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Effects of External Noise on Anomalous Diffusion in Hamiltonian Dynamical Systems
Abstract
The effect of external noise on diffusion processes in a Hamiltonian system with an external force is studied. This is represented
by the standard map with an external force. When external noise is applied to the system, we find strongly enhanced diffusion
in a parameter range in which anomalous diffusion occurs. Although chaotic orbits are not deterministic in a strict sense,
because the system includes noises, they exhibit Lévy flight due to the intermittent sticking to transient tori.
... For large noise 1/ξ → 0 the exponent evolves as γ → 1 because inner cores become blurred by noise and diffusion is dominant. For intermediate noise intensities some particles intermittently enter within the inner cores where they may remain for some time, while others travel long distances and this causes R 2 (t) to increase superdiffusively [16]. The very high values of γ up to 10 for intermediate noise intensities in Fig. 2(b) can be understood looking at the two curves (red squares and green triangles) in Fig. 2(a). ...
... Above a certain nonzero value of noise intensity the closed flow regimes become permeable to particle crossings which explains the presence of a maximum asymptotic diffusion coefficient. For intermediate noise intensities, D A ∼ ξ −α and the transport becomes superdiffusive [9,10,16]. However, as an important result we find that this dependence does not hold for ξ → 0. The scaling coefficient α was found to decrease with the jet amplitude. ...
The nontrivial dependence of the asymptotic diffusion on noise intensity has been studied for a Hamiltonian flow mimicking the Gulf Jet Stream. Three different diffusion regimes have been observed depending on the noise intensity. For intermediate noise the asymptotic diffusion decreases with noise intensity at a rate which is linearly dependent to the flow's meander amplitude. Increasing the noise the fluid transport passes through a superdiffusive regime and finally becomes diffusive again at large noise intensities. The presence of inner circulation regimes in the flow has been found to be determinant to explain the observed behavior.
... Since then many different applications and consequences of stickiness were investigated. The onset of anomalous transport is perhaps the most prominent one [Kar83,ISM00,ZT91,ZK94], what will be discussed carefully in Sec. 6.2. ...
Stickiness refers to chaotic orbits that stay in a particular region for a long time before escaping. For example, stickiness appears near the borders of an island of stability in the phase space of a 2-D dynamical system. This is pronounced when the KAM tori surrounding the island are destroyed and become cantori (see [Contopoulos, 2002]). We find the time scale of stickiness along the unstable asymptotic curves of unstable periodic orbits around an island of stability, that depends on several factors: (a) the largest eigenvalue |λ| of the asymptotic curve. If λ > 0 the orbits on the unstable asymptotic manifold in one direction (fast direction) escape faster than the orbits in the opposite direction (slow direction) (b) the distance from the last KAM curve or from the main cantorus (the cantorus with the smallest gaps) (c) the size of the gaps of the main cantorus and (d) the other cantori, islands and asymptotic curves. The most important factor is the size of the gaps of the main cantorus. Then we find when the various KAM curves are destroyed. The distance of the last KAM curve from the center of an island gives the size of the island. When the central periodic orbit becomes unstable, chaos is also formed around it, limited by a first KAM curve. Between the first and the last KAM curves there are still closed invariant curves. The sizes of the islands as functions of the perturbation, have abrupt changes at resonances. These functions have some universal features but also some differences. A new type of stickiness appears near the unstable asymptotic curves of unstable periodic orbits that extend far into the large chaotic sea. Such a stickiness lasts for long times, increasing the density of points close to the unstable asymptotic curves. However after a much longer time, the density becomes almost equal everywhere outside the islands of stability. We consider also stickiness near the asymptotic curves from new periodic orbits, and stickiness in Anosov systems and near totally unstable orbits. In systems that allow escapes to infinity the stickiness delays the escapes. An important astrophysical application is the case of barred-spiral galaxies. The spiral arms outside corotation consist mainly of sticky chaotic orbits. Stickiness keeps the spiral forms for times longer than a Hubble time, but after a much longer time most of the chaotic orbits escape to infinity.
We investigate the high-dimensional Hamiltonian chaotic dynamics in N coupled area-preserving maps. We show the existence of an enhanced trapping regime caused by trajectories performing a random walk inside the area corresponding to regular islands of the uncoupled maps. As a consequence, we observe long intermediate regimes of power law decay of the recurrence time statistics (with exponent γ = 0.5) and of ballistic motion. The asymptotic decay of correlations and anomalous diffusion depend on the stickiness of the N-dimensional invariant tori. Detailed numerical simulations show weaker stickiness for increasing N suggesting that such paradigmatic class of Hamiltonian systems asymptotically fulfill the demands of the usual hypotheses of strong chaos.
