Clustering of exponentially separating trajectories

It might be expected that trajectories of a dynamical system which has no negative Lyapunov exponent (implying exponential growth of small separations) will not cluster together. However, clustering can occur such that the density ρ(Δx) of trajectories within distance |Δx| of a reference trajectory has a power-law divergence, so that ρ(Δx) ∼ |Δx| ^−β when |Δx| is sufficiently small, for some 0 < β < 1. We demonstrate this effect using a random map in one dimension. We find no evidence for this effect in the chaotic logistic map, and argue that the effect is harder to observe in deterministic maps.     

Chaos in the perturbed Korteweg-de Vries equation with nonlinear terms of higher order

The dynamical behaviour of the generalized Korteweg-de Vries (KdV) equation under a periodic perturbation is investigated numerically. The bifurcation and chaos in the system are observed by applying bifurcation diagrams, phase portraits and Poincaré maps. To characterise the chaotic behaviour of this system, the spectra of the Lyapunov exponent and Lyapunov dimension of the attractor are also employed.     

The geometry of chaotic dynamics — a complex network perspective

Recently, several complex network approaches to time series analysis have been developed and applied to study a wide range of model systems as well as real-world data, e.g., geophysical or financial time series. Among these techniques, recurrence-based concepts and prominently ε-recurrence networks, most faithfully represent the geometrical fine structure of the attractors underlying chaotic (and less interestingly non-chaotic) time series. In this paper we demonstrate that the well known graph theoretical properties local clustering coefficient and global (network) transitivity can meaningfully be exploited to define two new local and two new global measures of dimension in phase space: local upper and lower clustering dimension as well as global upper and lower transitivity dimension. Rigorous analytical as well as numerical results for self-similar sets and simple chaotic model systems suggest that these measures are well-behaved in most non-pathological situations and that they can be estimated reasonably well using ε-recurrence networks constructed from relatively short time series. Moreover, we study the relationship between clustering and transitivity dimensions on the one hand, and traditional measures like pointwise dimension or local Lyapunov dimension on the other hand. We also provide further evidence that the local clustering coefficients, or equivalently the local clustering dimensions, are useful for identifying unstable periodic orbits and other dynamically invariant objects from time series. Our results demonstrate that ε-recurrence networks exhibit an important link between dynamical systems and graph theory.     

Local Geometry and Dynamical Behavior on Folded Basic Sets

We study new phenomena associated with the dynamics of higher dimensional non-invertible, hyperbolic maps f on basic sets of saddle type; the dynamics in this case presents important differences from the case of diffeomorphisms or expanding maps. We show that the stable dimension (i.e. the Hausdorff dimension of the intersection between local stable manifolds and the basic set) and the unstable dimension (similar definition) give a lot of information about the dynamical/ergodic properties of endomorphisms on folded basic sets. We prove a geometric flattening phenomenon associated to the stable dimension, i.e. we show that if the stable dimension is zero at a point, then the fractal Λ must be contained in a submanifold and f is expanding on Λ. We characterize folded attractors and folded repellers, as those basic sets with full unstable dimension, respectively with full stable dimension. We classify possible dynamical behaviors, and establish when is the system (Λ,f,μ) 1-sided or 2-sided Bernoulli for certain equilibrium measures μ on folded basic sets, for a class of perturbation maps.     

一种最简的并行忆阻器混沌系统

在提出的一种压控忆阻器的基础上, 构造了最简的并联忆阻器混沌系统, 分析其动力学特性, 得到了该系统的Lyapunov指数和Lyapunov维数, 给出了时域波形、相图、Lyapunov指数谱、分岔图、Poincaré映射等. 利用EWB软件设计了该新混沌系统的振荡电路并进行了仿真实验. 研究结果表明, 忆阻器的i-v特性在参数的变化时, 并不保持斜“8”字形, 会变为带尾巴的扇形. 该混沌系统与磁控忆阻器混沌系统不同, 系统只有一个平衡点, 初始条件在系统能振荡的情况下不影响系统状态. 电路实验仿真结果和数值仿真具有很好的一致性, 证实了该系统的存在性和物理上可实现性. 关键词： 忆阻器 混沌电路 并联 动力学行为     

Family-Vicsek scaling of detachment fronts in granular Rayleigh-Taylor instabilities during sedimentating granular/fluid flows

