Fluctuations and simple chaotic dynamics

We describe the effects of fluctuations on the period-doubling bifurcation to chaos. We study the dynamics of maps of the interval in the absence of noise and numerically verify the scaling behavior of the Lyapunov characteristic exponent near the transition to chaos. As previously shown, fluctuations produce a gap in the period-doubling bifurcation sequence. We show that this implies a scaling behavior for the chaotic threshold and determine the associated critical exponent. By considering fluctuations as a disordering field on the deterministic dynamics, we obtain scaling relations between various critical exponents relating the effect of noise on the Lyapunov characteristic exponent. A rule is developed to explain the effects of additive noise at fixed parameter value from the deterministic dynamics at nearby parameter values.     

Transition to chaos in continuous-time random dynamical systems

We consider situations where, in a continuous-time dynamical system, a nonchaotic attractor coexists with a nonattracting chaotic saddle, as in a periodic window. Under the influence of noise, chaos can arise. We investigate the fundamental dynamical mechanism responsible for the transition and obtain a general scaling law for the largest Lyapunov exponent. A striking finding is that the topology of the flow is fundamentally disturbed after the onset of noisy chaos, and we point out that such a disturbance is due to changes in the number of unstable eigendirections along a continuous trajectory under the influence of noise.     

Breakdown of chaos symmetry and intermittency in the double-well potential system

The chaos-chaos transition in a one-particle system in the symmetric double-well potential under an external periodic field is studied from the viewpoint of the breakdown of the chaos symmetry and the development of the intermittency characteristics. It is found that the similarity exponent introduced to analyze the intermittency characteristics satisfies a scaling law near the transition point.     

Phase separation and pattern formation in a binary Bose-Einstein condensate

The miscibility-immiscibility phase transition in binary Bose-Einstein condensates (BECs) can be controlled by a coupling between the two components. Here we propose a new scheme that uses coupling-induced pattern formation to test the Kibble-Zurek mechanism (KZM) of topological-defect formation in a quantum phase transition. For a binary BEC in a ring trap we find that the number of domains forming the pattern scales as a function of the coupling quench rate with an exponent as predicted by the KZM. For a binary BEC in an elongated harmonic trap we find a different scaling law due to the transition being spatially inhomogeneous. We perform a "quantum simulation" of the harmonically trapped system in a ring trap to verify the scaling exponent.     

Analytical realization of finite-size scaling for Anderson localization: Is there a transition in the 2D case?

Roughly half the numerical investigations of the Anderson transition are based on consideration of an associated quasi-1D system and postulation of one-parameter scaling for the minimal Lyapunov exponent. If this algorithm is taken seriously, it leads to unambiguous prediction of the 2D phase transition. The transition is of the Kosterlitz-Thouless type and occurs between exponential and power law localization (Pichard and Sarma, 1981). This conclusion does not contradict numerical results if raw data are considered. As for interpretation of these data in terms of one-parameter scaling, this is inadmissible: the minimal Lyapunov exponent does not obey any scaling. A scaling relation is valid not for a minimal, but for some effective Lyapunov exponent whose dependence on the parameters is determined by the scaling itself. If finite-sizedd scaling is based on the effective Lyapunov exponent, the existence of the 2D transition becomes indefinite, but still rather probable. Interpretation of the results in terms of the Gell-Mann-Low equation is also given.     

Amplification of intrinsic fluctuations by the Lorenz equations

Macroscopic systems (e.g., hydrodynamics, chemical reactions, electrical circuits, etc.) manifest intrinsic fluctuations of molecular and thermal origin. When the macroscopic dynamics is deterministically chaotic, the intrinsic fluctuations may become amplified by several orders of magnitude. Numerical studies of this phenomenon are presented in detail for the Lorenz model. Amplification to macroscopic scales is exhibited, and quantitative methods (binning and a difference-norm) are presented for measuring macroscopically subliminal amplification effects. In order to test the quality of the numerical results, noise induced chaos is studied around a deterministically nonchaotic state, where the scaling law relating the Lyapunov exponent to noise strength obtained for maps is confirmed for the Lorenz model, a system of ordinary differential equations.     

