Topological aspects of quantum chaos

Quantized classically chaotic maps on a toroidal two-dimensional phase space are studied. A discrete, topological criterion for phase-space localization is presented. To each eigenfunction is associated an integer, analogous to a quantized Hall conductivity, which tests the way the eigenfunction explores the phase space as some boundary conditions are changed. The correspondence between delocalization and chaotic classical dynamics is discussed, as well as the role of degeneracies of the eigenspectrum in the transition from localized to delocalized states. The general results are illustrated with a particular model.     

High-precision mapping of the magnetic field utilizing the harmonic function mean value property

The spatial distributions of the static magnetic field components and MR phase maps in space with homogeneous magnetic susceptibility are shown to be harmonic functions satisfying Laplace's equation. A mean value property is derived and experimentally confirmed on phase maps: the mean value on a spherical surface in space is equal to the value at the center of the sphere. Based on this property, a method is implemented for significantly improving the precision of MR phase or field mapping. Three-dimensional mappings of the static magnetic field with a precision of 10(-11) approximately 10(-12) T are obtained in phantoms by a 1.5-T clinical MR scanner, with about three-orders-of-magnitude precision improvement over the conventional phase mapping technique. In vivo application of the method is also demonstrated on human leg phase maps.     

Relaxation and localization in interacting quantum maps

Quantum relaxation is studied in coupled quantum baker's maps. The classical systems are exactly solvable Kolmogorov systems, for which the exponential decay to equilibrium is known. They model the fundamental processes of transport in classically chaotic phase space. The quantum systems, in the absence of global symmetry, show a marked saturation in the level of transport, as the suppression of diffusion in the quantum kicked rotor, and eigenfunction localization in the position basis. In the presence of a global symmetry we study another model that has classically an identical decay to equilibrium, but-quantally shows resonant transport, no saturation, and large fluctuations around equilibrium. We generalize the quantization to finite multibaker maps. As a byproduct we introduce some simple models of quantal tunneling between classically chaotic regions of phase space.     

Local and global statistical properties of deterministic chaos

Summary Locla and global statistical properties of a class of one-dimensional dissipative chaotic maps and a class of 2-dimensional conservative hyperbolic maps are investigated. This is achieved by considering the spectral properties of the Perron-Frobenius operator (the evolution operator for probability densities) acting on two different types of function space. In the first case, the function space is piecewise analytic, and includes functions having support over local regions of phase space. In the second case, the function space essentially consists of functions which are “globally? analytic,i.e. analytic over the given systems entire phase space. Each function space defines a space of measurable functions or observables, whose statistical moments and corresponding characteristic times can be exactly determined. Paper presented at the International Workshop ?Fluctuations in Physics and Biology: Stochastic Resonance, Signal Processing and Related Phenomena?, Elba, 5–10 June 1994.     

The Compound Poisson Limit Ruling Periodic Extreme Behaviour of Non-Uniformly Hyperbolic Dynamics

We prove that the distributional limit of the normalised number of returns to small neighbourhoods of periodic points of certain non-uniformly hyperbolic dynamical systems is compound Poisson. The returns to small balls around a fixed point in the phase space correspond to the occurrence of rare events, or exceedances of high thresholds, so that there is a connection between the laws of Return Times Statistics and Extreme Value Laws. The fact that the fixed point in the phase space is a repelling periodic point implies that there is a tendency for the exceedances to appear in clusters whose average sizes is given by the Extremal Index, which depends on the expansion of the system at the periodic point. We recall that for generic points, the exceedances, in the limit, are singular and occur at Poisson times. However, around periodic points, the picture is different: the respective point processes of exceedances converge to a compound Poisson process, so instead of single exceedances, we have entire clusters of exceedances occurring at Poisson times with a geometric distribution ruling its multiplicity. The systems to which our results apply include: general piecewise expanding maps of the interval (Rychlik maps), maps with indifferent fixed points (Manneville-Pomeau maps) and Benedicks-Carleson quadratic maps.     

Existence of chaos in a piecewise smooth two-dimensional contractive map

Piecewise smooth maps occur in a variety of physical systems. We show that in a two-dimensional continuous map a chaotic orbit can exist even when the map is contractive (eigenvalues less than unity in magnitude) at every point in the phase space. In this Letter we explain this peculiar feature of piecewise smooth continuous maps.     

