The growth rate for the number of singular and periodic orbits for a polygonal billiard

For any simply connected polygon in the plane, the number of billiard orbits which begin and end at a vertex grows subexponentially with respect to the length or to the number of reflections. This implies that the numbers of isolated periodic orbits and of families of parallel periodic orbits do grow subexponentially. The main technical device is a calculation showing that the topological entropy of the Poincaré map for the billiard flow is equal to zero.Supported in part by NSF Grant #DMS-8414400     

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.     

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.     

Integrability and ergodicity of classical billiards in a magnetic field

We consider classical billiards in plane, connected, but not necessarily bounded domains. The charged billiard ball is immersed in a homogeneous, stationary magnetic field perpendicular to the plane. The part of dynamics which is not trivially integrable can be described by a bouncing map. We compute a general expression for the Jacobian matrix of this map, which allows us to determine stability and bifurcation values of specific periodic orbits. In some cases, the bouncing map is a twist map and admits a generating function. We give a general form for this function which is useful to do perturbative calculations and to classify periodic orbits. We prove that billiards in convex domains with sufficiently smooth boundaries possess invariant tori corresponding to skipping trajectories. Moreover, in strong field we construct adiabatic invariants over exponentially large times. To some extent, these results remain true for a class of nonconvex billiards. On the other hand, we present evidence that the billiard in a square is ergodic for some large enough values of the magnetic field. A numerical study reveals that the scattering on two circles is essentially chaotic.     

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.     

Symbolic Analysis of the Hénon Map at a = 1.4 and b = 0.3

The symbolic dyhamics of the Lozi map is applied to the Hgnon map at a = 1.4 and b = 0.3. The fundamental forbidden zone, inside which points correspond to forbidden symbolic sequences, is constructed from a numerically determined bartition line. All the allowed periodic orbits up to period 7 are determined, and chaotic orbits are derived.     

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.     

A numerical method for determining bifurcation curves of mappings

A shooting type iterative method for determining bifurcation curves of mappings is developed in this paper. By using this method we are able to calculate bifurcation curves for orbits of periods ≤ 5 of a Hénon-like map in all details. We find an interesting phenomenon that two or more stable periodic orbits of the same or different periods may coexist for some combinations of parameter values. Two stable p5 orbits and their attracting regions are shown for (μ, b) = (1.536, 0.153).     

Controlling Spatiotemporal Chaos in Coupled Map Lattices to Periodic Orbits

Two methods are presented for controlling spatiotemporal chaotic motion in coupled map lattices to a kind of periodic orbit where the dynamicM variables of all lattice sites are equM and act periodically as time evolves. Stability analysis of the periodic orbits is presented. We prove that especially the second controlling method can stabilize all the periodic orbits we concern. Basin of attraction and noise problem are discussed.     

Controlling Spatiotemporal Chaos in Coupled Map Lattices to Periodic Orbits

Two methods are presented for controlling spatiotemporal chaotic motion in coupled map lattices to a kind of periodic orbit where the dynamical variables of all lattice sites are equal and act periodically as time evolves. Stability analysis of the periodic orbits is presented. We prove that especially the second controlling method can stabilize all the periodic orbits we concern. Basin of attraction and noise problem are discussed.     

Efficient topological chaos embedded in the blinking vortex system

We consider the particle mixing in the plane by two vortex points appearing one after the other, called the blinking vortex system. Mathematical and numerical studies of the system reveal that the chaotic particle mixing, i.e., the chaotic advection, is observed due to the homoclinic chaos, but the mixing region is restricted locally in the neighborhood of the vortex points. The present article shows that it is possible to realize a global and efficient chaotic advection in the blinking vortex system with the help of the Thurston-Nielsen theory, which classifies periodic orbits for homeomorphisms in the plane into three types: periodic, reducible, and pseudo-Anosov (pA). It is mathematically shown that periodic orbits of pA type generate a complicated dynamics, which is called topological chaos. We show that the combination of the local chaotic mixing due to the topological chaos and the dipole-like return orbits realize an efficient and global particle mixing in the blinking vortex system.     

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.     

动力学变换法探测连续系统不稳定周期轨道

本文利用动力学变换方法和庞加莱截面方法对两种连续混沌动力学系统进行不稳定周期轨道探测研究, 并对Lorenz系统进行了替代数据法检验.结果表明:基于庞加莱截面的动力学变换改进算法 可有效探测连续混沌动力学系统中的不稳定周期轨道.     

