On the Periodic Orbits of the Static,Spherically Symmetric Einstein-Yang-Mills Equations

In this paper we analyze the existence of the periodic orbits of the static, spherically symmetric Einstein–Yang–Mills Equations by using the qualitative theory of the ordinary differential equation. We prove that there are no periodic orbits restricted to some invariant set of codimension 1. Furthermore if there is a periodic orbit out of this invariant set, then there must be other periodic orbits, which are symmetric to the first one. We also have results on the non–existence of periodic orbits when the cosmological constant is negative.     

Localized waves in nonlinear oscillator chains

This paper reviews results about the existence of spatially localized waves in nonlinear chains of coupled oscillators, and provides new results for the Fermi-Pasta-Ulam (FPU) lattice. Localized solutions include solitary waves of permanent form and traveling breathers which appear time periodic in a system of reference moving at constant velocity. For FPU lattices we analyze the case when the breather period and the inverse velocity are commensurate. We employ a center manifold reduction method introduced by Iooss and Kirchgassner in the case of traveling waves, which reduces the problem locally to a finite dimensional reversible differential equation. The principal part of the reduced system is integrable and admits solutions homoclinic to quasi-periodic orbits if a hardening condition on the interaction potential is satisfied. These orbits correspond to approximate travelling breather solutions superposed on a quasi-periodic oscillatory tail. The problem of their persistence for the full system is still open in the general case. We solve this problem for an even potential if the breather period equals twice the inverse velocity, and prove in that case the existence of exact traveling breather solutions superposed on an exponentially small periodic tail.     

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.     

Hamiltonian Systems Admitting a Runge–Lenz Vector and an Optimal Extension of Bertrand’s Theorem to Curved Manifolds

Bertrand’s theorem asserts that any spherically symmetric natural Hamiltonian system in Euclidean 3-space which possesses stable circular orbits and whose bounded trajectories are all periodic is either a harmonic oscillator or a Kepler system. In this paper we extend this classical result to curved spaces by proving that any Hamiltonian on a spherically symmetric Riemannian 3-manifold which satisfies the same conditions as in Bertrand’s theorem is superintegrable and given by an intrinsic oscillator or Kepler system. As a byproduct we obtain a wide panoply of new superintegrable Hamiltonian systems. The demonstration relies on Perlick’s classification of Bertrand spacetimes and on the construction of a suitable, globally defined generalization of the Runge–Lenz vector.     

Kepler motion on single-sheet hyperboloid

The classical Kepler–Coulomb problem on the single-sheeted hyperboloid H ³ ₁ is solved in the framework of the Hamilton–Jacobi equation. We have proven that all the bounded orbits are closed and periodic. The paths are ellipses or circles for finite motion.     

More than six hundred new families of Newtonian periodic planar collisionless three-body orbits

The famous three-body problem can be traced back to Isaac Newton in the 1680 s. In the 300 years since this "three-body problem"was first recognized, only three families of periodic solutions had been found, until 2013 when ˇSuvakov and Dmitraˇsinovi′c [Phys.Rev. Lett. 110, 114301(2013)] made a breakthrough to numerically find 13 new distinct periodic orbits, which belong to 11 new families of Newtonian planar three-body problem with equal mass and zero angular momentum. In this paper, we numerically obtain 695 families of Newtonian periodic planar collisionless orbits of three-body system with equal mass and zero angular momentum in case of initial conditions with isosceles collinear configuration, including the well-known figure-eight family found by Moore in 1993, the 11 families found by ˇSuvakov and Dmitraˇsinovi′c in 2013, and more than 600 new families that have never been reported, to the best of our knowledge. With the definition of the average period T = T=Lf, where Lf is the length of the so-called "free group element", these 695 families suggest that there should exist the quasi Kepler's third law T* ≈ 2:433 ± 0:075 for the considered case, where T*= T|E|~(3/2) is the scale-invariant average period and E is its total kinetic and potential energy,respectively. The movies of these 695 periodic orbits in the real space and the corresponding close curves on the "shape sphere"can be found via the website: http://numericaltank.sjtu.edu.cn/three-body/three-body.htm.     

Heteroclinic Connections Between Periodic Orbits in Planar Restricted Circular Three Body Problem. Part II

We present a method for proving the existence of symmetric periodic, heteroclinic or homoclinic orbits in dynamical systems with the reversing symmetry. As an application we show that the Planar Restricted Circular Three Body Problem (PCR3BP) corresponding to the Sun-Jupiter-Oterma system possesses an infinite number of symmetric periodic orbits and homoclinic orbits to the Lyapunov orbits. Moreover, we show the existence of symbolic dynamics on six symbols for PCR3BP and the possibility of resonance transitions of the comet. This extends earlier results by Wilczak and Zgliczynski [12]. Electronic Supplementary Material: Supplementary material is available in the online version of this article at An erratum to this article is available at .     