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.     

Radon transforms of the Wigner operator on hyperplanes

The generalization of tomographic maps to hyperplanes is considered. We find that the Radon transform of the Wigner operator in multi-dimensional phase space leads to a normally ordered operator in binomial distribution---a mixed-state density operator. Reconstruction of the Wigner operator is also feasible. The normally ordered form and the Weyl ordered form of the Wigner operator are used in our derivation. The operator quantum tomography theory is expressed in terms of some operator identities, with the merit of revealing the essence of the theory in a simple and concise way.     

Holonomy structure of parameter manifolds

The Berry holonomy phase is usually attributed to homotopically nontrivial maps induced by the Hamiltonian in the space of orthonormal eigenstates constructed over a parameter manifold. We show that the issues of mappings and eigenstates should be addressed separately and that equivalence between them implies a trivial Berry phase.     

How efficiently Do three pointlike particles sample phase space?

We show that the continuous phase space of a hard particle system can be mapped onto a discrete but infinite phase space. For three pointlike particles confined to a ring, the evolution of the system maps onto a chaotic walk on a hexagonal lattice. This facilitates direct measurement of the departure of the system from its original configuration. In special cases of mass ratios the phase space becomes closed and finite (nonergodic). There are qualitative differences between this chaotic walk and a random walk, in particular a more rapid sampling of phase space.     

Stratifying monopoles and rational maps

A stratification of the moduli space of monopoles and of the space of rational maps into a flag variety is presented. It is shown that the map associating a rational map to a monopole preserves these strata. These strata explain some problems in the intepretation of the parameters of the moduli space in terms of superpositions of fundamental monopoles. This interpretation is not valid on the individual strata. The space of fundamental monopoles is described and shown to be the same as the corresponding space of rational maps.     

Towards the map of quantum gravity

In this paper we point out some possible links between different approaches to quantum gravity and theories of the Planck scale physics. In particular, connections between loop quantum gravity, causal dynamical triangulations, Ho?ava–Lifshitz gravity, asymptotic safety scenario, Quantum Graphity, deformations of relativistic symmetries and nonlinear phase space models are discussed. The main focus is on quantum deformations of the Hypersurface Deformations Algebra and Poincaré algebra, nonlinear structure of phase space, the running dimension of spacetime and nontrivial phase diagram of quantum gravity. We present an attempt to arrange the observed relations in the form of a graph, highlighting different aspects of quantum gravity. The analysis is performed in the spirit of a mind map, which represents the architectural approach to the studied theory, being a natural way to describe the properties of a complex system. We hope that the constructed graphs (maps) will turn out to be helpful in uncovering the global picture of quantum gravity as a particular complex system and serve as a useful guide for the researchers.     

Quantum chaos in the Wigner representation

The dynamics of a familiar model of stochastic behavior-the quantum kicked rotator-is analyzed in the Wigner representation. Exact nonlocal maps defined on a discrete phase space are derived. The basic dynamics of a quantum kicked rotator can be described satisfactorily by means of a simplified map that incorporates only the discrete nature of the phase space.     

Dynamics and transport in mean-field coupled, many degrees-of-freedom, area-preserving nontwist maps

Area-preserving nontwist maps, i.e., maps that violate the twist condition, arise in the study of degenerate Hamiltonian systems for which the standard version of the Kolmogorov-Arnold-Moser (KAM) theorem fails to apply. These maps have found applications in several areas including plasma physics, fluid mechanics, and condensed matter physics. Previous work has limited attention to maps in 2-dimensional phase space. Going beyond these studies, in this paper, we study nontwist maps with many-degrees-of-freedom. We propose a model in which the different degrees of freedom are coupled through a mean-field that evolves self-consistently. Based on the linear stability of period-one and period-two orbits of the coupled maps, we construct coherent states in which the degrees of freedom are synchronized and the mean-field stays nearly fixed. Nontwist systems exhibit global bifurcations in phase space known as separatrix reconnection. Here, we show that the mean-field coupling leads to dynamic, self-consistent reconnection in which transport across invariant curves can take place in the absence of chaos due to changes in the topology of the separatrices. In the context of self-consistent chaotic transport, we study two novel problems: suppression of diffusion and breakup of the shearless curve. For both problems, we construct a macroscopic effective diffusion model with time-dependent diffusivity. Self-consistent transport near criticality is also studied, and it is shown that the threshold for global transport as function of time is a fat-fractal Cantor-type set.     