When submillimetric particles are confined in a fluid such that a compact cluster of particles lie above the clear fluid, particles will detach from the lower boundary of the cluster and form an unstable separation front giving rise to growing fingers of falling particles. We study this problem using both experiments and hybrid granular/fluid mechanics models. In the case of particles from 50 to 500 microns in diameter falling in air, we study the horizontal density fluctuations at early times: the amplitude of the density difference between two points at a certain horizontal distance grows as a power law of time. This happens up to a saturation corresponding to a power law of the distance. The way in which the correlation length builds up to this saturation also follows a power law of time. We show that these decompaction fronts in sedimentation problems follow a Family-Vicsek scaling, characterize the dynamic and Hurst exponent of the lateral density fluctuations, respectively z ∼ 1 and ζ ∼ 0.75, and show how the prefactors depend on the grain diameter. We also show from similar simulations with a more viscous and incompressible fluid, that this feature is independent of the fluid compressibility or viscosity, ranging from air to water/glycerol mixtures.     

Duality relations for asymmetric exclusion processes

We derive duality relations for a class ofU _q [SU(2)]-symmetric stochastic processes, including among others the asymmetric exclusion process in one dimension. Like the known duality relations for symmetric hopping processes, these relations express certainm-point correlation functions inN-particle systems (Nm) in terms of sums of correlation functions of the same system but with onlym particles. For the totally asymmetric case we obtain exact expressions for some boundary density correlation functions. The dynamical exponent for these correlators isz=2, which is different from the dynamical exponent for bulk density correlations, which is known to bez=3/2.     

A dissipative network model with neighboring activation

We study the properties of spectrum and eigenstates of the Google matrix of a directed network formed by the procedure calls in the Linux Kernel. Our results obtained for various versions of the Linux Kernel show that the spectrum is characterized by the fractal Weyl law established recently for systems of quantum chaotic scattering and the Perron-Frobenius operators of dynamical maps. The fractal Weyl exponent is found to be ν ≈ 0.65 that corresponds to the fractal dimension of the network d ≈ 1.3. An independent computation of the fractal dimension by the cluster growing method, generalized for directed networks, gives a close value d ≈ 1.4. The eigenmodes of the Google matrix of Linux Kernel are localized on certain principal nodes. We argue that the fractal Weyl law should be generic for directed networks with the fractal dimension d < 2.     

Chaotic dynamics in the electroconvection system of a nematic liquid crystal

The transition from a steady domain structure to turbulence in the electroconvection system of a nematic under the action of a constant electric field is studied using the methods of optical and acoustic responses. The chaotic dynamics is investigated both by conventional methods (Fourier signal spectrum) and by methods of nonlinear dynamics. From the quantitative estimates of basic characteristics of the chaotic behavior (namely, the correlation dimension, leading Lyapunov exponent, K-entropy, and embedding dimension), one can conclude that temporal chaos arises in the system, giving rise to a strange attractor, as the control parameter increases at ɛ ≥ ɛ _c ≈ 0.5. The fact that the distribution of laminar domains in the liquid-crystal layer depends on their length under the conditions of developed turbulence indicates that the dynamics of the nematic demonstrates the intermittent behavior.     

Ergodic Jacobi Matrices and Conformal Maps

We study structural properties of the Lyapunov exponent γ and the density of states k for ergodic (or just invariant) Jacobi matrices in a general framework. In this analysis, a central role is played by the function w = − γ + iπ k as a conformal map between certain domains. This idea goes back to Marchenko and Ostrovskii, who used this device in their analysis of the periodic problem.     

Front propagation in chaotic and noisy reaction-diffusion systems: a discrete-time map approach

