间隙线性反馈控制混沌

提出了两种控制混沌的线性间隙反馈方法.该方法由控制相和非控制相组成,通过选取合适的反馈系数和控制相时间,可以获得各种不同的所需稳定的周期轨道.分别对一维的声光双稳系统和二维的类Henon吸引子进行计算机模拟,表明该方法可以使既定的系统按照给定的周期轨道演化,并且是大范围可控的. 关键词： 间隙线性反馈 混沌 类Henon映射 Lyapunov指数     

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.     

Statistical properties of actions of periodic orbits

We investigate statistical properties of unstable periodic orbits, especially actions for two simple linear maps (p-adic Baker map and sawtooth map). The action of periodic orbits for both maps is written in terms of symbolic dynamics. As a result, the expression of action for both maps becomes a Hamiltonian of one-dimensional spin systems with the exponential-type pair interaction. Numerical work is done for enumerating periodic orbits. It is shown that after symmetry reduction, the dyadic Baker map is close to generic systems, and the p-adic Baker map and sawtooth map with noninteger K are also close to generic systems. For the dyadic Baker map, the trace of the quantum time-evolution operator is semiclassically evaluated by employing the method of Phys. Rev. E 49, R963 (1994). Finally, using the result of this and with a mathematical tool, it is shown that, indeed, the actions of the periodic orbits for the dyadic Baker map with symmetry reduction obey the uniform distribution modulo 1 asymptotically as the period goes to infinity. (c) 2000 American Institute of Physics.     

非线性反馈控制单模激光Haken-Lorenz混沌系统

提出一种变量非线性反馈（VNF）方法控制混沌系统.介绍了该方法的控制原理以及反馈系数的选取原则,以单模激光Haken-Lorenz系统为例对非线性反馈控制方法进行了理论研究.仿真结果显示,通过恰当的选择反馈系数k,使系统的最大李雅普诺夫（Lyapunov）指数由正值转变为负值,相图中系统的轨迹由混沌吸引子转变为周期数为2n×3mp(n、m为整数)的周期轨道.通过与线性反馈控制结果对比发现,非线性反馈控制方法简便有效,控制速度快.     

Embedding dynamics for round-off errors near a periodic orbit

We study the propagation of round-off errors near the periodic orbits of a linear map conjugate to a planar rotation with rational rotation number. We embed the two-dimensional discrete phase space (a lattice) in a higher-dimensional torus, where points sharing the same round-off error are uniformly distributed within finitely many convex polyhedra. The embedding dynamics is linear and discontinuous, with algebraic integer coefficients. This representation affords efficient algorithms for classifying and computing the orbits and their exact densities, which we apply to the case of rational rotation number with denominator 7, corresponding to certain algebraic integers of degree three. We provide evidence that the hierarchical arrangement of orbits previously detected in quadratic cases [Lowenstein et al., Chaos 7, 49-66 (1997)] disappears, and that the growth of the number of orbits with the period is algebraic.(c) 2000 American Institute of Physics.     

Coordinate change eigenvalues for bimodal period doubling renormalisation

The coordinate change eigenvalues for the MacKay and van Zeijts period doubling renormalisation operator for bimodal 1 D maps are derived. They are found in numerical computations of the spectrum at all the periodic orbits of renormalisation of period up to five.     

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.     

Semiclassical Poincare map for integrable systems

The semiclassical Poincare map is applied to integrable systems and in particular to the rectangular billiard. The zeroes of the functional determinant are shown to give EBK quantization. The transfer operator is explicitly unitary and finite, resulting in a finite expansion of the Euler product over periodic orbits.     

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.     

Classical limit of the quantized hyperbolic toral automorphisms

The canonical quantization of any hyperbolic symplectomorphismA of the 2-torus yields a periodic unitary operator on aN-dimenional Hilbert space,N=1/h. We prove that this quantum system becomes ergodic and mixing at the classical limit (N,N prime) which can be interchanged with the time-average limit. The recovery of the stochastic behaviour out of a periodic one is based on the same mechanism under which the uniform distribution of the classical periodic orbits reproduces the Lebesgue measure: the Wigner functions of the eigenstates, supported on the classical periodic orbits, are indeed proved to become uniformly speread in phase space.     

No Quantum Ergodicity for Star Graphs

We investigate statistical properties of the eigenfunctions of the Schrödinger operator on families of star graphs with incommensurate bond lengths. We show that these eigenfunctions are not quantum ergodic in the limit as the number of bonds tends to infinity by finding an observable for which the quantum matrix elements do not converge to the classical average. We further show that for a given fixed graph there are subsequences of eigenfunctions which localise on pairs of bonds. We describe how to construct such subsequences explicitly. These structures are analogous to scars on short unstable periodic orbits.     

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.     

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.     

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.     

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.     

Gaps in the Spectrum of the Maxwell Operator with Periodic Coefficients

The periodic Maxwell operator is considered. Piecewise constant coefficients are constructed in such a way that the spectrum of operator has the gaps.     

Analytical study of chaos and applications

We summarize various cases where chaotic orbits can be described analytically. First we consider the case of a magnetic bottle where we have non-resonant and resonant ordered and chaotic orbits. In the sequence we consider the hyperbolic Hénon map, where chaos appears mainly around the origin, which is an unstable periodic orbit. In this case the chaotic orbits around the origin are represented by analytic series (Moser series). We find the domain of convergence of these Moser series and of similar series around other unstable periodic orbits. The asymptotic manifolds from the various unstable periodic orbits intersect at homoclinic and heteroclinic orbits that are given analytically. Then we consider some Hamiltonian systems and we find their homoclinic orbits by using a new method of analytic prolongation. An application of astronomical interest is the domain of convergence of the analytical series that determine the spiral structure of barred-spiral galaxies.     

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.     

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.