IntroductionContinuous Time RandomWalk ModelOrigin of Long-Time Tails in Hamiltonian SystemParadigmatic Fluid ModelDiscussionReferences
Static properties of lattices of Josephson junctions in an applied magnetic field h and with an external current gamma are studied. Effective stability criteria for the bound states of the magnetic flux and for emerging chaotic distributions of the magnetic field are suggested. It is shown that only stable periodic states survive when gamma is large and that they exist in narrow intervals of the applied magnetic field when conditions for emerging chaotic distributions are met. Thus we predict regularly distributed sharp peaks on a low chaotic background in the dependence of the critical external current gammac on h.
The Josephson map describes the nonlinear dynamics of systems characterized by the standard map with a uniform external bias superposed. The intricate structures of the phase-space portrait of the Josephson map are examined here on the basis of the associated tangent map. A numerical investigation of stochastic diffusion in the Josephson map is compared with the renormalized diffusion coefficient calculated using the characteristic function. The global stochasticity of the Josephson map occurs at far smaller values of the stochastic parameter than is the case of the standard map.
The authors analyze iterations of maps on an interval with an added noise term, in the neighbourhood of an intermittency threshold. They rigorously derive a universal scaling function for the laminar time expressed as a function of the distance from the threshold and the variance of the noise.
We investigate the dynamics generated from iterated maps and analyze the motion in terms of the probabilistic continuous-time random-walk (CTRW) approach. Two different CTRW models are considered: (i) Particles jump between sites (turning points) or (ii) particles move at a constant velocity between sites and choose a new direction at random. For both models we study the mean-squared displacement and the propagator P(r,t), the probability to be at location r at time t having started at the origin at t=0. Iterated maps are used to generate both dispersive and enhanced diffusion and the results are analyzed using the CTRW framework and scaling arguments. For the case of dispersive motion we discuss the problem of the stationary state and point out its relevance.
We study the interplay between a deterministic process of weak chaos, responsible for the anomalous diffusion of a variable x, and a white noise of intensity Ξ. The deterministic process of anomalous diffusion results from the correlated fluctuations of a statistical variable ξ between two distinct values +1 and -1, each of them characterized by the same waiting time distribution ψ(t), given by ψ(t)t-μ with 2<μ<3, in the long-time limit. We prove that under the influence of a weak white noise of intensity Ξ, the process of anomalous diffusion becomes normal at a time tc given by tc∼1/Ξβ(μ). Here β(μ) is a function of μ which depends on the dynamical generator of the waiting time distribution ψ(t). We derive an explicit expression for β(μ) in the case of two dynamical systems, a one-dimensional superdiffusive map and the standard map in the accelerating state. The theoretical prediction is supported by numerical calculations. (c) 1995 The American Physical Society
The effect of noise on systems which undergo period-doubling transitions to chaos is studied. With the aid of nonequilibrium field-theoretic techniques, a correlation-function expression for the Lyapunov parameter (which describes the sensitivity of the system to initial conditions) is derived and shown to satisfy a scaling theory. Since these transitions have previously been shown to exhibit universal behavior, this theory predicts universal effects for the noise. These predictions are in good agreement with numerical experiments.
Experimental studies are made of chaotic particle motion in a simple Stokes flow system. Surfaces-of-section that exhibit a generic mixture of regular and chaotic particle motions in good agreement with computer simulations are constructed experimentally. Deformations of line elements exhibit `whorl' and `tendril' structures of great complexity that can be correlated directly with the underlying particle dynamics and that agree well with numerical computations. In some cases, the laboratory studies are able to resolve dynamical features more accurately than the computer studies. Experiments demonstrating that the flows exhibit poor time reversal in regimes of chaotic particle motion are also performed.
The diffusion of chaotic orbits in a widespread chaotic sea is normal with a variance proportional to the time t even when islands of tori exist as long as there are no accelerator-mode islands. If an external driving force is applied,
however, the diffusion becomes anomalous with a variance tξ, ξ> 1. Indded, if one uses the coordinate system moving with the mean velocity of chaotic orbits due to the driving force,
the chaotic orbits become identical to the Lévy flight due to the intermittent sticking to the islands of tori. This is shown
numerically using the standard map with an external force, i.e., the Josephson map. The probability distribution function
of the coarse-grained velocity is determined explicitly and turns out to obey an anomalous scaling law characterized by the
exponent ξ.