We study the front propagation in reaction-diffusion systems whose reaction dynamics exhibits an unstable fixed point and chaotic or noisy behaviour. We have examined the influence of chaos and noise on the front propagation speed and on the wandering of the front around its average position. Assuming that the reaction term acts periodically in an impulsive way, the dynamical evolution of the system can be written as the convolution between a spatial propagator and a discrete-time map acting locally. This approach allows us to perform accurate numerical analysis. They reveal that in the pulled regime the front speed is basically determined by the shape of the map around the unstable fixed point, while its chaotic or noisy features play a marginal role. In contrast, in the pushed regime the presence of chaos or noise is more relevant. In particular the front speed decreases when the degree of chaoticity is increased, but it is not straightforward to derive a direct connection between the chaotic properties (e.g. the Lyapunov exponent) and the behaviour of the front. As for the fluctuations of the front position, we observe for the noisy maps that the associated mean square displacement grows in time as t ^1/2 in the pushed case and as t ^1/4 in the pulled one, in agreement with recent findings obtained for continuous models with multiplicative noise. Moreover we show that the same quantity saturates when a chaotic deterministic dynamics is considered for both pushed and pulled regimes. Received 17 July 2001     

Special electronic structures of inverse spinels LiMVO<Subscript>4</Subscript>(M = Ni and Cu): a first-principles study

We use the Ulam method to study spectral properties of the Perron-Frobenius operators of dynamical maps in a chaotic regime. For maps with absorption we show numerically that the spectrum is characterized by the fractal Weyl law recently established for nonunitary operators describing poles of quantum chaotic scattering with the Weyl exponent ν = d-1, where d is the fractal dimension of corresponding strange set of trajectories nonescaping in future times. In contrast, for dissipative maps we numerically find the Weyl exponent ν = d/2 where d is the fractal dimension of strange attractor. The Weyl exponent can be also expressed via the relation ν = d₀/2 where d₀ is the fractal dimension of the invariant sets. We also discuss the properties of eigenvalues and eigenvectors of such operators characterized by the fractal Weyl law.     

Gravitational Collapse and Ergodicity in Confined Gravitational Systems

The ergodic properties of many-body systems with repulsive-core interactions are the basis of classical statistical mechanics and are well established. This is not the case for systems of purely-attractive or gravitational particles. Here we consider two examples, (i) a family of one-dimensional systems with attractive power-law interactions, , and (ii) a system of N gravitating particles confined to a finite compact domain. For (i) we deduce from the numerically-computed Lyapunov spectra that chaos, measured by the maximum Lyapunov exponent or by the Kolmogorov–Sinai entropy, increases linearly for positive and negative deviations of ν from the case of a non-chaotic harmonic chain (ν = 2). For there is numerical evidence for two additional hitherto unknown phase-space constraints. For the theoretical interpretation of model (ii) we assume ergodicity and show that for a small-enough system the reduction of the allowed phase space due to any other conserved quantity, in addition to the total energy, renders the system asymptotically stable. Without this additional dynamical constraint the particle collapse would continue forever. These predictions are supported by computer simulations. PACS numbers: 05.45.Pq, Numerical simulation of chaotic systems, 05.20.−y, Classical statistical mechanics, 36.40.Qv, Stability and fragmentation of clusters, 95.10.Fh, Chaotic dynamics.     

Dynamical properties of constrained drops

In this communication we analyze the behavior of excited drops contained in spherical volumes. We study different properties of the dynamical systems, i.e. the maximum Lyapunov exponent MLE, the asymptotic distance in momentum space d _∞ and the normalized variance of the maximum fragment. It is shown that the constrained system behaves as undergoing a first-order phase transition at low densities while as a second-order one at high densities. The transition from liquid-like to vapor-like behavior is signaled both by the caloric curves, the thermal response functions and the MLE. The relationship between the MLE, d _∞, and the caloric curve is explored. Received: 28 March 2002 / Accepted: 17 May 2002     

Conversion of a chaotic attractor into a strange nonchaotic attractor in an one dimensional map and BVP oscillator

In this paper we investigate numerically the possibility of conversion of a chaotic attractor into a nonchaotic but strange attractor in both a discrete system (an one dimensional map) and in a continuous dynamical system — Bonhoeffer—van der Pol oscillator. In these systems we show suppression of chaotic property, namely, the sensitive dependence on initial states, by adding appropriate i) chaotic signal and ii) Gaussian white noise. The controlled orbit is found to be strange but nonchaotic with largest Lyapunov exponent negative and noninteger correlation dimension. Return map and power spectrum are also used to characterize the strange nonchaotic attractor.     

