Stability of minimal periodic orbits

Symplectic twist maps are obtained from a Lagrangian variational principle. It is well known that nondegenerate minima of the action correspond to hyperbolic orbits of the map when the twist is negative definite and the map is two-dimensional. We show that for more than two dimensions, periodic orbits with minimal action in symplectic twist maps with negative definite twist are not necessarily hyperbolic. In the proof we show that in the neighborhood of a minimal periodic orbit of period n, the nth iterate of the map is again a twist map. This is true even though in general the composition of twist maps is not a twist map.     

Stochastic Tools on Hilbert Manifolds: Interplay with Geometry and Physics

Projections via the action of a Hilbert Lie group of a class of semi-martingales (given by It? fields) defined on Hilbert manifolds are investigated. Using It? calculus, we show that the drift term arising in the projected process can be interpreted in terms of a regularised trace of the second fundamental form of the orbits. For group actions with finite dimensional orbit space, we introduce a notion of strongly harmonic functions resp. regularised Brownian motion, which project onto harmonic functions resp. onto Brownian motion, whenever the orbits are minimal (in a regularised sense). We relate this projection procedure of semi-martingales to the Faddeev-Popov procedure in gauge field theory. Received: 27 November 1995 / Accepted: 6 November 1996     

The quantum spectra analysis of the circular billiards in wells

We use a recently defined quantum spectral function and apply the method of closed-orbit theory to the 2D circular billiard system. The quantum spectra contain rich information of all classical orbits connecting two arbitrary points in the well. We study the correspondence between quantum spectra and classical orbits in the circular, 1/2 circular and 1/4 circular wells using the analytic and numerical methods. We find that the peak positions in the Fourier-transformed quantum spectra match accurately with the lengths of the classical orbits. These examples show evidently that semi-classical method provides a bridge between quantum and classical mechanics.     

Control of long-period orbits and arbitrary trajectories in chaotic systems using dynamic limiting

We demonstrate experimental control of long-period orbits and arbitrary chaotic trajectories using a new chaos control technique called dynamic limiting. Based on limiter control, dynamic limiting uses a predetermined sequence of limiter levels applied to the chaotic system to stabilize natural states of the system. The limiter sequence is clocked by the natural return time of the chaotic system such that the oscillator sees a new limiter level for each peak return. We demonstrate control of period-8 and period-34 unstable periodic orbits in a low-frequency circuit and provide evidence that the control perturbations are minimal. We also demonstrate control of an arbitrary waveform by replaying a sequence captured from the uncontrolled oscillator, achieving a form of delayed self-synchronization. Finally, we discuss the use of dynamic limiting for high-frequency chaos communications. (c) 2002 American Institute of Physics.     

Quasi-periodicity,global stability and scaling in a model of Hamiltonian round-off

We investigate the effects of round-off errors on the orbits of a linear symplectic map of the plane, with rational rotation number nu=p/q. Uniform discretization transforms this map into a permutation of the integer lattice Z(2). We study in detail the case q=5, exploiting the correspondence between Z and a suitable domain of algebraic integers. We completely classify the orbits, proving that all of them are periodic. Using higher-dimensional embedding, we establish the quasi-periodicity of the phase portrait. We show that the model exhibits asymptotic scaling of the periodic orbits and a long-range clustering property similar to that found in repetitive tilings of the plane. (c) 1997 American Institute of Physics.     

Orbits in the H(2)O molecule

We study the forms of the orbits in a symmetric configuration of a realistic model of the H(2)O molecule with particular emphasis on the periodic orbits. We use an appropriate Poincare 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. (c) 2001 American Institute of Physics.     

Phase-plane analysis of test particle orbits in regular black holes

We investigate phase-plane analysis of general relativistic orbits in a gravitational field of the Reissner–Nordstr?m-type regular black hole spacetime. We employ phase-plane analysis to obtain different phase-plane diagrams of the test particle orbits by varying charge q and dimensionless parameter β, where β contains angular momentum of the test particle. We compute numerical values of radii for the innermost stable orbits and corresponding values of energy required to place the test particle in orbits. Later on, we employ similar analysis on an Ayón–Beato–García(ABG) regular black hole and a comparison regarding key results is also included.     

