Global attractors and invariant measures for non-invertible planar piecewise isometric maps

The authors investigate dynamical behaviors of discrete systems defined by iterating non-invertible planar piecewise isometries, which are piecewisely defined maps that preserve Euclidean distance. After discussing subtleties for these kind of dynamical systems, they have characterized global attractors via invariant measures and via positive continuous functions on phase space. The main result of this Letter is that a compact set A is the global attractor for a piecewise isometry if and only if the Lebesgue measure restricted to A is invariant, while it is not invariant restricted to any measurable set B which contains A and whose Lebesgue measure is strictly larger than that of A.     

Scaling laws for invariant measures on hyperbolic and nonhyperbolic atractors

The analysis of dynamical systems in terms of spectra of singularities is extended to higher dimensions and to nonhyperbolic systems. Prominent roles in our approach are played by the generalized partial dimensions of the invariant measure and by the distribution of effective Liapunov exponents. For hyperbolic attractors, the latter determines the metric entropies and provides one constraint on the partial dimensions. For nonhyperbolic attractors, there are important modifications. We discuss them for the examples of the logistic and Hénon map. We show, in particular, that the generalized dimensions have singularities with noncontinuous derivative, similar to first-order phase transitions in statistical mechanics.     

不连续不可逆二维映象的动力学特性

总结两个保守映象不可逆地分段连续链接(称为类耗散系统)以及一个保守映象与一个耗散映象不可逆地分段连续链接(称为半耗散系统)情况下得到的五项共同动力学特征：不连续边界象集构成的随机网成为唯一的混沌轨道；由于某些相点具有两个逆象而导致的相空间塌缩(类耗散)；由于系统的不连续不可逆性质而出现的胖分形禁区网；在具有吸引子共存时占据不连续边界象集随机网和胖分形禁区网区域的点滴状吸引域以及由此导致的吸引子不可预言性；即使在传统强耗散存在的情况下点滴状吸引域仍由类耗散机制主宰.以一个累积-触发电路为例，说明这五项系统动 关键词： 随机网 禁区网 点滴状吸引域     

Phase order in one-dimensional piecewise linear discontinuous map

The phase order in a one-dimensional(1 D) piecewise linear discontinuous map is investigated. The striking feature is that the phase order may be ordered or disordered in multi-band chaotic regimes, in contrast to the ordered phase in continuous systems. We carried out an analysis to illuminate the underlying mechanism for the emergence of the disordered phase in multi-band chaotic regimes, and proved that the phase order is sensitive to the density distribution of the trajectories of the attractors. The scaling behavior of the net direction phase at a transition point is observed. The analytical proof of this scaling relation is obtained. Both the numerical and analytical results show that the exponent is 1, which is controlled by the feature of the map independent on whether the system is continuous or discontinuous. It extends the universality of the scaling behavior to systems with discontinuity. The result in this work is important to understanding the property of chaotic motion in discontinuous systems.     

Bifurcation phenomena in two-dimensional piecewise smooth discontinuous maps

In recent years the theory of border collision bifurcations has been developed for piecewise smooth maps that are continuous across the border and has been successfully applied to explain nonsmooth bifurcation phenomena in physical systems. However, there exist a large number of switching dynamical systems that have been found to yield two-dimensional piecewise smooth maps that are discontinuous across the border. In this paper we present a systematic approach to the problem of analyzing the bifurcation phenomena in two-dimensional discontinuous maps, based on a piecewise linear approximation in the neighborhood of the border. We first motivate the analysis by considering the bifurcations occurring in a familiar physical system-the static VAR compensator used in electrical power systems-and then proceed to formulate the theory needed to explain the bifurcation behavior of such systems. We then integrate the observed bifurcation phenomenology of the compensator with the theory developed in this paper. This theory may be applied similarly to other systems that yield two-dimensional discontinuous maps.     

Effect of noise on the relaxation to an invariant probability measure of nonhyperbolic chaotic attractors

We study the influence of external noise on the relaxation to an invariant probability measure for two types of nonhyperbolic chaotic attractors, a spiral (or coherent) and a noncoherent one. We find that for the coherent attractor the rate of mixing changes under the influence of noise, although the largest Lyapunov exponent remains almost unchanged. A mechanism of the noise influence on mixing is presented which is associated with the dynamics of the instantaneous phase of chaotic trajectories. This also explains why the noncoherent regime is robust against the presence of external noise.     