Statistical dynamics at critical bifurcations in Duffing-van der Pol oscillator

We study the characteristic features of certain statistical quantities near critical bifurcations such as onset of chaos, sudden widening and band-merging of chaotic attractor and intermittency in a periodically driven Duffing-van der Pol oscillator. At the onset of chaos the variance of local expansion rate is found to exhibit a self-similar pattern. For all chaotic attractors the variance Σ_n(q) of fluctuations of coarse-grained local expansion rates of nearby orbits has a single peak. However, multiple peaks are found just before and just after the critical bifurcations. On the other hand, Σ_n (q) associated with the coarse-grained state variable is zero far from the bifurcations. The height of the peak of Σ_n(q) is found to increase as the control parameter approached the bifurcation point. It is maximum at the bifurcation point. Power-law variation of maximal Lyapunov exponent and the mean value of the state variablex is observed near sudden widening and intermittency bifurcations while linear variation is seen near band-merging bifurcation. The standard deviation of local Lyapunov exponent λ(X,L) and the local mean valuex(L) of the coordinatex calculated after everyL time steps are found to approach zero in the limitL → ∞ asL ^-Β. Β is sensitive to the values of control parameters. Further weak and strong chaos are characterized using the probability distribution of ak-step difference quantity δx_k = x_i+k x _i.     

A Novel Measure Inspired by Lyapunov Exponents for the Characterization of Dynamics in State-Transition Networks

The combination of network sciences, nonlinear dynamics and time series analysis provides novel insights and analogies between the different approaches to complex systems. By combining the considerations behind the Lyapunov exponent of dynamical systems and the average entropy of transition probabilities for Markov chains, we introduce a network measure for characterizing the dynamics on state-transition networks with special focus on differentiating between chaotic and cyclic modes. One important property of this Lyapunov measure consists of its non-monotonous dependence on the cylicity of the dynamics. Motivated by providing proper use cases for studying the new measure, we also lay out a method for mapping time series to state transition networks by phase space coarse graining. Using both discrete time and continuous time dynamical systems the Lyapunov measure extracted from the corresponding state-transition networks exhibits similar behavior to that of the Lyapunov exponent. In addition, it demonstrates a strong sensitivity to boundary crisis suggesting applicability in predicting the collapse of chaos.     

The largest Lyapunov exponent of chaotic dynamical system in scale space and its application

The largest Lyapunov exponent is an important invariant of detecting and characterizing chaos produced from a dynamical system. We have found analytically that the largest Lyapunov exponent of the small-scale wavelet transform modulus of a dynamical system is the same as the system's largest Lyapunov exponent, both discrete map and continuous chaotic attractor with one or two positive Lyapunov exponents. This property has been used to estimate the largest Lyapunov exponent of chaotic time series with several kinds of strong additive noise.     

Probabilistic Averages of Jacobi Operators

I study the Lyapunov exponent and the integrated density of states for general Jacobi operators. The main result is that questions about these can be reduced to questions about ergodic Jacobi operators. I use this to show that for finite gap Jacobi operators, regularity implies that they are in the Cesàro–Nevai class, proving a conjecture of Barry Simon. Furthermore, I use this to study Jacobi operators with coefficients a(n) = 1 and b(n) = f(n ^ρ (mod 1)) for ρ > 0 not an integer.     

Polymer chain in a quenched random medium: slow dynamics and ergodicity breaking

The Langevin dynamics of a self-interacting chain embedded in a quenched random medium is investigated by making use of the generating functional method and one-loop (Hartree) approximation. We have shown how this intrinsic disorder causes different dynamical regimes. Namely, within the Rouse characteristic time interval the anomalous diffusion shows up. The corresponding subdiffusional dynamical exponents have been explicitly calculated and thoroughly discussed. For the larger time interval the disorder drives the center of mass of the chain to a trap or frozen state provided that the Harris parameter, (Δ/b ^d)N ^{2 - νd}≥1, where Δ is a disorder strength, b is a Kuhnian segment length, N is a chain length and ν is the Flory exponent. We have derived the general equation for the non-ergodicity function f (p) which characterizes the amplitude of frozen Rouse modes with an index p = 2πj/N. The numerical solution of this equation has been implemented and shown that the different Rouse modes freeze up at the same critical disorder strength Δ _c ∼ N ^{- γ} where the exponent γ ≈ 0.25 and does not depend from the solvent quality. Received 17 December 2002 Published online 23 May 2003 RID="a" ID="a"e-mail: vilgis@mpip-mainz.mpg.de