Double quantization of CP type orbits by generalized Verma modules

It is known that symmetric orbits in g^* for any simple Lie algebra g are equipped with a Poisson pencil generated by the Kirillov-Kostant-Souriau bracket and the reduced Sklyanin bracket associated to the “canonical” R-matrix. We realize quantization of the Poisson pencil CPⁿ type orbits (i.e. orbits in sl(n + 1)^* whose real compact form is CPⁿ) by means of q-deformed Verma modules.     

谐振子系统的量子-经典轨道、Berry相及Hannay角

从量子-经典轨道和几何相两方面, 研究了二维旋转平移谐振子系统的量子-经典对应. 通过广义规范变换得到了Lissajous经典周期轨道和Hannay角. 另外, 使用含时规范变换解析推导了旋转平移谐振子系统Schrödinger方程的本征波函数和Berry相, 得出结论: 原规范中的非绝热Berry相是经典Hannay角的-n倍. 最后, 使用SU(2)自旋相干态叠加, 构造一稳态波函数, 其波函数的概率云很好地局域于经典轨道上, 满足几何相位和经典轨道同时对应.     

Relativistic orbits of classical charged bodies in a spherically symmetric electrostatic field

The orbits of a relativistic charged body in a static, spherically symmetric electric field are calculated and classified in the classical theory. Contrary to the nonrelativistic problem, we find that there is a limiting minimal value for the angular momentumL _c . Should the actual angular momentum of a charged test body be lower than this limit, the test particle will spiral into the central point charge instead of having (precessing) Keplerian orbits.     

Strain-modulated excitonic gaps in mono-and bi-layer MoSe_2

Photoluminescence(PL) and Raman spectra under uniaxial strain were measured in mono- and bi-layer MoSe_2 to comparatively investigate the evolution of excitonic gaps and Raman phonons with strain. We observed that the strain dependence of excitonic gaps shows a nearly linear behavior in both flakes. One percent of strain increase gives a reduction of ~42 meV(~35 me V) in A-exciton gap in monolayer(bilayer) MoSe_2. The PL width remains little changed in monolayer MoSe_2 while it increases rapidly with strain in the bilayer case. We have made detailed discussions on the observed strain-modulated results and compared the difference between monolayer and bilayer cases. The hybridization between 4d orbits of Mo and 4p orbits of Se, which is controlled by the Se–Mo–Se bond angle under strain, can be employed to consistently explain the observations. The study may shed light into exciton physics in few-layer MoSe_2 and provides a basis for their applications.     

Propagating fronts in a bistable coupled map lattice

We consider the (traveling-wave-like) fronts which propagate with rational velocityp/q in a simple coupled map lattice for which the local map has two stable fixed points. We prove the uniqueness of such orbits up to time iterations, space translations, and permutations of the associated codes. A condition for their existence is also given, but it has to be checked in each case. We expect this condition to serve as a selection mechanism. The technique employed, the so-called (generalized) transfer matrix method, allows us to give explicit expressions for these solutions. These fronts are actually the observed orbits in the numerical simulations, as is shown with two examples: the case of velocity 1/2 and that of velocity 1.     

Structure of geodesics in the regular Hayward black hole space-time

The regular Hayward model describes a non-singular black hole space-time. By analyzing the behaviors of effective potential and solving the equation of orbital motion, we investigate the time-like and null geodesics in the regular Hayward black hole space-time. Through detailed analyses of corresponding effective potentials for massive particles and photons, all possible orbits are numerically simulated. The results show that there may exist four orbital types in the time-like geodesics structure: planetary orbits, circular orbits, escape orbits and absorbing orbits. In addition, when \(\ell \), a convenient encoding of the central energy density \(3/8\pi \ell ^{2}\), is 0.6M, and b is 3.9512M as a specific value of angular momentum, escape orbits exist only under \(b>3.9512M\). The precession direction is also associated with values of b. With \(b=3.70M\) the bound orbits shift clockwise but counter-clockwise with \(b=5.00M\) in the regular Hayward black hole space-time. We also find that the structure of null geodesics is simpler than that of time-like geodesics. There only exist three kinds of orbits (unstable circle orbits, escape orbits and absorbing orbits).     

