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.     

A theorem of Šarkovskii on the existence of periodic orbits of continuous endomorphisms of the real line

Two theorems are proved—the first and the more important of them due to ?arkovskii—providing complete and surprisingly simple answers to the following two questions: (i) given that a continuous mapT of an interval into itself (more generally, into the real line) has a periodic orbit of periodn, which other integers must occur as periods of the periodic orbits ofT? (ii) given thatn is the least odd integer which occurs as a period of a periodic orbit ofT, what is the “shape” of that orbit relative to its natural ordering as a finite subset of the real line? As an application, we obtain improved lower bounds for the topological entropy ofT.     

Paul阱中共线三离子体系的经典动力学

研究了在Paul阱囚禁场赝势作用下共线构形的三离子体系经典动力学特性.尽管这是一个非线性体系，但不存在混沌，即体系在任何能量下运动都是规则的，而相空间则由两个轨迹为对称和反对称周期(或准周期)轨道的KAM不变环面构成.体系的两条最简单的周期轨道S和A的周期随能量E的下降而增大，并在E趋于体系的最小值E_min=3.0时分别为反对称和对称谐振动. 关键词：     

Global analysis of periodic orbit bifurcations in coupled Morse oscillator systems: time-reversal symmetry,permutational representations and codimension-2 collisions

In this paper we study periodic orbit bifurcation sequences in a system of two coupled Morse oscillators. Time-reversal symmetry is exploited to determine periodic orbits by iteration of symmetry lines. The permutational representation of Tsuchiya and Jaffe is employed to analyze periodic orbit configurations on the symmetry lines. Local pruning rules are formulated, and a global analysis of possible bifurcation sequences of symmetric periodic orbits is made. Analysis of periodic orbit bifurcations on symmetry lines determines bifurcation sequences, together with periodic orbit periodicities and stabilities. The correlation between certain bifurcations is explained. The passage from an integrable limit to nointegrability is marked by the appearance of tangent bifurcations; our global analysis reveals the origin of these ubiquitous tangencies. For period-1 orbits, tangencies appear by a simple disconnection mechanism. For higher period orbits, a different mechanism involving 2-parameter collisions of bifurcations is found. (c) 1999 American Institute of Physics.     

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.     

Index and Stability of Symmetric Periodic Orbits in Hamiltonian Systems with Application to Figure-Eight Orbit

In this paper, using the Maslov index theory in symplectic geometry, we build up some stability criteria for symmetric periodic orbits in a Hamiltonian system, which is motivated by the recent discoveries in the n-body problem. The key ingredient is a generalized Bott-type iteration formula for periodic solution in the presence of finite group action on the orbit. For second order system, we prove, under general boundary conditions, the close formula for the relationship between the Morse index of an orbit in a Lagrangian system and the Maslov index of the fundamental solution for the corresponding orbit in its Hamiltonian system counterpart, and the boundary conditions cover the cases which appeared in the n-body problem. As an application we consider the stability problem of the celebrated figure-eight orbit due to Chenciner and Montgomery in the planar three-body problem with equal masses, and we clarify the relationship between linear stability and its variational nature on various loop spaces. The basic idea is as follows: the variational characterization of the figure-eight orbit provides information about its Morse index; based on its relation to the Maslov index, our stability criteria come into play. Partially supported by NSFC (No.10801127) and the knowledge innovation program of the Chinese Academy of Science. Partially supported by NSFC (No.s 10401025, 10571123 and 10731080) and NSFB-FBEC (No. KZ20 0610028015).     

Controlling chaos by periodic proportional pulses

It is shown that proportional pulses, X_i → kX_i, applied once every p iterations to chaotic dynamics, X_n+1 = f(X_n), may stabilize the dynamics at a periodic orbit. We give here an explicit procedure to find all such periodic points X_i, and to calculate the corresponding constants p and k.     