Charge transport transitions and scaling in disordered arrays of metallic dots

We examine the charge transport through disordered arrays of metallic dots using numerical simulations. We find power law scaling in the current-voltage curves for arrays containing no voids, while for void-filled arrays charge bottlenecks form and a single scaling is absent, in agreement with recent experiments. In the void-free case we also show that the scaling exponent depends on the effective dimensionality of the system. For increasing applied drives we find a transition from 2D disordered filamentary flow near threshold to a 1D smectic flow which can be identified experimentally using characteristics in the transport curves and conduction noise.     

Effect of noise on the neutral direction of chaotic attractor

A chaotic attractor from a deterministic flow must necessarily possess a neutral direction, as characterized by a null Lyapunov exponent. We show that for a wide class of chaotic attractors, particularly those having multiple scrolls in the phase space, the existence of the neutral direction can be extremely fragile in the sense that it is typically destroyed by noise of arbitrarily small amplitude. A universal scaling law quantifying the increase of the Lyapunov exponent with noise is obtained. A way to observe the scaling law in experiments is suggested.     

Josephson结的动力学行为(Ⅱ)

本文对射频电流驱动下的包含干涉项电流的Josephson结系统的阵发混沌行为进行了研究,得到的标度关系与理论预言的相符。此外,我们计算了一些参数下的里亚普诺夫指数以及由此引入定义的吸引子的维数d_L,并且还用盒子计数法计算了几组参数下奇异吸引子的柯尔莫哥洛夫容量d_c,其值与d_L相近。 关键词：     

Space-time renormalization at the onset of spatio-temporal chaos in coupled maps

The transition regime to spatio-temporal chaos via the quasiperiodic route as well as the period-doubling route is examined for coupled-map lattices. Space-time renormalization-group analysis is carried out and the scaling exponents for the coherence length, the Lyapunov exponent, and the size of the phase fluctuations are determined. Universality classes for the different types of coupling at various routes to chaos are identified.     

Universality and scaling of period-doubling bifurcations in a dissipative distributed medium

A homogeneous medium, consisting of nonlinear elements, demonstrating transition to chaos via period-doubling bifurcations, is considered. The coupling between the elements is supposed to be of a dissipative type, i.e. it tends to equalize their instantaneous states. Using the renormalization group approach, the following scaling law for weakly inhomogeneous states near the critical point is obtained: at each period doubling the spatial scale increases by β=√2. On the basis of this law the scaling hypotheses for the transition to chaos in the semi-infinite and finite systems are proposed. The scaling properties are verified by the numerical calculations with a simple model.     

A crisis of a stochastic web

In a kicked rotor subjected to a piecewise-continuous force field, it is observed that a stochastic web and the chaotic diffusion on it suddenly change to transients when an adjustable parameter drives the dissipation. This phenomenon appears to be a new crisis type, which occurs in systems where conservative dynamics may be converted to a dissipative one with a contraction rate showing linear time dependence. It is analytically and numerically shown that, in the crisis, the lifetime dependence obeys universal scaling law suggested by Grebogy, Ott, and Yorke [Phys. Rev. Lett. 57, 1284 (1986)], and the scaling exponent takes a special value, 1, due to the dissipation characteristics. Additionally presented is another power law that describes, from another viewpoint, the transition of a conservative stochastic web (which is a fat fractal) to a non-attracting thin fractal (the strange repeller).Received: 13 December 2003, Published online: 9 March 2004PACS: 05.45.Ac Low-dimensional chaos     

非周期力在混沌控制中的双重功效

探讨了非周期力（有界噪声或混沌驱动力）在非线性动力系统混沌控制中的影响.以一类典型的含有五次非线性项的Duffing-van der Pol系统为范例,通过对系统的轨道、最大Lyapunov指数、功率谱幅值及Poincar截面的分析,发现适当幅值的有界噪声或混沌信号,一方面可以消除系统对初始条件的敏感依赖性,抑制系统的混沌行为,将系统的混沌吸引子转化为奇怪非混沌吸引子;另一方面也可以诱导系统的混沌行为,将系统的周期吸引子转化为混沌吸引子.从而揭示了非周期力在混沌控制中的双重功效：抑制混沌和诱导混沌. 关键词： 混沌控制 有界噪声 混沌驱动力     

Scaling law and critical exponent for α0 at the 3D Anderson transition

We use high‐precision, large system‐size wave function data to analyse the scaling properties of the multifractal spectra around the disorder‐induced three‐dimensional Anderson transition in order to extract the critical exponents of the transition. Using a previously suggested scaling law, we find that the critical exponent ν is significantly larger than suggested by previous results. We speculate that this discrepancy is due to the use of an oversimplified scaling relation.     

