Sincethe
reducedsystemisequivalenttothereducedrotationalsystem
(see Appendix A.4), we use the analysis described briefly in
Section 1.1.
1.2.3. Polyads and polyad Hamiltonians
The term polyad quantum number is now widely used in
molecular spectroscopy to label a relatively isolated aggre-
gation of vibrational states [13]. For example, two stretching
modes of an AB2molecule often have nearly the same fre-
quencies ω1and ω3and can, therefore, be considered as a
1:1 system mentioned above. A stretching polyad of such
molecule is labeled by ns= n1+ n3, where n1and n3are
numbers of quanta in each of the stretching modes. The
polyadnumbercanbeextendedtoincludethebendingvibra-
tionaswell.Frequently,thereisanear1:2resonancebetween
stretching and bending modes. Then the polyad number can

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2869
be defined as
N1:1
2:1= n1+1
2n2+ n3.
Such number is often used for triatomic molecules [14]. If
the normal mode frequencies of AB2can be approximated
by integers k1:k2:k3, we can use the polyad number
Nk1:k2:k3= k1n1+ k2n2+ k3n3.
Accordingtothisgeneraldefinition,thenumberN2:1:2should
be used instead of the above N1:1
In this work, we will use the polyad number (cf.
Appendix A.1)
2:1.
n = N1:1:1= n1+ n2+ n3,
which we call the hyperpolyad number. We propose to clas-
sify quantum states first using n and then, if possible, other
quantum numbers. This principle works very well in the
ozone molecule where all known assigned vibrational lev-
els [15a,b]2can be easily and unambiguously grouped into
n-polyads, even though the resonance condition for ozone is
much closer to 5:3:5 or 2:1:2 than to plain 1:1:1.
Polyad Hamiltonians are used in spectroscopy to describe
internalstructureofpolyads.ThesespectroscopicHamiltoni-
ans are called effective and model to emphasize the absence
of explicit inter-polyad interaction terms and the liberty in
the choice of resonance condition, respectively. Parameters
of the polyad Hamiltonian are often treated as phenomeno-
logicalspectroscopicconstants,whosevaluesareobtainedby
fitting experimental data. To further confuse the uninitiated,
typical resonance conditions (=models) and related polyad
Hamiltonians are traditionally named after Fermi, Darling-
Dennyson, and others [16]. The simplest example is again
the 1:1 Hamiltonian, which is analogous to an effective rota-
tional Hamiltonian. (Recall that the “rotational polyad” is a
multiplet of levels with the same angular momentum quan-
tum number J, see Appendix A.4 and [17].)
Going back to Section 1.2.2 we can see that (i) the polyad
approximation amounts to the dynamical symmetry assump-
tion given by the resonance condition and followed by re-
duction, (ii) polyad Hamiltonians are nothing but reduced
Hamiltonians Hn, (iii) vibrational dynamics of the reduced
systemdefinesinternalpolyadstructure.Spectroscopistscon-
struct quantum polyad HamiltoniansˆHnusing terms which
preserve the polyad number. In order to commute with quan-
tum operator ˆ n, terms inˆHnare restricted to have creation-
annihilationoperatorsofcertaindegreeandtype.Thisisanal-
ogous to the classical construction of the ring of the dynam-
ical invariants (see Appendix A.2).
(1)
2The work [15b] gives essentially almost the same potential as in [15a].
The latter was obtained using MORBID, which made a few approxima-
tions in the kinetic energy operator. These were “absorbed” in the potential.
The new paper removed this deficiency using the “Exact Kinetic Energy”
operator.
1.2.4. Modes and periodic orbits
Thespectroscopicconceptof“mode”isoftenconfusingly
vague. The common practice of opposing “local” and “nor-
mal”modescanbeagoodexample.TheconceptofREhelps
to bring the situation in better order.
Considerthestandardboundmolecularsystemofsmallvi-
brationsdescribedintermsofsmalldisplacementsqandcor-
respondingconjugatemomentap.Thezero-ordervibrational
Hamiltonian is a sum of two positively definite quadratic
forms, the kinetic energy T and potential V. The phase space
variables (q, p) can be chosen so that both T(p) and V(q) are
diagonal. Such variables correspond to normal modes [18].
