Use of harmonic inversion techniques in the periodic orbit quantization of integrable systems

Harmonic inversion has already been proven to be a powerful tool for the analysis of quantum spectra and the periodic orbit orbit quantization of chaotic systems. The harmonic inversion technique circumvents the convergence problems of the periodic orbit sum and the uncertainty principle of the usual Fourier analysis, thus yielding results of high resolution and high precision. Based on the close analogy between periodic orbit trace formulae for regular and chaotic systems the technique is generalized in this paper for the semiclassical quantization of integrable systems. Thus, harmonic inversion is shown to be a universal tool which can be applied to a wide range of physical systems. The method is further generalized in two directions: firstly, the periodic orbit quantization will be extended to include higher order corrections to the periodic orbit sum. Secondly, the use of cross-correlated periodic orbit sums allows us to significantly reduce the required number of orbits for semiclassical quantization, i.e., to improve the efficiency of the semiclassical method. As a representative of regular systems, we choose the circle billiard, whose periodic orbits and quantum eigenvalues can easily be obtained. Received 24 February 2000 and Received in final form 22 May 2000     

Eigenvalues and eigenfunctions of a clover plate

We report a numerical study of the flexural modes of a plate using semi-classical analysis developed in the context of quantum systems. We first introduce the Clover billiard as a paradigm for a system inside which rays exhibit stable and chaotic trajectories. The resulting phase space explored by the ray trajectories is illustrated using the Poincare surface of section, and shows that it has both integrable and chaotic regions. Examples of the stable and the unstable periodic orbits in the geometry are presented. We numerically solve the biharmonic equation for the flexural vibrations of the Clover shaped plate with clamped boundary conditions. The first few hundred eigenvalues and the eigenfunctions are obtained using a boundary elements method. The Fourier transform of the eigenvalues show strong peaks which correspond to ray periodic orbits. However, the peaks corresponding to the shortest stable periodic orbits are not stronger than the peaks associated with unstable periodic orbits. We also perform statistics on the obtained eigenvalues and the eigenfunctions. The eigenvalue spacing distribution P(s) shows a strong peak and therefore deviates from both the Poisson and the Wigner distribution of random matrix theory at small spacings because of the C_4v symmetry of the Clover geometry. The density distribution of the eigenfunctions is observed to agree with the Porter-Thomas distribution of random matrix theory. Received 12 February 2001 and Received in final form 17 April 2001     

Periodic orbits and binary collisions in the classical three-body Coulomb problem

In the helium case of the classical three-body Coulomb problem in two dimensions with zero angular momentum, we develop a procedure to find periodic orbits applying two symbolic dynamics for one-dimensional and planar problems. Focusing our attention on binary collisions with these tools, a sequence of periodic orbits are predicted and are actually found numerically. A family of periodic orbits found has regularity in their actions. For this family of periodic orbits, it is shown that thanks to its regularity, a partial summation of the Gutzwiller trace formula with a daring approximation gives a Rydberg series of energy levels.     

Statistics of self-crossings and avoided crossings of periodic orbits in the Hadamard-Gutzwiller model

Employing symbolic dynamics for geodesic motion on the tesselated pseudosphere, the so-called Hadamard-Gutzwiller model, we construct extremely long periodic orbits without compromising accuracy. We establish criteria for such long orbits to behave ergodically and to yield reliable statistics for self-crossings and avoided crossings. Self-encounters of periodic orbits are reflected in certain patterns within symbol sequences, and these allow for analytic treatment of the crossing statistics. In particular, the distributions of crossing angles and avoided-crossing widths thus come out as related by analytic continuation. Moreover, the action difference for Sieber-Richter pairs of orbits (one orbit has a self-crossing which the other narrowly avoids and otherwise the orbits look very nearly the same) results to all orders in the crossing angle. These findings may be helpful for extending the work of Sieber and Richter towards a fuller understanding of the classical basis of quantum spectral fluctuations. Received 17 July 2002 Published online 29 November 2002     

The positronium negative ion: Classical properties aaand semiclassical quantization

Properties of collinear and planar periodic orbits for the positronium negative ion are examined with respect to the possibilities for semiclassical quantization. In contrast to other two-electron atomic systems as helium and H^- the relevant orbits for quantization are fully stable and permit a full torus quantization. However, for lower excitations the area of stability in phase-space is too small for a reliable torus quantization. Instead, a quasi-separability of the three-body system is used to apply effective one-dimensional (WKB) quantization. Received 19 January 2001     