非扩散洛伦兹系统的周期轨道

混沌系统的奇怪吸引子是由无数条周期轨道稠密覆盖构成的,周期轨道是非线性动力系统中除不动点之外最简单的不变集,它不仅能够体现出混沌运动的所有特征,而且和系统振荡的产生与变化密切相关,因此分析复杂系统的动力学行为时获取周期轨道具有重要意义.本文系统地研究了非扩散洛伦兹系统一定拓扑长度以内的周期轨道,提出一种基于轨道的拓扑结构来建立一维符号动力学的新方法,通过变分法数值计算轨道显得很稳定.寻找轨道初始化时,两条轨道片段能够被用作基本的组成单元,基于整条轨道的结构进行拓扑分类的方式显得很有效.此外,讨论了周期轨道随着参数变化时的形变情况,为研究轨道的周期演化规律提供了新途径.本研究可为在其他类似的混沌体系中找到并且系统分类周期轨道提供一种可借鉴的方法.     

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.     

A family of poisson structures on hermitian symmetric spaces

We investigate the compatibility of symplectic Kirillov-Kostant-Souriau structure and Poisson-Lie structure on coadjoint orbits of semisimple Lie group. We prove that they are compatible for an orbit compact Lie group iff the orbit is hermitian symmetric space. We prove also the compatibility statement for non-compact hermitian symmetric space. As an example we describe a structure of symplectic leaves onCP ⁿ for this family. These leaves may be considered as a perturbation of Schubert cells. Possible applications to infinite-dimensional examples are discussed.     

Bifurcation phenomena near homoclinic systems: A two-parameter analysis

The bifurcations of periodic orbits in a class of autonomous three-variable, nonlinear-differential-equation systems possessing a homoclinic orbit associated with a saddle focus with eigenvalues ( ±i,), where ¦/¦ < 1 (Sil'nikov's condition), are studied in a two-parameter space. The perturbed homoclinic systems undergo a countable set of tangent bifurcations followed by period-doubling bifurcations leading to periodic orbits which may be attractors if ¦/¦ < 1/2. The accumulation rate of the critical parameter values at the homoclinic system is exp(-2¦/¦). A global mechanism for the onset of homoclinicity in strongly contractive flows is analyzed. Cusp bifurcations with bistability and hysteresis phenomena exist locally near the onset of homoclinicity. A countable set of these cusp bifurcations with scaling properties related to the eigenvalues±i of the stationary state are shown to occur in infinitely contractive flows. In the two-parameter space, the periodic orbit attractor domain exhibits a spiral structure globally, around the set of homoclinic systems, in which all the different periodic orbits are continuously connected.     

A novel method for extracting oscillator strength of select rare-earth ion optical transitions in nanostructured dielectric materials

A new technique to obtain the oscillator strength of select rare-earth optical transitions in nanostructured dielectric materials (nanophosphors) is presented. It is based on the experimentally observed nanophosphor lifetime dependence on the embedding medium. A constant oscillator strength and parity-allowed electric dipole transitions of the RE ion emission are assumed. The oscillator strength is obtained from the slope of the 1/τ_ij vs. n(n²+2)² plot, where τ_ij is the radiative lifetime of transition between states i and j, and n is the index of refraction of the embedding medium. The use of the technique is illustrated for the Y ₂SiO₅:Ce nanophosphor.     

Act-and-wait time-delayed feedback control of autonomous systems

Recently an act-and-wait modification of time-delayed feedback control has been proposed for the stabilization of unstable periodic orbits in nonautonomous dynamical systems (Pyragas and Pyragas, 2016 [30]). The modification implies a periodic switching of the feedback gain and makes the closed-loop system finite-dimensional. Here we extend this modification to autonomous systems. In order to keep constant the phase difference between the controlled orbit and the act-and-wait switching function an additional small-amplitude periodic perturbation is introduced. The algorithm can stabilize periodic orbits with an odd number of real unstable Floquet exponents using a simple single-input single-output constraint control.     