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.     

Small-scale instabilities in dynamical systems with sliding

We demonstrate with a minimal example that in Filippov systems (dynamical systems governed by discontinuous but piecewise smooth vector fields) stable periodic motion with sliding is not robust with respect to stable singular perturbations. We consider a simple dynamical system that we assume to be a quasi-static approximation of a higher-dimensional system containing a fast stable subsystem. We tune a system parameter such that a stable periodic orbit of the simple system touches the discontinuity surface: this is the so-called grazing-sliding bifurcation. The periodic orbit remains stable, and its local return map becomes piecewise linear. However, when we take into account the fast dynamics the local return map of the periodic orbit changes qualitatively, giving rise to, for example, period-adding cascades or small-scale chaos.     

Multifractal Analysis of Weak Gibbs Measures for Intermittent Systems

 In this paper, we establish a multifractal formalism of weak Gibbs measures associated to potentials of weak bounded variation for certain nonhyperbolic systems. We apply our results to Manneville-Pomeau type maps and a piecewise conformal two-dimensional countable Markov map with indifferent periodic points which is related to a complex continued fraction. Received: 6 September 2001 / Accepted: 21 May 2002 Published online: 12 August 2002     

Stability properties of nonhyperbolic chaotic attractors with respect to noise

We study local and global stability of nonhyperbolic chaotic attractors contaminated by noise. The former is given by the maximum distance of a noisy trajectory from the noisefree attractor, while the latter is provided by the minimal escape energy necessary to leave the basin of attraction, calculated with the Hamiltonian theory of large fluctuations. We establish the important and counterintuitive result that both concepts may be opposed to each other. Even when one attractor is globally more stable than another one, it can be locally less stable. Our results are exemplified with the Holmes map, for two different sets of parameter, and with a juxtaposition of the Holmes and the Ikeda maps. Finally, the experimental relevance of these findings is pointed out.     

Generalized Physical and SRB Measures for Hyperbolic Diffeomorphisms

In this paper we introduce the notion of generalized physical and SRB measures. These measures naturally generalize classical physical and SRB measures to measures which are supported on invariant sets that are not necessarily attractors. We then perform a detailed case study of these measures for hyperbolic Hènon maps. For this class of systems we are able to develop a complete theory about the existence, uniqueness, finiteness, and properties of these natural measures. Moreover, we derive a classification for the existence of a measure of full dimension. We also consider general hyperbolic surface diffeomorphisms and discuss possible extensions of, as well as the differences to, the results for Hènon maps. Finally, we study the regular dependence of the dimension of the generalized physical/SRB measure on the diffeomorphism. For the proofs we apply various techniques from smooth ergodic theory including the thermodynamic formalism. 2000 Mathematics Subject Classification. Primary: 37C45, 37D20, 37D35, Secondary: 37A35, 37E30     

A novel scheme for image encryption based on 2D piecewise chaotic maps

In this paper, a hierarchy of two-dimensional piecewise nonlinear chaotic maps with an invariant measure is introduced. These maps have interesting features such as invariant measure, ergodicity and the possibility of K-S entropy calculation. Then by using significant properties of these chaotic maps such as ergodicity, sensitivity to initial condition and control parameter, one-way computation and random like behavior, we present a new scheme for image encryption. Based on all analysis and experimental results, it can be concluded that, this scheme is efficient, practicable and reliable, with high potential to be adopted for network security and secure communications. Although the two-dimensional piecewise nonlinear chaotic maps presented in this paper aims at image encryption, it is not just limited to this area and can be widely applied in other information security fields.     

Character of the map for switched dynamical systems for observations on the switching manifold

It has been proposed to obtain the discrete-time models of switching dynamical systems by observing the states at the switching instants. Apart from the lowering of dimension, such switching maps or impact maps offer advantage in modeling systems that exhibit chattering. In this Letter we derive the nature of the switching map for the special case of grazing orbits. We show that the map is discontinuous in the neighborhood of a grazing orbit, and that it has a square root slope singularity on one side of the discontinuity. We illustrate the above by obtaining the switching maps for two example systems: the Colpitt's oscillator in the electrical domain and the soft impact oscillator in the mechanical domain.     