On the Stability of Double Homoclinic Loops

We consider 2-degrees of freedom Hamiltonian systems with an involutive symmetry and a pair of orbits bi-asymptotic (homoclinic) to a saddle-center equilibrium (related to pairs of pure real, ±ν, and pure imaginary eigenvalues, ±ω i). We show that the stability of this double homoclinic loop is determined by the reflection coefficient of a one-dimensional scattering problem and ω/ν. We also show that the mechanism for losing stability is the creation of an infinite heteroclinic chain connecting a sequence of periodic orbits that accumulates at the double loop. Received: 10 November 1995 / Accepted: 5 June 1996     

Renormalization and Periodic Orbits¶for Hamiltonian Flows

We consider a renormalization group transformation for analytic Hamiltonians in two or more dimensions, and use this transformation to construct invariant tori, as well as sequences of periodic orbits with rotation vectors approaching that of the invariant torus. The construction of periodic and quasiperiodic orbits is limited to near-integrable Hamiltonians. But as a first step toward a non-perturbative analysis, we extend the domain of to include any Hamiltonian for which a certain non-resonance condition holds. Received: 5 October 1999 / Accepted: 2 February 2000     

Topological chaos and periodic braiding of almost-cyclic sets

In certain (2+1)-dimensional dynamical systems, the braiding of periodic orbits provides a framework for analyzing chaos in the system through application of the Thurston-Nielsen classification theorem. Periodic orbits generated by the dynamics can behave as physical obstructions that "stir" the surrounding domain and serve as the basis for this topological analysis. We provide evidence that, even in the absence of periodic orbits, almost-cyclic regions identified using a transfer operator approach can reveal an underlying structure that enables topological analysis of chaos in the domain.     

Basins of periodic orbits for elliptic maps of the torus

The behavior of basins of periodic orbits, for families of elliptic maps in the 2D torus depending on a parameter, is studied. We give an explicit formula for periodic orbits (i.e., central points of basins), considering also the occurrence of singular situations. Such a formula describes the evolution of basins, showing that onset and disappearance of periodic orbits cannot be reduced to a simple bifurcation scheme. Also, the stochastic features of the strange attractor at the border of ellipticity may be related to the dynamics of collapsing basins.     

Universal homoclinic bifurcations and chaos near double resonances

We study the dynamics near the intersection of a weaker and a stronger resonance inn-degree-of-freedom, nearly integrable Hamiltonian systems. For a truncated normal form we show the existence of (n–2)-dimensional hyperbolic invariant tori whose whiskers intersect inmultipulse homoclinic orbits with large splitting angles. The homoclinic obits are doubly asymptotic to solutions that diffuse across the weak resonance along the strong resonance. We derive a universalhomoclinic tree that describes the bifurcations of these orbits, which are shown to survive in the full normal form. We illustrate our results on a three-degree-of-freedom mechanical system.     

Regularisable and Minimal Orbits for Group Actions in Infinite Dimensions

We introduce a class of regularisable infinite dimensional principal fibre bundles which includes fibre bundles arising in gauge field theories like Yang-Mills and string theory and which generalise finite dimensional Riemannian principal fibre bundles induced by an isometric action. We show that the orbits of regularisable bundles have well defined, both heat-kernel and zeta function regularised volumes. We introduce a notion of μ-minimality ( ) for these orbits which extend the finite dimensional one. Our approach uses heat-kernel methods and yields both “heat-kernel” (obtained via heat-kernel regularisation) and “zeta function” (obtained via zeta function regularisation) minimality for specific values of the parameter μ. For each of these notions, we give an infinite dimensional version of Hsiang's theorem which extends the finite dimensional case, interpreting μ-minimal orbits as orbits with extremal (μ-regularised) volume. Received: 27 November 1995 / Accepted: 30 May 1997