Diffusion of chaotic orbits occurs in the widespread chaotic region of the standard map. Its conventional description by the diffusion constant breaks down if accelerator mode islands exist. We shall explore a new description of this anomalous diffusion due to the accelerator mode islands numerically and theoretically by use of Feller's theorem of recurrent events, and find that the probability distribution function P(v ; n) of the coarse-grained velocity v_{n} equiv (J_{n} - J_{0})/n for action J_{t} at time t = 0, 1, 2, obeys a scaling law P(v ; n) = n(delta}hat{p}(n({delta)) v) for large n with 1/2 > delta > 0, which has an inverse-power asymptotic form hat{p}(x)~|x|(-(1+beta)) for |x| -> infty with 2 > beta = 1/(1 - delta) > 1.
Irrational decimation schemes are constructed for functional integrals with the very interesting feature of preserving several distinct site Hamiltonians into the fixed-point limit. This method is applied to the quasiperiodicity transition to turbulence in order to compute the effects of external noise.
A new type of universality associated with the onset of a deterministic diffusion for systems described by iterative one-dimensional maps is reported. The diffusion coefficient D plays the role of an order parameter with a universal critical exponent. For the presence of external noise, the existence of a universal scaling function d is shown. An analytic expression is derived for d which is in good agreement with results of a numerical experiment.
The effect of external noise on the transition to chaos for maps of the interval which exhibit period-doubling bifurcations are considered. It is shown that the Liapunov characteristic exponent satisfies scaling in the vicinity of the transition. The critical exponent for noise is calculated with the use of Feigenbaum's renormalization group approach, and the scaling function for the Liapunov characteristic exponent is obtained numerically by iterating a map with additive noise.
Exact solutions to the Feigenbaum renormalization-group recursion relation, and the associated eigenvalue equations describing deterministic as well as stochastic perturbations, are found for the case of intermittency. These solutions are generated by a reformulation of the one-dimensional iterated map that exploits its topological equivalence to a translation. Direct resummation of series expansions gives the same results.
The aperiodic or chaotic behavior for one-dimensional maps just before a tangent bifurcation occurs appears as intermittency in which long laminarlike regions irregularly separated by bursts occur. Proceeding from the picture proposed by Pomeau and Manneville, numerical experiments and analytic calculations are carried out on various models exhibiting this behavior. The behavior in the presence of external noise is analyzed, and the case of a general power dependence of the curve near the tangent bifurcation is studied. Scaling relations for the average length of the laminar regions and deviations from scaling are determined. In addition, the probability distribution of path lengths, the stationary distribution of the maps, the correlation function and power spectrum of the map in the intermittent region, and the Lyapunov exponent are obtained.
A compact path-diagram method has been introduced for the calculation of velocity moments of a probability function. This method is complementary to the approach developed earlier by Rechester and White. It is applied to the Chirikov-Taylor model. Analytic expressions for velocity-space diffusion have been derived and compared with numerical computations. A numerical method for path summations has been developed which is more efficient than directly advancing the model equations, and is applicable for small field-amplitude values, where the direct stepping method is impractical.
Onset behavior of diffusion generated by a chaotic motion of a one-particle system in a sinusoidal potential, driven by a time-dependent external field, is investigated, especially near the Pomeau-Manneville intermittency transition. It is numerically found that the diffusion coefficient diverges as the system approaches the intermittency transition. A possible interpretation of such anomalous enhancement of the diffusion coefficient is proposed in connection with the appearance of a temporally long coherence of the phase point.
Stochastic motion in nonlinear oscillator systems is examined from a
primarily physical perspective, focusing on intrinsic stochasticity in
Hamiltonian systems and taking the effects of noise into account. The
analysis approach is justified and demonstrated using numerical methods.
Chapters are devoted to the introduction of basic concepts, canonical
perturbation theory, mappings and linear stability, the transition to
global stochasticity, stochastic motion and diffusion, systems with
three or more degrees of freedom, and dissipative systems. Applications
(to planetary motion, accelerators and beams, charged-particle
confinement, chemical dynamics, and quantum systems) and Hamiltonian
bifurcation theory are treated briefly in apendices.
Diffusive processes in one-dimensional discrete chaotic systems are
considered. Drift and diffusion coefficients are calculated, which show
critical behavior including logarithmic corrections. A Kubo formula for
the diffusion coefficient in terms of the time-correlation function is
given. Nondiffusive states may exist in certain parameter windows
showing drift with broken symmetry or strict localization.
Period-doubling bifurcations and states of periodic chaos describing
chaotic but nondiffusive drift occur.
