Fluctuation-diffusion relationship in chaotic dynamics

2004

Physical Review E - PHYS REV E, 2003

Results are reported concerning the transition to chaos in random dynamical systems. In particular, situations are considered where a periodic attractor coexists with a nonattracting chaotic saddle, which can be expected in any periodic window of a nonlinear dynamical system. Under noise, the asymptotic attractor of the system can become chaotic, as characterized by the appearance of a positive Lyapunov exponent. Generic features of the transition include the following: (1) the noisy chaotic attractor is necessarily nonhyperbolic as there are periodic orbits embedded in it with distinct numbers of unstable directions (unstable dimension variability), and this nonhyperbolicity develops as soon as the attractor becomes chaotic; (2) for systems described by differential equations, the unstable dimension variability destroys the neutral direction of the flow in the sense that there is no longer a zero Lyapunov exponent after the noisy attractor becomes chaotic; and (3) the largest Lyapu...

Physics of Fluids - PHYS FLUIDS, 1991

The statistics of systems with good chaotic properties obey a formal fluctuation-response relation which gives the average linear response of a dynamical system to an external perturbation in terms of two-time correlation functions. Unfortunately, except for particularly simple cases, the appropriate form of correlation function is unknown because an analytic expression for the invariant density is lacking. The simplest situation is that in which the probability distribution is Gaussian. In that case, the fluctuation-response relation is a linear relation between the response matrix and the two-time two-point correlation matrix. Some numerical computations have been carried out in low-dimensional models of hydrodynamic systems. The results show that fluctuation-response relation for Gaussian distributions is not a useful approximation. Nevertheless, these calculations show that, even for non-Gaussian statistics, the response function and the two-time correlations can have similar qu...

2011

Abstract We analyse deterministic diffusion in a simple, one-dimensional setting consisting of a family of four parameter dependent, chaotic maps defined over the real line. When iterated under these maps, a probability density function spreads out and one can define a diffusion coefficient. We look at how the diffusion coefficient varies across the family of maps and under parameter variation.

Physics Reports, 1982

IEEE Transactions on Circuits and Systems, 1983

Abs~act-Chaotic motion refers to complicated trajectories in dynamical systems. It occurs even in deterministic systems governed by simple differential equations and its presence has been experimentally verified for many systems in several disciplines.-A technique-due to Melnikov provides an analytical tool for measuring chaos caused by horseshoes in certain systems.. The .phenomenon .of .,Amold diffusion is another type of complicated behavior. Since 1964, it has been playing an important role for Hamiltonian systems in physics. We present a tutorial treatment of this work and its place in dynamical systems theory, with an emphasis on results that can be checked in specific systems. A generalization of the Melnikov technique has been recently developed to treat n-degree of freedom Hamiltonian systems when n > 3. We extend the Melnikov technique to certain non-Hamiltonian systems of ordinary differential equations. The extension is made with a view to applications in the physical sciences and engineering.

Journal of Physics A-mathematical and General, 1997

We study some new universal aspects of diffusion in chaotic systems, especially such having very large Lyapunov coefficients on the chaotic (indecomposable, topologically transitive) component. We do this by discretizing the chaotic component on the Surface-of-Section in a (large) number $N$ of simplectically equally big cells (in the sense of equal relative invariant ergodic measure, normalized so that the total measure of the chaotic component is unity). By iterating the transition of the chaotic orbit through SOS, where $j$ counts the number of iteration (discrete time), and assuming complete lack of correlations even between consecutive crossings (which can be justified due to the very large Lyapunov exponents), we show the universal approach of the relative measure of the occupied cells, denoted by $\rho(j)$, to the asymptotic value of unity, in the following way: $\rho(j) = 1 - (1-\frac{1}{N})^j$, so that in the limit of big $N$, $N\to \infty$, we have, for $j/N$ fixed, the exponential law $\rho(j) \approx 1 - \exp (-j/N)$. This analytic result is verified numerically in a variety of specific systems: For a plane billiard (Robnik 1983, $\lambda=0.375$), for a 3-D billiard (Prosen 1997, $a=-1/5, b=-12/5$), for ergodic logistic map (tent map), for standard map ($k=400$) and for hydrogen atom in strong magnetic field ($\epsilon=-0.05$) the agreement is almost perfect (except, in the latter two systems, for some long-time deviations on very small scale), but for H\'enon-Heiles system ($E=1/6$) and for the standard map ($k=3$) the deviations are noticed although they are not very big (only about 1%). We have tested the random number generators (Press et al 1986), and confirmed that some are almost perfect (ran0 and ran3), whilst two of them (ran1 and ran2) exhibit big deviations.

Journal of Physics A: Mathematical and General, 1990

The diffusion process of Hamiltonian map lattice models is numerically studied. For weak non-integrability, the diffusion coefficient of the model has stretched exponential dependence on the non-integrability, which is consistent with Nekhoroshev's bound. Up to a certain size, the exponent of the stretched exponential decreases with the system size, showing that diffusion is enhanced with the increase of the size. As the size gets larger than the correlation length the exponent approaches a finite value. The diffusion coefficient of a model with global interaction is also studied. It again is enhanced with the system size.

Physica D: Nonlinear Phenomena, 2001

We examine some new diagnostics for the behavior of a field ρ evolving in an advective-diffusive system. One of these diagnostics is approximately the Fourier second moment (denoted as χ 2 ) and the other is the linear entropy or field intensity S, the latter being significantly easier to compute or measure. We establish that as a result of chaos the increasing structure in ρ is accompanied by χ increasing exponentially rapidly in time at a rate given by ρ-dependent Lyapunov exponents Λ i and dominated by the largest one Λ max . Noise or diffusive coarse-graining of ρ causes χ to decrease as χ 2 ≈ 1 4 Dt, where D is a measure of the diffusion. When both effects are present the competition between the processes leads to metastability for χ followed by a final diffusive tail. The initial stages may be chaotic or diffusive depending upon the value of Λ −1 max 2Dχ 2 (0) but the metastable value of χ 2 is given by χ 2 * = i Λ i /2D irrespective. SinceṠ = −2Dχ 2 , similar analysis applies to S, and in particular there exists a metastable decay rate for S given byṠ * = i Λ i . These arguments are verified for a simple case, the Arnol'd Cat Map with added diffusive noise.

2019

We consider the cumulant generating function of the logarithm of the distance between two infinitesimally close trajectories of a chaotic system. Its long-time behavior is given by the generalized Lyapunov exponent γ(k) providing the logarithmic growth rate of the k-th moment of the distance. The Legendre transform of γ(k) is a large deviations function that gives the probability of rare fluctuations where the logarithmic rate of change of the distance is much larger or much smaller than the mean rate defining the first Lyapunov exponent. The only non-trivial zero of γ(k) is at minus the correlation dimension of the attractor which for incompressible flows reduces to the space dimension. We describe here general properties constraining the form of γ(k) and the Gallavotti-Cohen type relations that hold when there is symmetry under time-reversal. This demands studying joint growth rates of infinitesimal distances and volumes. We demonstrate that quartic polynomial approximation for γ(...