Periodic solutions for planar four-body problems

For Newtonian four-body problems with equal masses, we use variational minimizing methods to prove the existence of non-collision periodic solutions such that all bodies move on two symmetric trajectories with opposite orientation; here the methods and results are simpler and more general than those of Chen [K.-C. Chen, Action minimizing orbits in the parallelogram four-body problem with equal masses, Arch. Ration. Mech. Anal. 158 (2001) 293–318].     

Quasiclassical dynamics of resonantly driven Rydberg states

We present a semiclassical analysis of the dynamics of Rydberg states of atomic hydrogen driven by a resonant microwave field of linear polarization. The semiclassical quasienergies of the atom in the field are found to be in very good agreement with the exact quantum solutions. The ionization rates of individual eigenstates of the atom dressed by the field reflect their quasiclassical dynamics along classical periodic orbits in the near integrable regime, but exhibit a transition to nonspecific rates when global chaos takes over in phase space. We concentrate both on the principal resonance where the unperturbed Kepler frequency is equal to the driving field frequency and on the higher primary resonance The latter case allows for the construction of nondispersive wave packets which propagate along Kepler ellipses of intermediate eccentricity. Received: 23 June 1998 / Accepted: 10 November 1998     

The symmetric periodic orbits for the two-electron atom

We analyse the existence of symmetric periodic orbits of the two-electron atom. The results obtained show that there exist six families of periodic orbits that can be prolonged from a continuum of periodic symmetric orbits. The main technique applied in this study is the continuation method of Poincaré.     

Perturbation theory for periodic orbits in a class of infinite dimensional Hamiltonian systems

We consider a class of Hamiltonian systems describing an infinite array of coupled anharmonic oscillators, and we study the bifurcation of periodic orbits off the equilibrium point. The family of orbits we construct can be parametrized by their periods which belong to Cantor sets of large measure containing certain periods of the linearized problem as accumulation points. The infinitely many holes forming a dense set on which the existence of a periodic orbit cannot be proven originate from a dense set of resonances that are present in the system. We also have a result concerning the existence of solutions of arbitrarily large amplitude.Address after September 1990     

耦合不连续系统同步转换过程中的多吸引子共存

本文研究了耦合不连续系统的同步转换过程中的动力学行为, 发现由混沌非同步到混沌同步的转换过程中特殊的多吸引子共存现象. 通过计算耦合不连续系统的同步序参量和最大李雅普诺夫指数随耦合强度的变化, 发现了较复杂的同步转换过程: 临界耦合强度之后出现周期非同步态(周期性窗口); 分析了系统周期态的迭代轨道,发现其具有两类不同的迭代轨道: 对称周期轨道和非对称周期轨道, 这两类周期吸引子和同步吸引子同时存在, 系统表现出对初值敏感的多吸引子共存现象. 分析表明, 耦合不连续系统中的周期轨道是由于局部动力学的不连续特性和耦合动力学相互作用的结果. 最后, 对耦合不连续系统的同步转换过程进行了详细的分析, 结果表明其同步呈现出较复杂的转换过程.     

Dynamical ordering of symmetric non-Birkhoff periodic points in reversible monotone twist mappings

We study the coexistence of symmetric non-Birkhoff periodic orbits of C(1) reversible monotone twist mappings on the cylinder. We prove the equivalence of the existence of non-Birkhoff periodic orbits and that of transverse homoclinic intersections of stable and unstable manifolds of the fixed point. We derive the positional relation of symmetric Birkhoff and non-Birkhoff periodic orbits and obtain the dynamical ordering of symmetric non-Birkhoff periodic orbits. An extension of the Sharkovskii ordering to two-dimensional mappings has been carried out. In the proof of various properties of the mappings, reversibility plays an essential role. (c) 2002 American Institute of Physics.     

Periodic orbit quantization of the anisotropic Kepler problem

The periodic orbit quantization on the anisotropic Kepler problem is tested. By computing the stability and action of some 2000 of the shortest periodic orbits, the eigenvalue spectrum of the anisotropic Kepler problem is calculated. The aim is to test the following claims for calculating the quantum spectrum of classically chaotic systems: (1) Curvature expansions of quantum mechanical zeta functions offer the best semiclassical estimates; (2) the real part of the cycle expansions of quantum mechanical zeta functions cut at appropriate cycle length offer the best estimates; (3) cycle expansions are superfluous; and (4) only a small subset of cycles (irreducible cycles) suffices for good estimates for the eigenvalues. No evidence is found to support any of the four claims.     