Saddle-node bifurcation of invariant cones in 3D piecewise linear systems

In this work, the existence of a saddle-node bifurcation of invariant cones in three-dimensional continuous homogeneous piecewise linear systems is considered. First, we prove that invariant cones for this class of systems correspond one-to-one to periodic orbits of a continuous piecewise cubic system defined on the unit sphere. Second, let us give the conditions for which the sphere is foliated by a continuum of periodic orbits. The principal idea is looking for the periodic orbits of the continuum that persist when this situation is perturbed. To do this, we establish the relationship between the invariant cones of the three-dimensional system and the periodic orbits of two planar hybrid piecewise linear systems. Next, we define two functions whose zeros provide the invariant cones that persist under the perturbation. These functions will be called Melnikov functions and their properties allow us to state some results about the existence of invariant cones and other results about the existence of saddle-node bifurcations of invariant cones, which is the principal goal of this paper.     

Generalized dimensions,entropies, and Liapunov exponents from the pressure function for strange sets

For conformal mixing repellers such as Julia sets and nonlinear one-dimensional Cantor sets, we connect the pressure of a smooth transformation on the repeller with its generalized dimensions, entropies, and Liapunov exponents computed with respect to a set of equilibrium Gibbs measures. This allows us to compute the pressure by means of simple numerical algorithms. Our results are then extended to axiom-A attractors and to a nonhyperbolic invariant set of the line. In this last case, we show that a first-order phase transition appears in the pressure.     

Smooth analog of standard mapping

Twenty years ago Bullett [1] published an article [1] where he found the invariant curves of standard mapping, having replaced the sinusoidal force by its smooth analog, a piecewise linear saw. His studies discovered an unexpected effect: at certain values of the perturbation parameter, unsplit separatrices of integer and fractional resonances arise among global invariant curves, while chaotic layers, which are usually attendant to these separatrices, are absent. Interestingly, the system remains nonintegrable in this case and the separatrices persist, confining momentum global diffusion under the condition of strong local chaos. For reasons not well understood, this important effect and its consequences had gone largely unnoticed until Ovsyannikov [2] independently proved a similar theorem for integer resonances in terms of the same model of symmetric piecewise linear 2D mapping. Since then, piecewise linear maps and their related continuous systems have become a subject of extensive research. Both Bullett and Ovsyannikov restricted analysis to the invariant curves of the new type, since the two branches of split separatrices form chaotic trajectories that are impossible to treat analytically. To the author’s knowledge, examples of persisting separatrices other than those given in [1, 2] have not been reported. In this work, the author presents numerical and analytical results that directly or indirectly concern the effect of separatrix persistence in the absence of attendant dynamic chaos. Issues remaining to be understood are noted.     

Chaotic itinerancy based on attractors of one-dimensional maps

A general methodology is described for constructing systems that have a slowly converging Lyapunov exponent near zero, based on one-dimensional maps with chaotic attractors. In certain parameter ranges, these relatively simple systems display the properties of intermittent dynamics known as chaotic itinerancy. We show that in addition to the local sensitivity characteristic of chaotic dynamics, these itinerant systems display a global sensitivity, in the sense that fine-scale additive noise may significantly change the natural measure on the large scale.     

Large Deviations for Countable to One Markov Systems

In this paper, we study large deviation properties for countable to one Markov systems associated to weak Gibbs measures for non-Hölder potentials. Furthermore, we establish multifractal large deviation laws for countable to one piecewise conformal Markov systems, which are derived systems constructed over hyperbolic regions for certain nonhyperbolic systems exhibiting intermittency. We apply our results to higher-dimensional number theoretical transformations.     

Information flow and maximum entropy measures for 1-D maps

The invariant measures of maximal metric entropy are constructed explicitly for some maps of the interval, by iterating the maps backward. The construction illustrates in a particularly clear way the information flow in simple systems, as well as recently conjectured relationships between dimensions of invariant measures, Lyapunov exponents, and entropies. maps, it is conjectured that the natural measure is the invariant measure with strongest mixing.