Unconventional field and angle dependences of the Shubnikov-de Haas oscillations spectra in the quasi two-dimensional organic superconductor (BEDO-TTF) 2ReO 4H 2O

We report on the inter-layer oscillatory conductance of the two-dimensional organic superconductor (BEDO-TTF)₂ReO₄H₂O measured in static and pulsed magnetic fields of up to 15 and 52 T, respectively. In agreement with previous in-plane studies, two Shubnikov-de Haas oscillation series linked to the two electron and the hole orbits are observed. The influence of the magnitude and orientation of the magnetic field with respect to the conducting plane is studied in the framework of the conventional two- and three-dimensional Lifshits-Kosevich (LK) model. Deviations of the data from this model are observed in low fields strongly tilted with respect to the normal to the conducting plane. In this latter case, the observed behaviour is consistent with an unexplained lowering of the cyclotron effective mass. At high magnetic field, the oscillatory data could have been compatible with the occurrence of a magnetic breakdown orbit built from the hole and electron orbits. However, the increase of the cyclotron effective mass, linked to the electron orbits, as the magnetic field increases above ∼12 T is consistent with a field-induced phase transition. In the lower field range, where the conventional LK model holds, the analysis of the angle dependence of the oscillations amplitude suggests significant renormalisation of the effective Landé factor. Received 22 August 2000 and Received in final form 20 December 2000     

Double lunar swing-by orbits revisited

The double lunar swing-by orbits are a special kind of orbits in the Earth-Moon system.These orbits repeatedly pass through the vicinity of the Moon and change their shapes due to the Moon’s gravity.In the synodic frame of the circular restricted three-body problem consisting of the Earth and the Moon,these orbits are periodic,with two close approaches to the Moon in every orbit period.In this paper,these orbits are revisited.It is found that these orbits belong to the symmetric horseshoe periodic families which bifurcate from the planar Lyapunov family around the collinear libration point L3.Usually,the double lunar swing-by orbits have k=i+j loops,where i is the number of the inner loops and j is the number of outer loops.The genealogy of these orbits with different i and j is studied in this paper.That is,how these double lunar swing-by orbits are organized in the symmetric horseshoe periodic families is explored.In addition,the 2n lunar swing-by orbits(n≥2)with 2n close approaches to the Moon in one orbit period are also studied.     

Delayed feedback control of time-delayed chaotic systems: Analytical approach at Hopf bifurcation

This Letter is concerned with bifurcation and chaos control in scalar delayed differential equations with delay parameter τ. By linear stability analysis, the conditions under which a sequence of Hopf bifurcation occurs at the equilibrium points are obtained. The delayed feedback controller is used to stabilize unstable periodic orbits. To find the controller delay, it is chosen such that the Hopf bifurcation remains unchanged. Also, the controller feedback gain is determined such that the corresponding unstable periodic orbit becomes stable. Numerical simulations are used to verify the analytical results.     

Higher-order <Emphasis Type="Bold">?</Emphasis> corrections in the semiclassical quantization of chaotic billiards

In the periodic orbit quantization of physical systems, usually only the leading-order ? contribution to the density of states is considered. Therefore, by construction, the eigenvalues following from semiclassical trace formulae generally agree with the exact quantum ones only to lowest order of ?. In different theoretical work the trace formulae have been extended to higher orders of ?. The problem remains, however, how to actually calculate eigenvalues from the extended trace formulae since, even with ? corrections included, the periodic orbit sums still do not converge in the physical domain. For lowest-order semiclassical trace formulae the convergence problem can be elegantly, and universally, circumvented by application of the technique of harmonic inversion. In this paper we show how, for general scaling chaotic systems, also higher-order ? corrections to the Gutzwiller formula can be included in the harmonic inversion scheme, and demonstrate that corrected semiclassical eigenvalues can be calculated despite the convergence problem. The method is applied to the open three-disk scattering system, as a prototype of a chaotic system. Received 10 September 2001 and Received in final form 3 January 2002     

Travelling Breathers with Exponentially Small Tails in a Chain of Nonlinear Oscillators