Phase space structure of the hydrogen atom in a circularly polarized microwave field

We consider the problem of the hydrogen atom interacting with a circularly polarized microwave field, modeled as a perturbed Kepler problem. A remarkable feature of this system is that the electron can follow what we term erratic orbits before ionizing. In an erratic orbit the electron makes multiple large distance excursions from the nucleus with each excursion being followed by a close approach to the nucleus, where the interaction is large. Here we are interested in the mechanisms that explain this observation. We find that the manifolds associated with certain hyperbolic periodic orbits may play an important role, despite the fact that, in some respects, the dynamics is almost Keplerian. A study of some relevant invariant objects is carried out for different system parameters. The consequences of our findings for ionization of an electron by the external field are also discussed.     

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.     

Resonant capture of multiple planet systems under dissipation and stable orbital configurations

Migration of planetary systems caused by the action of dissipative forces may lead the planets to be trapped in a resonance. In this work we study the conditions and the dynamics of such resonant trapping. Particularly, we are interested in finding out whether resonant capture ends up in a long-term stable planetary configuration. For two planet systems we associate the evolution of migration with the existence of families of periodic orbits in the phase space of the three-body problem. The family of circular periodic orbits exhibits a gap at the 2:1 resonance and an instability and bifurcation at the 3:1 resonance. These properties explain the high probability of 2:1 and 3:1 resonant capture at low eccentricities. Furthermore, we study the resonant capture of three-planet systems. We show that such a resonant capture is possible and can occur under particular conditions. Then, from the migration path of the system, stable three-planet configurations, either symmetric or asymmetric, can be determined.     

Geodesics around oscillatons

Oscillatons are spherically symmetric solutions to the Einstein–Klein–Gordon equations. These solutions are non-singular, asymptotically flat, and with periodic time dependency. In this paper, we investigate the geodesic motion of particles moving around of an oscillatonic field. Bound orbits are found for particular values of the particles' angular momentum L and their initial radial position r ₀. It is found that the radial coordinate of such particles oscillates in time and we are able to predict the corresponding oscillating period as well as its amplitude. We carry out this study for the quadratic V(φ) = m _Φ Φ²/2 scalar field potential. We discuss possible ways to follow in order to connect this kind of studies with astrophysical observations.     

Semiclassical expansion of quantum characteristics for many‐body potential scattering problem

In quantum mechanics, systems can be described in phase space in terms of the Wigner function and the star‐product operation. Quantum characteristics, which appear in the Heisenberg picture as the Weyl's symbols of operators of canonical coordinates and momenta, can be used to solve the evolution equations for symbols of other operators acting in the Hilbert space. To any fixed order in the Planck's constant, many‐body potential scattering problem simplifies to a statistical‐mechanical problem of computing an ensemble of quantum characteristics and their derivatives with respect to the initial canonical coordinates and momenta. The reduction to a system of ordinary differential equations pertains rigorously at any fixed order in ?. We present semiclassical expansion of quantum characteristics for many‐body scattering problem and provide tools for calculation of average values of time‐dependent physical observables and cross sections. The method of quantum characteristics admits the consistent incorporation of specific quantum effects, such as non‐locality and coherence in propagation of particles, into the semiclassical transport models. We formulate the principle of stationary action for quantum Hamilton's equations and give quantum‐mechanical extensions of the Liouville theorem on conservation of the phase‐space volume and the Poincaré theorem on conservation of 2p‐forms. The lowest order quantum corrections to the Kepler periodic orbits are constructed. These corrections show the resonance behavior.     

Frequency modulation indicator, Arnold’s web and diffusion in the Stark-Quadratic-Zeeman problem

We notice that the fundamental frequencies of a slightly perturbed integrable Hamiltonian system are not time-constant inside a resonance but frequency modulated, as is evident from pendulum models and wavelet analysis. Exploiting an intrinsic imprecision inherent to the numerical frequency analysis algorithm itself, hence transforming a drawback into an opportunity, we define the Frequency Modulation Indicator, a very sensitive tool in detecting where fundamental frequencies are modulated, localizing so the resonances without having to resort, as in other methods, to the integration of variational equations. For the Kepler problem, the space of the orbits with a fixed energy has the topology of the product of two 2-spheres. The perturbation Hamiltonian, averaged over the mean anomaly, has surely a maximum and a minimum, to which correspond two periodic orbits in physical space. Studying the neighbourhood of these two elliptic stable points, we are able to define adapted action-angle variables, for example, the usual but “SO(4)-rotated” Delaunay variables. The procedure, implemented in the program KEPLER, is performed transparently for the user, providing a general scheme suited for generic perturbation. The method is then applied to the Stark-Quadratic-Zeeman problem, displaying very clearly the Arnold web of the resonances. Sectioning transversely one of the resonance strips so highlighted and performing a numerical frequency analysis, one is able to locate with great precision the thin stochastic layer surrounding a separatrix. Another very long (10⁸ revolutions) frequency analysis on an orbit starting here reveals, as expected, a well defined pattern, which ensures that the integration errors do not eject the point out of the layer, and moreover a very slow drift in the frequency values, clearly due to Arnold diffusion.