Superpersistent chaotic transients in physical space: advective dynamics of inertial particles in open chaotic flows under noise

Superpersistent chaotic transients are characterized by an exponential-like scaling law for their lifetimes where the exponent in the exponential dependence diverges as a parameter approaches a critical value. So far this type of transient chaos has been illustrated exclusively in the phase space of dynamical systems. Here we report the phenomenon of noise-induced superpersistent transients in physical space and explain the associated scaling law based on the solutions to a class of stochastic differential equations. The context of our study is advective dynamics of inertial particles in open chaotic flows. Our finding makes direct experimental observation of superpersistent chaotic transients feasible. It also has implications to problems of current concern such as the transport and trapping of chemically or biologically active particles in large-scale flows.     

Zurek-Kibble mechanism for the spontaneous vortex formation in Nb-Al/Al(ox)/Nb Josephson tunnel junctions: new theory and experiment

New scaling behavior has been both predicted and observed in the spontaneous production of fluxons in quenched Nb-Al/Al(ox)/Nb annular Josephson tunnel junctions (JTJs) as a function of the quench time, tau(Q). The probability f(1) to trap a single defect during the normal-metal-superconductor phase transition clearly follows an allometric dependence on tau(Q) with a scaling exponent sigma = 0.5, as predicted from the Zurek-Kibble mechanism for realistic JTJs formed by strongly coupled superconductors. This definitive experiment replaces one reported by us earlier, in which an idealized model was used that predicted sigma = 0.25, commensurate with the then much poorer data. Our experiment remains the only condensed matter experiment to date to have measured a scaling exponent with any reliability.     

Noisy one-dimensional maps near a crisis. II. General uncorrelated weak noise

The escape rate for one-dimensional noisy maps near a crisis is investigated. A previously introduced perturbation theory is extended to very general kinds of weak uncorrelated noise, including multiplicative white noise as a special case. For single-humped maps near the boundary crisis at fully developed chaos an asymptotically exact scaling law for the rate is derived. It predicts that transient chaos is stabilized by basically any noise of appropriate strength provided the maximum of the map is of sufficiently large order. A simple heuristic explanation of this effect is given. The escape rate is discussed in detail for noise distributions of Lévy, dichotomous, and exponential type. In the latter case, the rate is dominated by an exponentially leading Arrhenius factor in the deep precritical regime. However, the preexponential factor may still depend more strongly than any power law on the noise strength.     

Statistical properties of a dissipative kicked system: Critical exponents and scaling invariance

A new universal empirical function that depends on a single critical exponent (acceleration exponent) is proposed to describe the scaling behavior in a dissipative kicked rotator. The scaling formalism is used to describe two regimes of dissipation: (i) strong dissipation and (ii) weak dissipation. For case (i) the model exhibits a route to chaos known as period doubling and the Feigenbaum constant along the bifurcations is obtained. When weak dissipation is considered the average action as well as its standard deviation are described using scaling arguments with critical exponents. The universal empirical function describes remarkably well a phase transition from limited to unlimited growth of the average action.     

Noisy one-dimensional maps near a crisis. I. Weak Gaussian white and colored noise

We study one-dimensional single-humped maps near the boundary crisis at fully developed chaos in the presence of additive weak Gaussian white noise. By means of a new perturbation-like method the quasi-invariant density is calculated from the invariant density at the crisis in the absence of noise. In the precritical regime, where the deterministic map may show periodic windows, a necessary and sufficient condition for the validity of this method is derived. From the quasi-invariant density we determine the escape rate, which has the form of a scaling law and compares excellently with results from numerical simulations. We find that deterministic transient chaos is stabilized by weak noise whenever the maximum of the map is of orderz>1. Finally, we extend our method to more general maps near a boundary crisis and to multiplicative as well as colored weak Gaussian noise. Within this extended class of noises and for single-humped maps with any fixed orderz>0 of the maximum, in the scaling law for the escape rate both the critical exponents and the scaling function are universal.