Hamiltonian systems with detuned 1:1:2 resonance: Manifestation of bidromy

Abstract

We consider a generalization of the 1:1:2 resonant swing–spring [see H. Dullin, A. Giacobbe, R.H. Cushman, Physica D 190 (2004) 15] which is suggested both by the symmetries of this system and by its physical and in particular molecular realizations [see R.H. Cushman, H.R. Dullin, A. Giacobbe, D.D. Holm, M. Joyeux, P. Lynch, D.A. Sadovskií, B.I. Zhilinskií, Phys. Rev. Lett. 93 (2004) 024302-1–024302-4]. Our generic integrable system is detuned off the exact Fermi resonance 1:2. The three-dimensional (3D) image of its energy–momentum map EM consists either of two or three qualitatively different non-intersecting 3D regions: a regular region at low vibrational excitation, a region with monodromy similar to that studied for the exact resonance, and in some cases—an intermediate region in which the 3D set of regular values of EM is partially self-overlapping while remaining connected. In the presence of this latter region, the system has an interesting property which we called bidromy. We analyze monodromy and bidromy for a concrete integrable classical Hamiltonian system of three coupled oscillators and for its quantum analog. We also show that the bifurcation involved in the transition from the regular region to the region with monodromy can be regarded as a special resonant equivariant analog of the Hamiltonian Hopf bifurcation.

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Hamiltonian systems with detuned 1:1:2
resonance: Manifestation of bidromy
D.A. Sadovskiı ´, B.I. Zhilinskiı ´*
Universite ´ du Littoral, UMR 8101 du CNRS, 59140 Dunkerque, France
Received 30 May 2006; accepted 26 September 2006
Available online 13 November 2006
Abstract
We consider a generalization of the 1:1:2 resonant swing–spring [see H. Dullin, A. Giacobbe, R.H.
Cushman, Physica D 190 (2004) 15] which is suggested both by the symmetries of this system and by
its physical and in particular molecular realizations [see R.H. Cushman, H.R. Dullin, A. Giacobbe,
D.D. Holm, M. Joyeux, P. Lynch, D.A. Sadovskiı ´, B.I. Zhilinskiı ´, Phys. Rev. Lett. 93 (2004) 024302-
1–024302-4]. Our generic integrable system is detuned off the exact Fermi resonance 1:2. The
three-dimensional (3D) image of its energy–momentum map EM consists either of two or three qual-
itatively different non-intersecting 3D regions: a regular region at low vibrational excitation, a region
with monodromy similar to that studied for the exact resonance, and in some cases—an intermediate
region in which the 3D set of regular values of EM is partially self-overlapping while remaining con-
nected. In the presence of this latter region, the system has an interesting property which we called
bidromy. We analyze monodromy and bidromy for a concrete integrable classical Hamiltonian sys-
tem of three coupled oscillators and for its quantum analog. We also show that the bifurcation
involved in the transition from the regular region to the region with monodromy can be regarded
as a special resonant equivariant analog of the Hamiltonian Hopf bifurcation.
? 2006 Elsevier Inc. All rights reserved.
Pacs: 33.20?t; 45.20?d; 05.45a
Keywords: Molecular vibrations; Integrable model; Nonlinear resonance; Quantum monodromy; Bidromy
0003-4916/$ - see front matter ? 2006 Elsevier Inc. All rights reserved.
doi:10.1016/j.aop.2006.09.011
*Corresponding author. Fax: +33328658244.
E-mail address: zhilin@univ-littoral.fr (B.I. Zhilinskiı ´).