A horseshoe for the doubling operator: Topological dynamics for metric universality

The doubling operator, properly defined on the space of smooth maps on the interval at the boundary of chaos, yields a dynamical system in this function space. Even if one restricts oneself to the space of real analytic maps, there is evidence that the dynamics of the doubling operator contains a horseshoe whose symbolic dynamics is described by the one-sided shift on two symbols. We indicate also how some of the global aspects of this dynamics could be recognized in a physical experiment on the transition to chaos.     

Robust dangerous border-collision bifurcations in piecewise smooth systems

Physical and computer experiments involving systems describable by piecewise smooth continuous maps that are nondifferentiable on some surface in phase space exhibit novel types of bifurcations in which an attracting fixed point exists before and after the bifurcation. The striking feature of these bifurcations is that they typically lead to "unbounded behavior" of orbits as a system parameter is slowly varied through its bifurcation value. This new type of border-collision bifurcation is fundamental and robust. A method that prevents such "dangerous border-collision bifurcations" is given. These bifurcations may be found in a variety of experiments including circuits.     

New Representation for Tensor Product of a Kind of Squeeze Operators

Using the IWOP (integration within ordered product) technique, we construct a new state vector representation in two-mode Fock space. The tensor product of a kind of squeeze operators can then be well identified in the new representation, which manifestly shows that these squeeze operators are quantum maps imaged by certain symplectic transformation in classical phase space.     

一种混沌映射的相空间去噪方法

对于混沌映射来说,它们的频谱比混沌流的频谱更广阔,与噪声频谱的重叠率更高,所以混沌流的去噪方法对它们并不适用. 在半盲分析法的框架下,混沌系统的参数估计问题终将归结为最小二乘估计问题. 本文从最小二乘拟合的角度出发估计混沌映射的演化参数,进而通过相空间重构以及投影操作,实现对观测信号的噪声抑制. 实验结果表明,该算法的去噪效果优于扩展卡尔曼滤波器（extended Kalman filter,EKF）和无先导卡尔曼滤波器（unscneted Kalman filter,UKF）,并且能够较大程度地将信号源的混沌特征量还原. 关键词： 混沌 噪声抑制 相空间重构 投影     

A biologically inspired spatiotemporal saliency attention model based on entropy value

A biologically inspired spatiotemporal saliency attention model based on entropy value is proposed in this paper. This model includes a dynamic attention phase and a static attention phase. In the dynamic attention phase, low-level visual features are extracted from current and some previous frames. Every feature map is resized into some different sizes. The feature maps in same size and same feature for all the frames are used to calculate the entropy value map. All the entropy maps are normalized and are fused into a dynamic saliency map. In the static attention phase, same features are extracted and form multi-scale feature maps by center-surround differences in current frame, and then those feature maps are transformed into conspicuity maps, which are linearly combined into a static saliency map. Our model decides salient regions based on a spatiotemporal saliency map which is generated by integration of the dynamic and the static saliency map. Experimental results indicate that: when there is noise among the frames or there is change of illumination among the frames, our model is excellent to Shi's model and Marat's model; when the moving objects do not belong to the static salient regions, our model is better than Ban's model.     

若干二维映象中V型阵发层流相区的特征

通过几个实例的分析,说明二维映象中V型阵发的层流相所占据的相空间区域常常具有一维特征.这些一维相空间区域对应于已经由边界碰撞分岔消失的周期点的“鬼魂”留下的一段稳定流形.它们构成类似于一维映象中阵发发生后层流相迭代通过的“隧道”.隧道的开口有明确的位置.V型阵发的湍流相所对应相空间区域具有二维特征.解析讨论了这些特征的形成机制. 关键词： V型阵发 层流相 流形