Globally enumerating unstable periodic orbits for observed data using symbolic dynamics

The unstable periodic orbits of a chaotic system provide an important skeleton of the dynamics in a chaotic system, but they can be difficult to find from an observed time series. We present a global method for finding periodic orbits based on their symbolic dynamics, which is made possible by several recent methods to find good partitions for symbolic dynamics from observed time series. The symbolic dynamics are approximated by a Markov chain estimated from the sequence using information-theoretical concepts. The chain has a probabilistic graph representation, and the cycles of the graph may be exhaustively enumerated with a classical deterministic algorithm, providing a global, comprehensive list of symbolic names for its periodic orbits. Once the symbolic codes of the periodic orbits are found, the partition is used to localize the orbits back in the original state space. Using the periodic orbits found, we can estimate several quantities of the attractor such as the Lyapunov exponent and topological entropy.     

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.     

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.     

On the periodic orbits of perturbed Hooke Hamiltonian systems with three degrees of freedom

We study periodic orbits of Hamiltonian differential systems with three degrees of freedom using the averaging theory. We have chosen the classical integrable Hamiltonian system with the Hooke potential and we study periodic orbits which bifurcate from the periodic orbits of the integrable system perturbed with a non-autonomous potential.     

Stability transformation: a tool to solve nonlinear problems

We present an analysis of the properties as well as the diverse applications and extensions of the method of stabilisation transformation. This method was originally invented to detect unstable periodic orbits in chaotic dynamical systems. Its working principle is to change the stability characteristics of the periodic orbits by applying an appropriate global transformation of the dynamical system. The theoretical foundations and the associated algorithms for the numerical implementation of the method are discussed. This includes a geometrical classification of the periodic orbits according to their behaviour when the stabilisation transformations are applied. Several refinements concerning the implementation of the method in order to increase the numerical efficiency allow the detection of complete sets of unstable periodic orbits in a large class of dynamical systems. The selective detection of unstable periodic orbits according to certain stability properties and the extension of the method to time series are discussed. Unstable periodic orbits in continuous-time dynamical systems are detected via introduction of appropriate Poincaré surfaces of section. Applications are given for a number of examples including the classical Hamiltonian systems of the hydrogen and helium atom, respectively, in electromagnetic fields. The universal potential of the method is demonstrated by extensions to several other nonlinear problems that can be traced back to the detection of fixed points. Examples include the integration of nonlinear partial differential equations and the numerical determination of Markov-partitions of one-parametric maps.     

混沌体系中寻找周期轨迹的方法

一个不可积混沌体系，由于扰动而遭到破坏时，存活的周期轨迹体现了体系的本质特征，是 体系的运动骨架.在一定程度上， 可以由周期轨迹来量子化不可积体系，这充分说明了 周期轨迹的重要性.而寻找周期轨迹，也就成为研究混沌体系动力学特性以及对混沌体系进 行量子化的关键问题.结合具体实例，给出了3种常用的寻找周期轨迹方法，并详细探讨了各 种方法的优缺点和适用范围. 关键词： 周期轨迹 数值方法 混沌     

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.     

光学二次谐波浑沌控制

用变量延时反馈控制法对光学二次谐波系统的浑沌进行了有效的控制.通过对系统的最大李雅普诺夫指数分析,给出了确定可控参数区的方法.证明适当的延时量和反馈强度可以使浑沌得到稳定的控制,被控制系统的轨道是初始系统浑沌吸引子中的不稳定周期轨道.     

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.     

量子能谱中的长程关联

从可积系统求迹公式出发,运用Einstein-Brillouin-Keller(EBK)量子化条件,导出了二维无关联振子系统周期轨道作用量量子化条件,由此发现了量子能级与周期轨道之间的对应关系.这种对应关系表明,如果两条能级对应的周期轨道的拓扑相同,这两条能级对回归函数的贡献相干.回归谱中的一个峰是量子能谱中一组与具有相同拓扑的周期轨道相对应的能级之间相干的结果,这一组能级间存在着长程关联.     

On a simple recursive control algorithm automated and applied to an electrochemical experiment

We review a simple recursive proportional feedback (RPF) control strategy for stabilizing unstable periodic orbits found in chaotic attractors. The method is generally applicable to high-dimensional systems and stabilizes periodic orbits even if they are completely unstable, i.e., have no stable manifolds. The goal of the control scheme is the fixed point itself rather than a stable manifold and the controlled system reaches the fixed point in d+1 steps, where d is the dimension of the state space of the Poincare map. We provide a geometrical interpretation of the control method based on an extended phase space. Controllability conditions or special symmetries that limit the possibility of using a single control parameter to control multiply unstable periodic orbits are discussed. An automated adaptive learning algorithm is described for the application of the control method to an experimental system with no previous knowledge about its dynamics. The automated control system is used to stabilize a period-one orbit in an experimental system involving electrodissolution of copper. (c) 1997 American Institute of Physics.     