The presence of symmetry often simplifies the task of diag-
onalization. Thus ozone and molecules AB2have only one
asymmetric displacement q3which defines the normal mode
ν3unambiguously. The form of the two symmetric modes ν1
and ν2depends on the particular T(p) and V(q).
The dynamical concept of mode begins with the theorem
of Weinstein [19]. Consider a stable equilibrium of a Hamil-
tonian system with K degrees of freedom, and suppose that
harmonic frequencies are incommensurate, i.e., there are no
resonances. Then near the limit of linearization, i.e., at ener-
gieshclosetotheequilibriumenergyh0,thesystemhasaset
ofKenergy-dependentfamiliesofperiodictrajectorieswhich
are defined by the nonlinear terms of the local Hamiltonian.
Thesefamiliesarebasicvibrationalmodesofthesystemnear
thegivenequilibrium.TodistinguishthemfromtheKnormal
modes defined above, it was suggested to use the term non-
linear normal modes [20a–c]. We realize immediately that,
in fact, these modes are nothing but vibrational RE.
In a resonant system, the number of nonlinear normal
modes (=RE) can be a priori greater than the number
of normal modes (coordinates) [20a–c]. Again, the pres-
ence of symmetry can greatly simplify the task of char-
acterizing these RE. Thus recall the textbook example of
the eight RE of the two-dimensional H´ enon–Heiles system
[10,20a–c], a nonlinear 1:1 resonant oscillator with symme-
try D3, and its molecular analogue—the H+
[11].
As the energy h gets further from h0, nonlinear normal
modes (=RE) can bifurcate, and in particular, their number
can increase. These bifurcations correspond to bifurcations
oftheequilibriaofthereducedsystem.Thenew(familiesof)
periodic trajectories are, therefore, also RE. Different sets
of RE correspond to qualitatively different internal polyad
structures. The classic example of an RE-bifurcation is the
so-called normal-to-local mode transition, which happens in
ozone when h is very close to h0. We will discuss this bifur-
cation in detail.
At very low energies, the ozone molecule has three non-
linear normal modes which correlate with the three normal
modes, symmetric and antisymmetric stretching ν1and ν3,
andbendingν2.Atslightlyhigherenergies,italsohastwoex-
tra equivalent RE, which spectroscopists call “bond length”
modes or local modes [21]. These modes are close to the
vibration of the individual O O bonds.
3molecular ion

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Like any stable periodic trajectories, stable RE of period
TRE(n) can undergo period-s bifurcations which involve tra-
jectories of period sTRE(n). Within the framework of the
polyad approximation, we can only describe period-1 bifur-
cations. Such bifurcations affect the internal structure of the
hyperpolyad but not its validity. Cascading higher-period bi-
furcations signal the destruction of hyperpolyads and the on-
set of chaos. However, when suggesting the limits of the hy-
perpolyad approximation for the classical system, we should
also consider that the quantum analogue system is, gener-
ally, more robust to chaos, and our approximation has a good
chance to stretch further than we would expect classically.
1.2.5. Assignment of quantum states
Classical textbooks on molecular vibrations and spec-
troscopy [16,18] usually suggest normal modes for vibra-
tional energy level assignment. More recently, the local
modesofanumberofhydrogen-bondedmoleculesandozone
were suggested as more “physical” [21]. On the other hand,
bothnomenclaturescanbeconsideredformallyequivalentin
the limit of small distortions where local and normal modes
are linear combinations of each other and there is linear rela-
tionship between the corresponding quantum numbers. The
currentspectroscopicpracticeistopresentbothlocalandnor-
mal mode quantum numbers and specify their relationship.
Dynamical approach to assigning quantum states is based
on localization near RE or, in general, other dynamically
invariant subspaces. Consider a sufficiently stable relative
equilibrium Π, such as the local mode RE of ozone. Take a
projection Πqof the periodic orbit Π in the original phase
spaceR2K
(q,p)ontheconfigurationspaceRK
wavefunctionislocalizednearΠq,itsnodesfollowΠq.Inthe
limitingcasethenumberofsuchnodesNΠequalsthepolyad
quantum number N. When NΠis less than N, other degrees
offreedomareinvolvedinthedirectiontransversaltoΠ.Yet,
if NΠis sufficiently close to N, the node pattern can still re-
mainaregularlatticewhichfollowsΠ.ForsuchstatesNΠis
clearly a good quantum number. Of course, not all states fall
into such category. Other states can, possibly, be assigned in
termsofotherstableRE,andsomestronglydelocalizedstates
wouldremainwithoutanymeaningfuldynamicalassignment
exceptforthehyperpolyadnumberN.Thus,ourmainpropo-
sition is to begin with the hyperpolyad assignment of all
states, and then classify them further where it is possible.
