Some sufficient conditions for stabilizing periodic orbits without the odd-number property by delayed feedback control in continuous-time systems

One of the important topics in the study of delayed feedback control (DFC) for chaotic systems is stability analysis. In the present Letter, we give some sufficient conditions for stabilizing periodic orbits by the DFC without the odd-number property in continuous-time systems. Our results naturally connect the stability condition for inversely unstable orbits and the odd-number limitation.     

Transition to turbulence for doubly periodic flows

We construct a typical model for the Poincaré map of doubly periodic flows, which presents numerically a transition to chaotic behavior. After the frequency locking phenomenon, we observe two types of transitions to turbulence. The first one involves successive subharmonic instabilities of a periodic solution. The second one occurs after the disappearance of a periodic solution and can be either intermittent or discontinuous with hysteresis.     

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.     

Wigner function for discrete phase space: Exorcising ghost images

We construct, using simple geometrical arguments, a Wigner function defined on a discrete phase space of arbitrary integer Hilbert-space dimension that is free of redundancies. “Ghost images” plaguing other Wigner functions for discrete phase spaces are thus revealed as artifacts. It allows to devise a corresponding phase-space propagator in an unambiguous manner. We apply our definitions to eigenstates and propagator of the quantum baker map. Scars on unstable periodic points of the corresponding classical map become visible with unprecedented resolution.     

The periodic orbits of an area preserving twist map

We study the oscillation properties of periodic orbits of an area preserving twist map. The results are inspired by the similarity between the gradient flow of the associated action-function, and a scalar parabolic PDE in one space dimension. The Conley-Zehnder Morse theory is used to construct orbits with prescribed oscillatory behavior.     

Chaos in discrete fractional difference equations

Recently, the discrete fractional calculus (DFC) is receiving attention due to its potential applications in the mathematical modelling of real-world phenomena with memory effects. In the present paper, the chaotic behaviour of fractional difference equations for the tent map, Gauss map and 2x(mod 1) map are studied numerically. We analyse the chaotic behaviour of these fractional difference equations and compare them with their integer counterparts. It is observed that fractional difference equations for the Gauss and tent maps are more stable compared to their integer-order version.     

Exact treatment of mode locking for a piecewise linear map

A piecewise linear map with one discontinuity is studied by analytic means in the two-dimensional parameter space. When the slope of the map is less than unity, periodic orbits are present, and we give the precise symbolic dynamic classification of these. The localization of the periodic domains in parameter space is given by closed expressions. The winding number forms a devil's terrace, a two-dimensional function whose cross sections are complete devils's staircases. In such a cross section the complementary set to the periodic intervals is a Cantor set with dimensionD=0.     

Characteristic classes of star products on Marsden–Weinstein reduced symplectic manifolds

In this note we consider a quantum reduction scheme in deformation quantization on symplectic manifolds proposed by Bordemann, Herbig and Waldmann based on BRST cohomology. We explicitly construct the induced map on equivalence classes of star products which will turn out to be an analogue to the Kirwan map in the Cartan model of equivariant cohomology. As a byproduct, we shall see that every star product on a (suitable) reduced manifold is equivalent to a reduced star product.     

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     

Remarks on KdV-type flows on star-shaped curves

We study the relation between the centro-affine geometry of star-shaped planar curves and the projective geometry of parametrized maps into RP¹. We show that projectivization induces a map between differential invariants and a bi-Poisson map between Hamiltonian structures. We also show that a Hamiltonian evolution equation for closed star-shaped planar curves, discovered by Pinkall, has the Schwarzian KdV equation as its projectivization. (For both flows, the curvature evolves by the KdV equation.) Using algebro-geometric methods and the relation of group-based moving frames to AKNS-type representations, we construct examples of closed solutions of Pinkall’s flow associated with periodic finite-gap KdV potentials.     

Periodic,incommensurate and chaotic states in a continuum statistical mechanics model

We study the thermodynamic behavior of a continuum system with competing periodicities. We show that in addition to commensurate and incommensurate phases, there exist configurations which are chaotic in nature and exhibit no long-range order. These phases are metastable and characterized by an order parameter with a continuous spectrum. By transforming the problem of determining the ground states of the system into a classical mechanics problem, we construct a two-dimensional area-preserving map which can be used to study the qualitative nature of the orbits. Our results might be of relevance to adsorbed monolayers on periodic substrates.     

Noise-induced asymptotic periodicity in a piecewise linear map

We examine asymptotically periodic density evolution in one-dimensional maps perturbed by noise, associating the macroscopic state of these dynamical systems with a phase space density. For asymptotically periodic systems density evolution becomes periodic in time, as do some macroscopic properties calculated from them. The general formalism of asymptotic periodicity is examined and used to calculate time correlations along trajectories of these maps as well as their limiting conditional entropy. The time correlation is shown to naturally decouple into periodic and stochastic components. Finally, asymptotic periodicity is studied in a noise-perturbed piecewise linear map, focusing on how the variation of noise amplitude can cause a transition from asymptotic periodicity to asymptotic stability in the density evolution of this system.     

