Metamorphoses of Hamiltonian Systems with Symmetries
K. Efstathiou
Lecture Notes in Mathematics, vol. 1864, Springer-Verlag, 2005
ISBN: 978-3-540-24316-8
DOI: 10.1007/b105138
Preface and Table of Contents (156kb PDF download from springer.com)
K. Efstathiou
Lecture Notes in Mathematics, vol. 1864, Springer-Verlag, 2005
ISBN: 978-3-540-24316-8
DOI: 10.1007/b105138
Preface and Table of Contents (156kb PDF download from springer.com)
K. Efstathiou and G. Contopoulos
Chaos, 11, pp. 327-334, 2001
DOI: 10.1063/1.1356068
We study the forms of the orbits in a symmetric configuration of a realistic model of the H2O molecule with particular emphasis on the periodic orbits. We use an appropriate Poincaré surface of section (PSS) and study the distribution of the orbits on this PSS for various energies. We find both ordered and chaotic orbits. The proportion of ordered orbits is almost 100% for small energies, but decreases abruptly beyond a critical energy. When the energy exceeds the escape energy there are still nonescaping orbits around stable periodic orbits. We study in detail the forms of the various periodic orbits, and their connections, by providing appropriate stability and bifurcation diagrams.
K. Efstathiou and N. Voglis
Physica D, 158(1-4), pp. 151-163, 2001
DOI: 10.1016/S0167-2789(01)00299-8
A method is proposed for accurate evaluation of the rotation and the twist numbers of invariant circles in two degrees of freedom Hamiltonian systems or two-dimensional symplectic maps. The method uses the recurrence of orbits to overcome the problems usually arising because of the multivalued character of the angles (due to modulo 2pi) that have to be added in order to evaluate the above numbers. Furthermore, best convergent demoninators Qn of these numbers can be estimated and we show that under a proper treatment of the sequences of Qn iterations the accuracy is of the order of 1/Qn^4.
G. Contopoulos and K. Efstathiou
Cel. Mech., 88(2), pp. 163-183, 2004
DOI: 10.1023/B:CELE.0000016816.87061.11
Many physical systems can be modeled as scattering problems. For example, the motions of stars escaping from a galaxy can be described using a potential with two or more escape routes. Each escape route is crossed by an unstable Lyapunov orbit. The region between the two Lyapunov orbits is where the particle interacts with the system. We study a simple dynamical system with escapes using a suitably selected surface of section. The surface of section is partitioned in different escape regions which are defined by the intersections of the asymptotic manifolds of the Lyapunov orbits with the surface of section. The asymptotic curves of the other unstable periodic orbits form spirals around various escape regions. These manifolds, together with the manifolds of the Lyapunov orbits, govern the transport between different parts of the phase space. We study in detail the form of the asymptotic manifolds of a central unstable periodic orbit, the form of the escape regions and the infinite spirals of the asymptotic manifolds around the escape regions. We compute the escape rate for different values of the energy. In particular, we give the percentage of orbits that escape after a finite number of iterations. In a system with escapes one cannot define a Poincaré recurrence time, because the available phase space is infinite. However, for certain domains inside the lobes of the asymptotic manifolds there is a finite ‘minimum recurrence time’. We find the minimum recurrence time as a function of the energy.
K. Efstathiou, D.A. Sadovskií and R.H. Cushman
Proc. Roy. Soc. London Ser. A, 459(2040), pp. 2997-3019, 2003
DOI: 10.1098/rspa.2003.1158
We consider G × R-invariant Hamiltonians H on complex projective 2-space, where G is a point group and R is the time-reversal group. We find the symmetry-induced stationary points of H and classify them in terms of their linear stability. We then determine those points that can undergo a linear Hamiltonian Hopf bifurcation.
K. Efstathiou and D.A. Sadovskií
Nonlinearity, 17(2), pp. 415-446, 2004
DOI: 10.1088/0951-7715/17/2/003
We study a class of three degree of freedom (3-DOF) Hamiltonian systems that share certain characteristics with the 2-DOF Hénon–Heiles Hamiltonian. Our systems represent a 1:1:1 resonant three-oscillator whose principal nonlinear perturbation is the cubic potential term xyz with tetrahedral symmetry. After normalizing and reducing the 1:1:1 oscillator symmetry, we show that near the limit of linearization all our systems can be described as a one-parametric family. Such reduced systems have been suggested earlier by Hecht (1960 J. Mol. Spectrosc. 5 355) and later by Patterson (1985 J. Chem. Phys. 83 4618) to model triply degenerate vibrations of tetrahedral molecules. We describe relative equilibria (RE) of these systems, classify all qualitatively different family members, and discuss bifurcations of RE involved in the transitions from one region of regular parameter values to the other.