Annals of Physics 322 (2007) 164–200
www.elsevier.com/locate/aop

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1. Introduction
Hamiltonian systems with 1:1:2 resonance became recently of renewed acute interest
after it was pointed out in [1] that they exhibited an interesting ‘‘plane switching’’ phenom-
enon, called so after the behavior which can be observed in the classical 1:1:2 resonant
swing–spring or elastic pendulum with axial symmetry O(2) [2,3] (see Fig. 1, left). It was
also shown in [3] that this behavior was intrinsically related to the monodromy [5,6] of inte-
grable approximations to these systems. Furthermore, a quantum analog of the 1:1:2 res-
onant swing–spring was studied in [4], where its quantum monodromy [8,7] was
interpreted as a particular one-dimensional defect of the three-dimensional (3D) lattice
[9] whose nodes represented quantum eigenstates within the 3D image of the energy–mo-
mentum map EM of the corresponding classical system. The evidence that the CO2mol-
ecule (see Fig. 1) constitutes an almost perfect quantum 1:1:2 resonant swing–spring [10]
opened perspectives of experimental manifestations of monodromy in molecules. In par-
ticular, the quantum plane switching phenomenon was demonstrated for CO2molecule
[11].
1.1. Motivation for studying systems with detuned Fermi resonance
The Lie symmetry group G of the integrable approximation to the 1:1:2 resonant
swing–spring system is a 2-torus group which includes the axial S1symmetry (rotation
about the vertical axis in Fig. 1) and the S1symmetry defined by the flow of the 1:1:2 har-
monic oscillator Hamiltonian. It can be easily shown (see Section 2) that any exactly 1:1:2
resonant system is exceptional within the class of G-invariant systems. Furthermore, while
the frequencies of the 1:1 subsystem are exactly degenerate due to the axial symmetry, the
1:2 ratio between these frequencies and the frequency of the third oscillator, known as
Fig. 1. Elastic pendulum and normal mode vibrations of the CO2molecule. Note that the first three modes,
symmetric stretching m1and the two components of bending m2constitute the Fermi resonant oscillator system
while the asymmetric stretch vibration m3is not in resonance and can be averaged out, see more in [10,4].
D.A. Sadovskiı ´, B.I. Zhilinskiı ´ / Annals of Physics 322 (2007) 164–200
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obviously, the bifurcating RE becomes a singular point after our S1action gets reduced.
In order to underline the nontrivial way in which the S1symmetry becomes involved, we
speak of resonant equivariant generalization of the Hamiltonian Hopf bifurcation to RE.
In what follows we consider the 3D-image of the EM map as a one-parameter family of
2D-images parameterized by the value n of the polyad integral. To understand the global
picture we split the whole range of n > 0 into regions where the EM images (for fixed n) do
Fermi resonance, is not exact in real systems. So both symmetry arguments and physical
realizations suggest studying a system with detuned Fermi resonance [12,13]. In this work
we use the example of an integrable approximation to a classical system of three coupled
nonlinear oscillators whose Hamiltonian may be regarded as an effective vibrational Ham-
iltonian of a linear triatomic molecule, such as CO2in Fig. 1.
1.2. Dynamical regimes of systems with detuned Fermi resonance
Our system has three first integrals, the integral N of the 1:1:2 resonant harmonic oscil-
lator with value n, which we call polyad integral, the angular momentum L with value ‘,
and Hamiltonian H whose value, energy, we will denote by E; the image of the energy–mo-
mentum map EM of our system is a 3D region in the space R3with coordinates (‘,E,n).
The detuning term is part of the linearized Hamiltonian, i.e., this term is quadratic in (q,p).
Since the main Fermi resonance term is of degree 3, it is clear that at very small n when our
oscillator system is near its equilibrium q = p = 0, detuning is more important than the
resonance interaction. In this region, which we call regular, the system can be globally
described using three pairs of action-angle variables associated with its three normal oscil-
lator modes. At the same time, detuning can be neglected at sufficiently high n. Dynamics
in this high n region is qualitatively the same as that in the case of the exact resonance 1:1:2
studied in [3] and the joint eigenvalue spectrum lattice of the corresponding quantum sys-
tem should be equivalent to that described in [4]. In particular, our system should have
monodromy and consequently, it cannot be described using global action-angle variables.