We study the existence of travelling breathers in Klein-Gordon chains, which consist of one-dimensional networks of nonlinear oscillators in an anharmonic on-site potential, linearly coupled to their nearest neighbors. Travelling breathers are spatially localized solutions which appear time periodic in a referential in translation at constant velocity. Approximate solutions of this type have been constructed in the form of modulated plane waves, whose envelopes satisfy the nonlinear Schrödinger equation (M. Remoissenet, Phys. Rev. B 33, n.4, 2386 (1986), J. Giannoulis and A. Mielke, Nonlinearity 17, p. 551–565 (2004)). In the case of travelling waves (where the phase velocity of the plane wave equals the group velocity of the wave packet), the existence of nearby exact solutions has been proved by Iooss and Kirchgässner, who have obtained exact solitary wave solutions superposed on an exponentially small oscillatory tail (G. Iooss, K. Kirchgässner, Commun. Math. Phys. 211, 439–464 (2000)). However, a rigorous existence result has been lacking in the more general case when phase and group velocities are different. This situation is examined in the present paper, in a case when the breather period and the inverse of its velocity are commensurate. We show that the center manifold reduction method introduced by Iooss and Kirchgässner is still applicable when the problem is formulated in an appropriate way. This allows us to reduce the problem locally to a finite dimensional reversible system of ordinary differential equations, whose principal part admits homoclinic solutions to quasi-periodic orbits under general conditions on the potential. For an even potential, using the additional symmetry of the system, we obtain homoclinic orbits to small periodic ones for the full reduced system. For the oscillator chain, these orbits correspond to exact small amplitude travelling breather solutions superposed on an exponentially small oscillatory tail. Their principal part (excluding the tail) coincides at leading order with the nonlinear Schrödinger approximation.     

Electron-pair densities of group 2 atoms in their and terms

Electron-pair intracule (relative motion) and extracule (center-of-mass motion) densities are studied in both position and momentum spaces for the ¹ P and ³ P terms of the group 2 atoms Be (atomic number Z =4), Mg (Z =12), Ca (Z =20), Sr (Z =38), Ba (Z =56), and Ra (Z =88). In position space, the ¹ P - ³ P difference in the intracule densities shows that the probability of a small interelectronic distance is larger in the triplet for all the six atoms, as reported for the lightest Be atom in the literature. The position-space extracule density clarifies that the triplet electrons are more likely to be at opposite positions with respect to the nucleus than the singlet electrons for all the atoms. In momentum space, the singlet generally has a larger probability of a small relative momentum between two electrons as a na?ve manifestation of the Fermi hole in the triplet. The extracule density in momentum space shows that the ¹ P term has a distribution larger in a large center-of-mass momentum region than the ³ P term. Received: 26 August 1998 / Received in final form: 1 February 1999     

Triaxial octupole deformations and shell structure

Manifestations of pronounced shell effects are discovered when non-axial octupole deformations are added to a harmonic oscillator model. The degeneracies of the quantum spectra are in good agreement with the corresponding main periodic orbits and winding number ratios which are found by classical analysis. Pis’ma Zh. éksp. Teor. Fiz. 69, No. 8, 525–530 (25 April 1999) Published in English in the original Russian journal. Edited by Steve Torstveit.     

Quantum effects of periodic orbits for the kicked top

We analyze traces of powers of the time evolution operator of a periodically kicked top. Semiclassically, such traces are related to periodic orbits of the classical map. We derive the semiclassical traces in a coherent state basis and show how the periodic orbits can be recovered via a Fourier transform. A breakdown of the stationary phase approximation is detected. The quasi energy spectrum remains elusive due to lack of knowledge of sufficiently many periodic orbits. Divergencies of periodic orbit formulas are avoided by appealing to the finiteness of the quantum mechanical Hilbert space. The traces also enter the coefficients of the characteristic polynominal of the Floquet operator. Statistical properties of these coefficients give rise to a new criterion for the distinction of chaos and regular motion.     

Periodic orbits in magnetic billiards

We propose a simple method to calculate periodic orbits in two-dimensional systems with no symbolic dynamics. The method is based on a line by line scan of the Poincaré surface of section and is particularly useful for billiards. We have applied it to the Square and Sinai's billiards subjected to a uniform orthogonal magnetic field and we obtained about 2000 orbits for both systems using absolutely no information about their symbolic dynamics. Received 21 September 1999 and Received in final form 13 April 2000     

The Al-Pd-Mn quasicrystalline approximant -phase revisited

A detailed investigation of the Fourier space of several Al-Pd-Mn samples with composition Al-72.6 at. %, Pd-22.9 at. %, Mn-4.5 at. % is reported. In the phase diagram of the Al-Pd-Mn ternary alloy, this composition corresponds to the so-called ξ' phase which was described as an icosahedral quasicrystalline approximant. By re-examining the Fourier space by means of X-ray diffraction (powder patterns and single crystal precession patterns), complex structures in close relation with the ξ'-phase have been observed. These long-range order complex structures are described as resulting from a periodic perturbation of the ξ' structure along the c direction. Two states with periodicities c (3 + τ) and c (5 + τ) have been observed in this study (τ: golden mean). Structural models based on periodic arrangements of “defects” layers separating layers of phase are proposed. These two states are certainly intermediate states between the phase and the metastable decagonal quasicrystalline phase. Received 11 April 2002 / Received in final form 24 June 2002 Published online 17 September 2002     