K. Efstathiou, D.A. Sadovskií and B.I. Zhilinskií
SIAM J. Appl. Dyn. Systems, 3(3), pp. 261-351, 2004
DOI: 10.1137/030600015
We study relative equilibria (RE) of a nonrigid molecule, which vibrates about a well-defined equilibrium configuration and rotates as a whole. Our analysis unifies the theory of rotational and vibrational RE. We rely on the detailed study of the symmetry group action on the initial and reduced phase space of our system and consider the consequences of this action for the dynamics of the system. We develop our approach on the concrete example of a four-atomic molecule A4 with tetrahedral equilibrium configuration, a dynamical system with six vibrational degrees of freedom. Further applications and illustrations of our results can be found in [van Hecke et al., Eur. Phys. J. D At. Mol. Opt. Phys., 17 (2001), pp. 13–35].
K. Efstathiou, M. Joyeux and D.A. Sadovskií
Phys. Rev. A, 69, 032504, 2004
DOI: 10.1103/PhysRevA.69.032504
We introduce and analyze a model system based on a deformation of a spherical pendulum that can be used to reproduce large amplitude bending vibrations of flexible triatomic molecules with two stable linear equilibria. On the basis of our model and the recent vibrational potential [J. Chem. Phys. 115, 3706 (2001)], we analyze the HCN/CNH isomerizing molecule. We find that HCN/CNH has no monodromy and introduce the second global bending quantum number for this system at all energies where the potential is expected to work. We also show that LiNC/LiCN is a qualitatively different system with monodromy.
K. Efstathiou, R.H. Cushman and D.A. Sadovskií
Physica D, 194(3-4), pp. 250-274, 2004
DOI: 10.1016/j.physd.2004.03.003
We consider the hydrogen atom in crossed electric and magnetic fields. We prove that near the Stark and Zeeman limits the system goes through two qualitatively different Hamiltonian Hopf bifurcations. We explain in detail the geometry of the bifurcations.
K. Efstathiou, D.A. Sadovskií and R.H. Cushman
Advances in Mathematics, 209(1), pp. 241-273, 2007
DOI: 10.1016/j.aim.2006.05.006
We give an analytic proof of the fractional monodromy theorem for the 1:-2 oscillator system with S^1 symmetry formulated by N. N. Nekhoroshev, D. A. Sadovskií, and B. I. Zhilinskií in C. R. Acad. Sci. Paris, Ser. I, 335 (2002) 985–988. Our proof is based on an analytic description of the Hamiltonian flow on the fibers of the integral map of this system.
K. Efstathiou, D.A. Sadovskií and B.I. Zhilinskií
Proc. Roy. Soc. London Ser. A, 463(2083), pp. 1771-1790, 2007
DOI: 10.1098/rspa.2007.1843
We consider perturbations of the hydrogen atom by sufficiently small homogeneous static electric and magnetic fields of all possible mutual orientations. Normalizing with regard to the Keplerian symmetry, we uncover resonances and conjecture that the parameter space of this family of dynamical systems is stratified into zones centred on the resonances. The 1:1 resonance corresponds to the orthogonal field limit, studied earlier by Cushman & Sadovskií (Cushman & Sadovskií 2000, Physica D 142, 166–196). We describe the structure of the 1:1 zone, where the system may have monodromy of different kinds, and consider briefly the 1:2 zone.
H.W. Broer, K. Efstathiou and E. Subramanian
Nonlinearity, 21(1), pp. 13-49, 2008
DOI: 10.1088/0951-7715/21/1/002
We consider arbitrarily large networks of pulse coupled oscillators with non-zero delay where the coupling is given by the Mirollo-Strogatz function. We prove that such systems have unstable attractors (saddle periodic orbits whose stable set has non-empty interior) in an open parameter region for three or more oscillators. The evolution operator of the system can be discontinuous and we propose an improved model with continuous evolution operator.
H.W. Broer, K. Efstathiou and E. Subramanian
Nonlinearity, 21(6), pp. 1385-1410, 2008
DOI: 10.1088/0951-7715/21/6/014
We consider networks of pulse coupled linear oscillators with non-zero delay where the coupling between the oscillators is given by the Mirollo-Strogatz function. We prove the existence of heteroclinic cycles between unstable attractors for a network of four oscillators and for an open set of parameter values.
K. Efstathiou, O.V. Lukina and D.A. Sadovskií
Phys. Rev. Lett., 101(25), 253003, 2008
DOI: 10.1103/PhysRevLett.101.253003
We study a perturbation of the hydrogen atom by small homogeneous static electric and magnetic fields in a specific mutual alignment with angle approximately π/3 which results in the 1:2 resonance of the linearized Keplerian n-shell approximation. The stratified image of the energy-momentum map (bifurcation diagram) of the classical integrable approximation has for most such field configurations the same typical structure which we describe. The structure of the corresponding quantum energy spectrum, which we describe in detail, is in certain ways an analogue of the well known degeneracy found by Herrick [Phys. Rev. A 26, 323 (1982)] for the quadratic Zeeman effect and present also for many near orthogonal field configurations.