1.3. Transition between different dynamical regimes
It is commonly known that in a parametric family of systems with two degrees of free-
dom, transition from a regular system to a system with monodromy is often associated
with Hamiltonian Hopf bifurcation [14]. In our situation, the most directly relevant gener-
alization of this bifurcation to three degrees of freedom is that of a bifurcation of a relative
equilibrium (RE) which is an orbit of an S1action. However, in all cases studied up to now
[15–20] the bifurcating orbit was a regular orbit of a locally free S1action and after the S1
symmetry was reduced the RE became an equilibrium undergoing a standard Hamiltonian
Hopf bifurcation. So, though clearly important for many applications, such generalization
is somewhat trivial. The major difference of our situation is that the bifurcating orbit is an
isolated critical orbit of the particular S1action of the 1:1:2 resonant harmonic oscillator
system. This orbit corresponds to the pure ‘‘spring’’ motion described by the oscillator
with frequency 2. (Note that in the CO2molecule it corresponds to the symmetric stretch
vibration m1.) All other orbits are generic and are two times longer. The short orbit is stable
in the regular region (at low n); it turns unstable as n increases past ncrit. We show that this
bifurcation is in many ways reminiscent of the trivial generalization of the Hamiltonian
Hopf bifurcation to RE. However, it cannot be reduced to the standard case because,
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a subcritical bifurcation occurs has been observed in a number of different systems
[16,17,20,23] and seems to be a common feature. In all these cases the set REMof regular
values in the image of the EM map can be represented using a cell unfolding surface
SEM. Different points of this surface which have the same energy E and momenta
(n,‘, ...) represent different regular connected components (normally tori) of the fiber
EM?1ðE; n;‘;...Þ. Such representation is especially simple in the two-dimensional case.
not vary qualitatively. We are not interested in the trivial exceptional value n = 0. We are
interested only in modifications of the EM image closely related to the bifurcation of the
‘‘spring’’ RE. We do not take into account possible saddle-node bifurcations [27–29] which
can lead to the appearance of new RE and to more complex intra-polyad dynamics but do
not involve the local neighborhood of the ‘‘spring’’ RE.
1.4. Supercritical transition
We aim at describing the whole of the image and fibers of the EM map of our system,
and therefore we should go beyond studying the resonant bifurcation of the ‘‘spring’’ RE
as a purely local individual phenomenon. Much like the standard Hamiltonian Hopf bifur-
cation [14], this bifurcation can be of two types depending on the parameters of the high
order terms in the Hamiltonian. Keeping the terminology, we call them supercritical and
subcritical.
The global context of the supercritical bifurcation is quite simple. There is no other
related phenomena and the 3D-image of the EM map can be split (depending on n value)
into two non-intersecting 3D-domains and one exceptional 2D-section. One of these 3D-
domains belongs to the region 0 < n < ncrit, the other domain lies in the region n > ncrit.
The exceptional section of the EM map for n = ncritincludes the image of the fiber corre-
sponding to the short ‘‘spring’’ RE at the moment of bifurcation. Note that because of
somewhat less stringent conditions on the Hamiltonian parameters, this kind of global
structure of the EM image seems to be prevailing in real molecular systems. CO2is certain-
ly of this kind. However the bifurcation happens there at very low value of n which is far
below n ¼3
other hand, the CS2molecule with much larger detuning provides a good example.
2of the ground state of the quantum analog and hence is unobservable. On the
1.5. Subcritical transition and bidromy
The subcritical scenario of the transition from the regular normal mode region to the
monodromy region is more complicated. In this case, the resonant Hopf-like bifurcation
of the short ‘‘central’’ or ‘‘spring’’ RE at n = ncritis preceded by two equivalent-by-sym-
metry cusp bifurcations at n = n0< ncritwhich give rise to new RE. In the small region with
n0< n < ncrit, the 3D set REMof regular EM values folds over itself while remaining connect-
ed thus forming a third transitional region. All this happens ‘‘semi-globally’’ in the small
immediate vicinity of the spring RE and the whole sequence of bifurcations is predefined
or ‘‘organized’’ by the form of the Hamiltonian. Most importantly, this sequence of bifur-
cations and the geometry of the image and fibers of the EM map in the transitional region
are structurally stable with regard to any sufficiently small equivariant variations of the
Hamiltonian.
The fact that the inverse map EM?1becomes multi-component shortly before or after
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