In the Lagrangian representation, the problem of advection of a passive marker particle by a prescribed flow defines a dynamical system. For two-dimensional incompressible flow this system is Hamiltonian and has just one degree of freedom. For unsteady flow the system is non-autonomous and one must in general expect to observe chaotic particle motion. These ideas are developed and subsequently corroborated through the study of a very simple model which provides an idealization of a stirred tank. In the model the fluid is assumed incompressible and inviscid and its motion wholly two-dimensional. The agitator is modelled as a point vortex, which, together with its image(s) in the bounding contour, provides a source of unsteady potential flow. The motion of a particle in this model device is computed numerically. It is shown that the deciding factor for integrable or chaotic particle motion is the nature of the motion of the agitator. With the agitator held at a fixed position, integrable marker motion ensues, and the model device does not stir very efficiently. If, on the other hand, the agitator is moved in such a way that the potential flow is unsteady, chaotic marker motion can be produced. This leads to efficient stirring. A certain case of the general model, for which the differential equations can be integrated for a finite time to produce an explicitly given, invertible, area-preserving mapping, is used for the calculations. The paper contains discussion of several issues that put this regime of chaoticadvection in perspective relative to both the subject of turbulent advection and to recent work on critical points in the advection patterns of steady laminar flows. Extensions of the model, and the notion of chaotic advection, to more realistic flow situations are commented upon.
The large-scale motion of one-dimensional discrete systemX
t+1=X
t+f
a(X
t), (t=0, 1,2,...;f
a(X+1)=f
a(X)) is studied. This motion can be asymptotically decoposed into a drift and a diffusion (chaos-induced diffustion). We derive the formulae for the drift velocityv as the average jump number per step and for the diffusion coefficientD in terms of the jump number's time correlation function. It is shown that the coarsegrained probability distribution is asymptotically gaussian. Considering piecewise linear models and the sinusoidal one, we study the onset behavior of diffusion and its gross behavior. It is found thatD is proportional to the length of the region with a non-zero jump number, if the critical dynamics is well-defined. Otherwise we have logarithmic or inverse power corrections to the simple law originating from an intermittency type behavior or from band splitting phenomena. Maps with a smooth maximum (as e.g. the sinusoidal) exhibit several additional types of trajectories: running modes with broken symmetry, localized trajectories, regular periodic or chaotic, non-diffusive or diffusive ones. These dynamical states appear in a nested window structure, which is described.
A numerical observation of stochastic diffusion in the standard map has been carried out in the domain of small stochastic parameter Ac<A<1.0, where Ac is given as (2π)−1×0.9716 = 0.1546. Multiple periodic accelerator modes manifest their pronounced sharp resonant effect in the stochastic diffusion process even in the region of small stochastic parameter, where the fundamental accelerator modes are not allowed to exist.
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.
The transport of passive impurities in nearly two-dimensional, time-periodic Rayleigh-Bénard convection is studied experimentally and numerically. The transport may be described as a one-dimensional diffusive process with a local effective diffusion constant D*(x) that is found to depend linearly on the local amplitude of the roll oscillation. The transport is independent of the molecular diffusion coefficient and is enhanced by 1–3 orders of magnitude over that for steady convective flows. The local amplitude of oscillation shows strong spatial variations, causing D*(x) to be highly nonuniform. Computer simulations of a simplified model show that the basic mechanism of transport is chaotic advection in the vicinity of oscillating roll boundaries. Numerical estimates of D* are found to agree semiquantitatively with the experimental results. Chaotic advection is shown to provide a well-defined transition from the slow, diffusion-limited transport of time-independent cellular flows to the rapid transport of turbulent flows.
Continuous-time sporadic dynamics that are intermediate between regular and random behaviors are discussed. A characterization of such processes is provided by a scale-dependent entropic quantity, and is applied to a model of Levy motion introduced by Klafter et al. (1987). The study suggests that sporadicity can be a feature of some physical systems exhibiting anomalous diffusion.
The phase space for Hamiltonians of two degrees of freedom is usually divided into stochastic and integrable components. Even when well into the stochastic regime, integrable orbits may surround small stable regions or islands. The effect of these islands on the correlation function for the stochastic trajectories is examined. Depending on the value of the parameter describing the rotation number for the elliptic fixed point at the center of the island, the long-time correlation function may decay as t^-5 or exponentially, but more commonly it decays much more slowly (roughly as t^-1). As a consequence these small islands may have a profound effect on the properties of the stochastic orbits. In particular, there is evidence that the evolution of a distribution of particles is no longer governed by a diffusion equation.
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