Global view on a nonlinear oscillator subject to time-delayed feedback control

We apply time-delayed feedback control to stabilise unstable periodic orbits of an amplitude-phase oscillator. The control acts on both, the amplitude and the frequency of the oscillator, and we show how the phase of the control signal influences the dynamics of the oscillator. A comprehensive bifurcation analysis in terms of the control phase and the control strength reveals large stability regions of the target periodic orbit, as well as an increasing number of unstable periodic orbits caused by the time delay of the feedback loop. Our results provide insight into the global features of time-delayed control schemes.     

Twofold-broken rational tori and the uniform semiclassical approximation

We investigate broken rational tori consisting of a chain of four (rather than two) periodic orbits. The normal form that describes this configuration is identified and used to construct a uniform semiclassical approximation, which can be utilized to improve trace formulae. An accuracy gain can be achieved even for the situation when two of the four orbits are ghosts. This is illustrated for a model system, the kicked top. Received 3 August 1999     

Propagation of travelling waves in sub-excitable systems driven by noise and periodic forcing

It has been reported that traveling waves propagate periodically and stably in sub-excitable systems driven by noise [Phys. Rev. Lett. 88, 138301 (2002)]. As a further investigation, here we observe different types of traveling waves under different noises and periodic forces, using a simplified Oregonator model. Depending on different noises and periodic forces, we have observed different types of wave propagation (or their disappearance). Moreover, reversal phenomena are observed in this system based on the numerical experiments in the one-dimensional space. We explain this as an effect of periodic forces. Thus, we give qualitative explanations for how stable reversal phenomena appear, which seem to arise from the mixing function of the periodic force and the noise. The output period and three velocities (normal, positive and negative) of the travelling waves are defined and their relationship with the periodic forces, along with the types of waves, are also studied in sub-excitable system under a fixed noise intensity. Electronic supplementary material Supplementary Online Material     

One family of 13315 stable periodic orbits of non-hierarchical unequal-mass triple systems

The three-body problem can be traced back to Newton in 1687,but it is still an open question today.Note that only a few periodic orbits of three-body systems were found in 300 years after Newton mentioned this famous problem.Although triple systems are common in astronomy,practically all observed periodic triple systems are hierarchical(similar to the Sun,Earth and Moon).It has traditionally been believed that non-hierarchical triple systems would be unstable and thus should disintegrate into a stable binary system and a single star,and consequently stable periodic orbits of non-hierarchical triple systems have been expected to be rather scarce.However,we report here one family of 135445 periodic orbits of non-hierarchical triple systems with unequal masses;13315 among them are stable.Compared with the narrow mass range(only 10-5)in which stable"Figure-eight"periodic orbits of three-body systems exist,our newly found stable periodic orbits have fairly large mass region.We find that many of these numerically found stable non-hierarchical periodic orbits have mass ratios close to those of hierarchical triple systems that have been measured with astronomical observations.This implies that these stable periodic orbits of non-hierarchical triple systems with distinctly unequal masses quite possibly can be observed in practice.Our investigation also suggests that there should exist an infinite number of stable periodic orbits of non-hierarchical triple systems with distinctly unequal masses.Note that our approach has general meaning:in a similar way,every known family of periodic orbits of three-body systems with two or three equal masses can be used as a starting point to generate thousands of new periodic orbits of triple systems with distinctly unequal masses.     

A variational approach to connecting orbits in nonlinear dynamical systems

We propose a variational method for determining homoclinic and heteroclinic orbits including spiral-shaped ones in nonlinear dynamical systems. Starting from a suitable initial curve, a homotopy evolution equation is used to approach a true connecting orbit. The procedure is an extension of a variational method that has been used previously for locating cycles, and avoids the need for linearization in search of simple connecting orbits. Examples of homoclinic and heteroclinic orbits for typical dynamical systems are presented. In particular, several heteroclinic orbits of the steady-state Kuramoto–Sivashinsky equation are found, which display interesting topological structures, closely related to those of the corresponding periodic orbits.