Periodic Sturm-Liouville problems related to two Riccati equations of constant coefficients

We consider two closely related Riccati equations of constant parameters whose particular solutions are used to construct the corresponding class of supersymmetrically coupled second-order differential equations. We solve analytically these parametric periodic problems along the whole real axis. Next, the analytically solved model is used as a case study for a powerful numerical approach that is employed here for the first time in the investigation of the energy band structure of periodic not necessarily regular potentials. The approach is based on the well-known self-matching procedure of James (1949) [9] and implements the spectral parameter power series solutions introduced by Kravchenko (2008) [10]. We obtain additionally an efficient series representation of the Hill discriminant based on Kravchenko’s series.     

Chains of Infinite Order,Chains with Memory of Variable Length,and Maps of the Interval

We show how to construct a topological Markov map of the interval whose invariant probability measure is the stationary law of a given stochastic chain of infinite order. In particular we characterize the maps corresponding to stochastic chains with memory of variable length. The problem treated here is the converse of the classical construction of the Gibbs formalism for Markov expanding maps of the interval.     

Solving the Difference Initial-Boundary Value Problems by the Operator Exponential Method

Abstract

We suggest a modification of the operator exponential method for the numerical solving the difference linear initial boundary value problems. The scheme is based on the representation of the difference operator for given boundary conditions as the perturbation of the same operator for periodic ones. We analyze the error, stability and efficiency of the scheme for a model example of the one-dimensional operator of second difference.     

Stability of temporal solitons in uniform and “managed” quadratic nonlinear media with opposite group-velocity dispersions at fundamental and second harmonics

The problem of the stability of solitons in second-harmonic-generating media with normal group-velocity dispersion (GVD) in the second-harmonic (SH) field, which is generic to available χ⁽²⁾ materials, is revisited. Using an iterative numerical scheme to construct stationary soliton solutions, and direct simulations to test their stability, we identify a full soliton-stability range in the space of the system’s parameters, including the coefficient of the group-velocity-mismatch (GVM). The soliton stability is limited by an abrupt onset of growth of tails in the SH component, the relevant stability region being defined as that in which the energy loss to the tail generation is negligible under experimentally relevant conditions. We demonstrate that the stability domain can be readily expanded with the help of two “management” techniques (spatially periodic compensation of destabilizing effects) - the dispersion management (DM) and GVM management. In comparison with their counterparts in optical fibers, DM solitons in the χ⁽²⁾ medium feature very weak intrinsic oscillations.     

Signatures of quantum stability in a classically chaotic system

We experimentally and numerically investigate the quantum accelerator mode dynamics of an atom optical realization of the quantum delta-kicked accelerator, whose classical dynamics are chaotic. Using a Ramsey-type experiment, we observe interference, demonstrating that quantum accelerator modes are formed coherently. We construct a link between the behavior of the evolution's fidelity and the phase space structure of a recently proposed pseudoclassical map, and thus account for the observed interference visibilities.     

Computing Invariant Densities and Metric Entropy

We present a method for accurately computing the metric entropy (or, equivalently, the Lyapunov exponent) of the absolutely continuous invariant measure μ for a piecewise analytic expanding Markov map T of the interval. We construct atomic signed measures μ _M supported on periodic orbits up to period M, and prove that super-exponentially fast. We illustrate our method with several examples. Received: 25 July 1999 / Accepted: 7 January 2000     

Cyclic Cocycles on Twisted Convolution Algebras

We give a construction of cyclic cocycles on convolution algebras twisted by gerbes over discrete translation groupoids. For proper étale groupoids, Tu and Xu (Adv Math 207(2):455–483, 2006) provide a map between the periodic cyclic cohomology of a gerbe-twisted convolution algebra and twisted cohomology groups which is similar to the construction of Mathai and Stevenson (Adv Math 200(2):303–335, 2006). When the groupoid is not proper, we cannot construct an invariant connection on the gerbe; therefore to study this algebra, we instead develop simplicial techniques to construct a simplicial curvature 3-form representing the class of the gerbe. Then by using a JLO formula we define a morphism from a simplicial complex twisted by this simplicial curvature 3-form to the mixed bicomplex computing the periodic cyclic cohomology of the twisted convolution algebras.     

Hash function based on the generalized Henon map

A new Hash function based on the generalized Henon map is proposed. We have obtained a binary sequence with excellent pseudo-random characteristics through improving the sequence generated by the generalized Henon map, and use it to construct Hash function. First we divide the message into groups, and then carry out the Xor operation between the ASCII value of each group and the binary sequence, the result can be used as the initial values of the next loop. Repeat the procedure until all the groups have been processed, and the final binary sequence is the Hash value. In the scheme, the initial values of the generalized Henon map are used as the secret key and the messages are mapped to Hash values with a designated length. Simulation results show that the proposed scheme has strong diffusion and confusion capability, good collision resistance, large key space, extreme sensitivity to message and secret key, and it is easy to be realized and extended.