K. Efstathiou, O.V. Lukina and D.A. Sadovskií
J. Phys. A, 42(5), 055209, 2009
DOI: 10.1088/1751-8113/42/5/055209
We consider perturbations of the hydrogen atom by sufficiently small homogeneous static electric and magnetic fields in near orthogonal configurations. Normalization of the Keplerian symmetry reveals that in the parameter space such systems belong in a ‘zone’ of systems close to the 1:1 resonance, the latter corresponding to the exactly orthogonal configuration. Integrable approximations obtained from second normalization of systems in the 1:1 zone are classified into several different qualitative types, many of which possess nontrivial monodromy. We compute monodromy of the complete three-dimensional energy-momentum map, compare the joint quantum spectrum to classical bifurcation diagrams, and show the effect of second normalization to the joint spectrum.
K. Efstathiou and D. Sugny
J. Phys. A, 43, 085216, 2010
DOI: 10.1088/1751-8113/43/8/085216
We consider two degree of freedom integrable Hamiltonian systems with bifurcation diagrams containing swallowtail structures. The global properties of the action coordinates in such systems together with the parallel transport of the period lattice and corresponding quantum cells in the joint spectrum are described in detail. The relation to the concept of bidromy which was introduced in (Sadovskií & Zhilinskií, Annals of Physics, 2007) is discussed.
E. Assémat, K. Efstathiou, M. Joeyux, and D. Sugny
Phys. Rev. Lett., 104(11), 113002, 2010
DOI: 10.1103/PhysRevLett.104.113002
We introduce the notion of fractional bidromy which is the combination of fractional monodromy and bidromy, two recent generalizations of Hamiltonian monodromy. We consider the vibrational spectrum of the HOCl molecule which is used as an illustrative example to show the presence of nontrivial fractional bidromy. To our knowledge, this is the first example of a molecular system where such a generalized monodromy is exhibited.
K. Efstathiou and D.A. Sadovskií
Rev. Mod. Phys., 82(3), pp. 2099-2154, 2010
DOI: 10.1103/RevModPhys.82.2099
The hydrogen atom perturbed by sufficiently small homogeneous static electric and magnetic fields of arbitrary mutual alignment is a specific perturbation of the Kepler system with three degrees of freedom and three parameters. Normalization of the Keplerian symmetry reveals that the parameter space is stratified into resonant zones of systems, each zone with an internal dynamical stratification of its own [Efstathiou et al, Proc. Royal Soc. London A 463, 1771-1790 (2007)]. , the bundle of invariant tori of individual systems within zones is characterized globally and the qualitative dynamical stratification is uncovered. The techniques involved in this analysis are illustrated with the example of the 1:1 resonance zone (near orthogonal fields) whose structure is known at present. Applications in the corresponding quantum system are also described.
H.W. Broer, K. Efstathiou and O.V. Lukina
Discr. Cont. Dyn. Sys. - Ser. S, 3(4), pp. 517-532, 2010
DOI: 10.3934/dcdss.2010.3.517
We prove the existence of fractional monodromy for two degree of freedom integrable Hamiltonian systems with one-parameter families of curled tori under certain general conditions. We describe the action coordinates of such systems near curled tori and we show how to compute fractional monodromy using the notion of the rotation number.
D.G.M. Beersma, H.W. Broer, K. Efstathiou, K.A. Gargar, and I. Hoveijn
Physica D, 240(19), pp. 1516-1527, 2011
DOI: 10.1016/j.physd.2011.06.019
Almost all organisms show some kind of time periodicity in their behavior. In mammals, the neurons of the suprachiasmatic nucleus form a biological clock regulating the activity-inactivity cycle of the animal. The main question is how this clock is able to entrain to the natural 24-hour light-dark cycle by which it is stimulated. Such a system is usually modelled as a collection of mutually coupled 2-state (active-inactive) phase oscillators with an external stimulus (Zeitgeber). In this article however, we investigate the entrainment of a single pacer cell to the ensemble of other pacer cells. Moreover the stimulus of the ensemble is taken to be periodic. The pacer cell interacts with its environment by phase delay at the end of its activity interval and phase advance at the end of its inactivity interval. We develop a mathematical model for this system, naturally leading to a circle map depending on parameters like the intrinsic period and phase delay and advance. The existence of resonance tongues in a circle map shows that an individual pacer cell is able to synchronize with the ensemble. We furthermore show how the parameters in the model can be related to biological observable quantities. Finally we give several directions of further research.
K. Efstathiou and D.A. Sadovskií
In «Geometric Mechanics and Symmetry: the Peyresq Lectures», J. Montaldi and T. Ratiu eds., Cambridge University Press, 2005
DOI: 10.1017/CBO9780511526367.005
K. Efstathiou
PhD thesis, Université du Littoral, Dunkerque, 2004