Calculation of Coefficients of Fractional Parentage for Large-Basis Harmonic-Oscillator Shell Model

A new procedure for large-scale calculations of the coefficients of fractional parentage (CFPs) for a single j-orbit with isospin is presented. The approach is based on a simple enumeration scheme for antisymmetric A-particle states and an efficient method for constructing the eigenvectors of an idempotent matrix. We investigate the characteristics of the introduced CFP basis and the application of this procedure to the ab initio harmonic-oscillator shell-model approach. The results of CFP calculations for the j=1/2,…,41/2 orbits are presented (the full sets of one-particle and two-particle CFPs up to the j=9/2 orbit are obtained). The new computer code for calculation of the CFPs proves to be very quick, efficient, and numerically stable and produces results possessing only small numerical uncertainties.     

Dispersion relation for electromagnetic waves in anisotropic media

Electromagnetic wave propagation in anisotropic dielectric media with two generic matrices εij and μij of permittivity and permeability is studied. In the framework of a metric-free electrodynamics approach, a compact tensorial dispersion relation is derived. The derivation does not require the corresponding matrices to be symmetric, positive definite, nor even invertible. The resulting formula is useful for a theoretical and experimental study of electromagnetic wave propagation in a wide class of linear media.     

矩形弹子球中的量子谱和经典周期轨道

利用SU(2)相干态的表示,我们构造了二维矩形弹子球中与经典周期轨道对应的波函数.经典周期轨道和量子波函数之间的关系可以通过物理图像清晰的表示出来.另外,利用周期轨道理论,我们计算了二维矩形弹子球体系的量子谱的傅立叶变换ρ(L).变换谱|ρN(L)|2对L图像中的峰可以和粒子在二维矩形腔中运动的经典轨迹的长度相比较.量子谱中的每一条峰正好对应一条经典周期轨道的长度,表明量子力学和经典力学的对应关系.     

The design and artificial realization of a controller of pulse coupling feedback

In this paper a controller of pulse coupling feedback (PCF) is designed to control chaotic systems. Control principles and the technique to select the feedback coefficients are introduced. This controller is theoretically studied with a three dimensional (3D) chaotic system. The artificial simulation results show that the chaotic system can be stabilized to different periodic orbits by using the PCF method, and the number of the periodic orbits are 2ⁿ× 3^mp (n and m are integers). Therefore, this control method is effective and practical.     

Limit Cycles Bifurcating from a k-dimensional Isochronous Center Contained in ?n with k ? n

The goal of this paper is double. First, we illustrate a method for studying the bifurcation of limit cycles from the continuum periodic orbits of a k-dimensional isochronous center contained in ℝ ⁿ with n ⩾ k, when we perturb it in a class of differential systems. The method is based in the averaging theory. Second, we consider a particular polynomial differential system in the plane having a center and a non-rational first integral. Then we study the bifurcation of limit cycles from the periodic orbits of this center when we perturb it in the class of all polynomial differential systems of a given degree. As far as we know this is one of the first examples that this study can be made for a polynomial differential system having a center and a non-rational first integral. The first and third authors are partially supported by a MCYT/FEDER grant MTM2005-06098-C01, and by a CIRIT grant number 2005SGR-00550. The second author is partially supported by a FAPESP–BRAZIL grant 10246-2. The first two authors are also supported by the joint project CAPES–MECD grant HBP2003-0017.     

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     

On the stability of the three classes of Newtonian three-body planar periodic orbits

Currently,the fifteen new periodic orbits of Newtonian three-body problem with equal mass were found by Suvakov and Dmitra sinovi[Phys Rev Lett,2013,110:114301]using the gradient descent method with double precision.In this paper,these reported orbits are checked stringently by means of a reliable numerical approach(namely the"Clean Numerical Simulation",CNS),which is based on the arbitrary-order Taylor series method and data in arbitrary-digit precision with a procedure of solution verification.It is found that seven among these fifteen orbits greatly depart from the periodic ones within a long enough interval of time,and are thus most possibly unstable at least.It is suggested to carefully check whether or not these seven unstable orbits are the so-called"computational periodicity"mentioned by Lorenz in 2006.This work also illustrates the validity and great potential of the CNS for chaotic dynamic systems.