Chaotic Solutions of a Typical Nonlinear Oscillator in a Double Potential Trap

We have obtained a general unstable chaotic solution of a typical nonlinear oscillator in a double potential trap with weak periodic perturbations by using the direct perturbation method. Theoretical analysis reveals that the stable periodic orbits are embedded in the Melnikov chaotic attractors. The corresponding chaotic region and orbits in parameter space are described by numerical simulations.     

Periodic Orbits Around Kerr Sen Black Holes

We investigate periodic orbits and zoom-whirl behaviors around a Kerr Sen black hole with a rational number q in terms of three integers(z,w,v),from which one can immediately read off the number of leaves(or zooms),the ordering of the leaves,and the number of whirls.The characteristic of zoom-whirl periodic orbits is the precession of multi-leaf orbits in the strong-field regime.This feature is analogous to the counterpart in the Kerr space-time.Finally,we analyze the impact of the charge parameter b on the zoom-whirl periodic orbits.Compared to the periodic orbits around the Kerr black hole,it is found that typically lower energies are required for the same orbits in the Kerr Sen black hole.     

The dynamical feature of transition of a Hamiltonian system to a dissipative system

The mechanism of generation and annihilation of attractors during transition from a Hamiltonian system to a dissipative system is studied numerically using the dissipative standard map. The transient process related to the formation of attracting basins of periodic attractors is studied by discussing the evolution of the KAM tori of the standard map. The result shows that as damping increases, attractors are mainly generated from elliptic orbits of the Hamiltonian system and annihilated by colliding with unstable periodic orbits originating from the corresponding hyperbolic orbits of the Hamiltonian system. The transient process also exhibits the general feature of bifurcation.     

Application of the diffraction trace formula to the three-disk scattering system

The diffraction trace formula derived previously and the spectral determinant are tested on the open three-disk scattering system. The system contains a generic and exponentially growing number of diffraction periodic orbits. In spite of this it is shown that even the scattering resonances with large imaginary part can be reproduced semiclassically. The nontrivial interplay of the diffraction periodic orbits with the usual geometrical orbits produces the fine structure of the complicated spectrum of scattering resonances, which are beyond the resolution of the conventional periodic orbit theory.     

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.     

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.     

Effects of Colored Noise on Periodic Orbits in a One-Dimensional Map

Noise can induce inverse period-doubling transition and chaos. The effects of the colored noise on periodic orbits, of the different periodic sequences in the logistic map, are investigated. It is found that the dynamical behaviors of the orbits, induced by an exponentially correlated colored noise, are different in the mergence of transition, and the effects of the noise intensity ontheir dynamical behaviors are different from the effects of the correlation time of noise. Remarkably, the noise can induce new periodic orbits, namely, two new orbits emerge in the period-four sequence at the bifurcation parameter value μ=3.5, four new orbits in the period-eight sequence at μ=3.55, and three new orbits in the period-six sequence at μ= 3.846, respectively.Moreover, the dynamical behaviors of the new orbits clearly show theresonance like response to the colored noise.     

Unstable periodic orbits and templates of the Rossler system: Toward a systematic topological characterization

The Rossler system has been exhaustively studied for parameter values (a in [0.33,0.557],b=2,c=4). Periodic orbits have been systematically extracted from Poincare maps and the following problems have been addressed: (i) all low order periodic orbits are extracted, (ii) encoding of periodic orbits by symbolic dynamics (from 2 letters up to 11 letters) is achieved, (iii) some rules of growth and of pruning of the periodic orbits population are obtained, and (iv) the templates of the attractors are elaborated to characterize the attractors topology. (c) 1995 American Institute of Physics.     

GLOBAL CHAOS CONTROL OF NON-AUTONOMOUS SYSTEMS

This study describes a global approach of controlling chaos to reduce tedious waiting time caused by using conventional local controllers. With Euler's method, a non-autonomous system is approximated by a non-linear difference system and then an approximate global Poincaré map function is derived from the difference system by iterating one or more periods of a periodic excitation. Based on the map function, unstable periodic orbits embedded in a chaotic motion can be detected and a global controller for a targeted unstable periodic orbit is designed. The global controller makes all the unstable periodic orbits vanish except a targeted periodic orbit. Furthermore, a Lyapunov's direct method is applied to confirm that the global controller can asymptotically stabilize the unique periodic orbit. For practical applications, system models are usually unknown. To obtain a mathematical model, non-linear system identification based on the harmonic balance principle is applied to an unknown chaotic system of a noisy environment. Simulation results demonstrate that the global controller successfully regularizes a chaotic motion even if the chaotic trajectory is far from the targeted periodic orbit.