q.Whenaquantum
1.3. Outline
The paper has three main directions. In order to uncover
theexistenceofhyperpolyads,wefirststudynumerically(see
Section 3) the main periodic orbits of the vibrational sys-
tem of ozone with Hamiltonian in [15a,b]. We then show the
relation of these orbits to relative equilibria. We normalize
the initial three-mode Hamiltonian of [15a] and analyze RE
as equilibria of the reduced system (Section 4). Finally we
compute quantum levels in two ways, using local lineariza-
tionnearstableperiodicorbits(Section3.3)andbyquantiza-
tion of the global normal form (Section 4.5). We reproduce
adequately all quantum levels assigned in [15a–c] and thus
demonstrate the validity of the polyad approximation. Sub-
sequently, we address the main problem faced in [15a,b,22],
namely the assignment of quantum levels computed numer-
ically. We suggest a direct method of computing the hyper-
polyad number for a given wavefunction (Section 5).
This work should, of course, be regarded in the context of
numerouspublicationsonthevibrationallevelsanddynamics
of triatomic and polyatomic molecules [23,24a–f], including
more recent work on ozone [15a–c,25,26]. In particular, we
like to mention the work by Lu and Kellman [14], who study
ozone on the basis of a 2:1:2 model polyad Hamiltonian.
Theyfocusmainlyonstretchingpolyads(cf.Section4.3)and
apply the standard analysis based on the angular momentum
analogy [6,12,5].
Contrary to [14], we derive our polyad approximation
fromthefullvibrationalHamiltonian,whichdescribesallex-
perimentally known states of ozone. The same computation
was independently attempted by Joyeux [27], who obtained
even better reproduction of the numerical quantum energies
of [15a,b]. However, far from trying to compete with [15a–
c] in accuracy of reproduced quantum energies, we consider
ourclassicalstudyandthesubsequentquantizationasabasic
qualitative tool of dynamical characterization of computed
states. We like to focus mostly on the global hyperpolyad
structure and on the problem of level assignment.
2. Ozone molecule
The most abundant isotopomer of ozone molecule has
threeidenticalatomsandisoscelesequilibriumconfiguration
with two equal bond lengths r12= r23= reand the bond
angle αe. Vibrations of this molecule are described most nat-
urally in terms of two dimensionless bond length displace-
ments ξ1and ξ2, such that
r12= re(1 + ξ1),
and the bond angle displacement α. In these coordinates, the
kinetic energy term T in the Hamiltonian
r23= re(1 + ξ2),
H =
1
mr2
e
T(ξ,η,α,pα) + V(ξ1,α,ξ2) (2a)
has the form
T = η2
1+ η2
sin(αe+ α)
(1 + ξ1)(1 + ξ2)pα{(1 + ξ1)η1+ (1 + ξ2)η2},
2+(1 + ξ1)2+ (1 + ξ2)2− (1 + ξ1)(1 + ξ2)cos(αe+ α)
(1 + ξ1)2(1 + ξ2)2
p2
α+ cos(αe+ α)η1η2
−
(2b)

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2871
where η1, η2, and pαare momenta conjugate to ξ1, ξ2, and
α, respectively. The potential function V(ξ1,α,ξ2) was de-
termined very accurately in the recent work by Tyuterev et
al. [15a] from fitting all known experimental data. This po-
tential has good global properties and the dissociation en-
ergy of 9450cm−1which is in good agreement with experi-
ment. An improvement of this potential was reported later in
[15b].
The Hamiltonian (2) is invariant with respect to bond per-
mutation C2and time reversal T . These two finite symme-
try operations generate a group of order four with structure
Z2× Z2. Appendix B gives a detailed description of this
group and of its implications.
For the normalization purposes we represent the Hamil-
tonian (2) as a power series H(q, p) in the near equilibrium
normal mode displacements q = (q1,q2,q3) and conjugate
momenta p = (p1,p2,p3)
H =ω1
+?H1+ ?2H2+ ?3H3+ ···.