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.     

Clustering of Periodic Orbits and Ensembles of Truncated Unitary Matrices

In present article we consider a combinatorial problem of counting and classification of periodic orbits in dynamical systems on an example of the baker’s map. Periodic orbits of a chaotic system can be organized into a set of clusters, where orbits from a given cluster traverse approximately the same points of the phase space but in a different time-order. We show that counting of cluster sizes in the baker’s map can be turned into a spectral problem for matrices from truncated unitary ensemble (TrUE). We formulate a conjecture of universality of the spectral edge in the eigenvalues distribution of TrUE and utilize it to derive asymptotics of the second moment of cluster distribution in the regime when both the orbit lengths and the parameter controlling closeness of the orbit actions tend to infinity. The result obtained allows to estimate the size of average cluster for various numbers of encounters in periodic orbit.     

Universal behaviour in families of area-preserving maps

We have investigated numerically the behaviour, as a perturbation parameter is varied, of periodic orbits of some reversible area-preserving maps of the plane. Typically, an initially stable periodic orbit loses its stability at some parameter value and gives birth to a stable orbit of twice the period. An infinite sequence of such bifurcations is accomplished in a finite parameter range. This period-doubling sequence has a universal limiting behaviour: the intervals in parameter between successive bifurcations tend to a geometric progression with a ratio of $\text{1}\text{δ}\text{=}\text{1}\text{8.721097200}$…, and when examined in the proper coordinates, the pattern of periodic points reproduces itself, asymptotically, from one bifurcation to the next when the scale is expanded by α = ?4.018076704… in one direction, and by β = 16.363896879… in another. Indeed, the whole map, including its dependence on the parameter, reproduces itself on squaring and rescaling by the three factors α, β and δ above. In the limit we obtain a universal one-parameter, area-preserving map of the plane. The period-doubling sequence is found to be connected with the destruction of closed invariant curves, leading to irregular motion almost everywhere in a neighbourhood.     

Synchronized family dynamics in globally coupled maps

The dynamics of a globally coupled, logistic map lattice is explored over a parameter plane consisting of the coupling strength, varepsilon, and the map parameter, a. By considering simple periodic orbits of relatively small lattices, and then an extensive set of initial-value calculations, the phenomenology of solutions over the parameter plane is broadly classified. The lattice possesses many stable solutions, except for sufficiently large coupling strengths, where the lattice elements always synchronize, and for small map parameter, where only simple fixed points are found. For smaller varepsilon and larger a, there is a portion of the parameter plane in which chaotic, asynchronous lattices are found. Over much of the parameter plane, lattices converge to states in which the maps are partitioned into a number of synchronized families. The dynamics and stability of two-family states (solutions partitioned into two families) are explored in detail. (c) 1999 American Institute of Physics.     

Dual polygonal billiards and necklace dynamics

We study the orbits of the dual billiard map about a polygonal table using the technique of necklace dynamics. Our main result is that for a certain class of tables, called the quasi-rational polygons, the dual billiard orbits are bounded. This implies that for the subset of rational tables (i.e. polygons with rational vertices) the dual billiard orbits are periodic.Partially supported by NSF Grant DMS 88-02643     

二维均匀耦合映象格子中的时空周期图案

目的——构造二维均匀耦合映象格子中的时空周期图案;方法——通过一维耦合映象格子模型的相空间中已知低空间周期轨道,直接构造二维均匀耦合映象格子模型中一系列空间周期轨道,而不必求解其模型方程,并对构造轨道的稳定性进行分析;结果——L²×L²雅可比矩阵可化简为几个2×2矩阵组成的对角矩阵;结论——所构造轨道的稳定性不可能比原来轨道的稳定性高. 关键词： 耦合映象格子 时空周期图案 雅可比矩阵     

On the stability of delayed feedback controllers for discrete time systems

We consider the stability of delayed feedback control (DFC) scheme for multi-dimensional discrete time systems. We first construct a map whose fixed points correspond to the periodic orbits of the uncontrolled system. Then the stability of the DFC is analyzed as the stability of the corresponding equilibrium point of the constructed map. For each periodic orbit, we construct a characteristic polynomial whose Schur stability corresponds to the stability of DFC scheme.