The totally symmetric coordinates q1 and q2 depend on
ξ1+ ξ2and α; the symmetric stretch q1has a predominant
contribution due to ξ1+ ξ2while the bending coordinate q2
depends mostly on α. The anti-symmetric stretch coordinate
q3is proportional to (ξ1− ξ2). The zero order term in the
series H(q, p) is in the standard diagonal quadratic form.
Note, that unlike in some other AB2molecules, such as wa-
ter, the symmetric stretch frequency ω1is larger than that of
the asymmetric stretch. In this study we expanded H(q, p) to
degree 6 (order ?4); cubic and quartic terms in this series are
listed in Table 1.
2(p2
1+ q2
1) +ω2
2(p2
2+ q2
2) +ω3
2(p2
3+ q2
3)
3. Relative equilibria as principal periodic orbits
We begin with direct numerical study of periodic or-
bits which correspond to relative equilibria (RE). As fol-
lows from Section 1.2, we should be interested primarily
in the energy–action characteristics of these orbits. Indeed,
action corresponds to classical polyad number n and the
polyad structure is given by the RE energies at fixed n.
We argue that our periodic orbit analysis justifies the ba-
sis for using the 1:1:1 resonance model. This agrees with
Lu and Kellman [14], who did not find low energy bifur-
cations in their 2:1:2 model except for the local mode bi-
furcation, which can be equally well described by the 1:1:1
model.
3.1. Energy–action characteristics
Continuation of periodic orbits of molecular systems has
been done by a number of authors, notably Prosmiti and
Farantos [28]. We do it in a somewhat different context [29].
We opted for the very well-developed continuation package
Table 1
Hamiltonian of ozone with vibrational potential of Ref. [15a] expressed
using normal mode coordinates
Order CoefficientTermCoefficient Term
?0
1132.3600
1
2(q2
1
2(q2
1
2(q2
1+ p2
2+ p2
3+ p2
1)
2)
3)
Symmetric stretch
714.6150
Bending
1086.9425
Asymmetric stretch
?1
−40.26499
29.41346
−8.49418
−195.22643
8.92844
65.82223
6.36668
−24.45093
0.23935
2.21248
4.36140
q3
q3
q1q2
q1q2
q2
1q2
q2q2
q1p2
q1p2
1
−10.30595
20.49931
−5.25573
−13.48096
−41.43463
−11.77458
13.26030
4.98473
q1p2
q2p2
q2p2
q2p2
3
21
2
2
33
q1p1p2
3
q2p1p2
1
q3p2p3
2
q3p1p3
?2
q4
q4
q4
q3
q1q3
q2
q2
q1q2q2
q2
q2
q2
q2
q2
q2
1
0.16508
−0.03366
0.39659
q2
q2
q2
q1q2p2
1p2
2p2
2p2
1
21
3
2
−4.01685
−1.56543
−0.77267
12.92855
−13.31552
−0.26197
1.19930
1.80520
−0.13517
−0.23128
0.42515
1q2
−0.35362
−2.29488
−0.15429
−1.04417
1.58181
0.94469
−0.36182
0.47273
0.27819
0.74005
−0.41046
3
2
q1q2p1p2
1q2
1q2
2
q1q3p1p3
q2
2p1p2
q2
3p1p2
q1q2p2
q1q2p2
q2
1p1p2
q2q3p1p3
3
3
2q2
3p2
1p2
1p2
2p2
3p2
3
2
21
2
3
3
q2q3p2p3
1
q1q3p2p3
CONTENT [30],3which can be easily adapted for our pur-
poses by introducing a free dummy parameter λ in the equa-
tions of motion
˙ qi=∂H
∂pi,
˙ pi= −∂H
∂qi
+ λpi,i = 1,2,3.
We normally start from a normal mode and continue the pe-
riodic orbit by allowing the period to change and calculating
the energy and action. The parameter λ is also varied but
maintained around zero. In this way one is able to produce
a graph of the energy against the action for every periodic
orbit, see Fig. 1.
3.2. Bifurcation of periodic orbits
Atverylowenergies,thethreeperiodicorbits(POs)corre-
late with the three normal modes, see Fig. 1; the bending PO
has the minimum energy at given fixed action n, the symmet-
ric stretch PO stays on top, and the asymmetric stretch PO
3ThispackageisbasedontheotherversatilecontinuationpackageAUTO
and provides extremely flexible and powerful graphic interface.