Quartic polynomial approximation for fluctuations of separation of trajectories in chaos and correlation dimension

Abstract

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(k) does not violate the Marcinkiewicz theorem on invalidity of polynomial form for the generating function. We propose that this quartic approximation will fit many experimental situations, not having the effective time-reversibility and the short correlation time properties of the quadratic Grassberger-Procaccia estimates. We take the existing γ(k) for turbulent channel flow and demonstrate that the quartic fit is nearly perfect. The violation of time-reversibility for the Lagrangian trajectories of the incompressible Navier-Stokes turbulence below the viscous scale is considered. We demonstrate how the fit can be used for finding the correlation dimensions of strange attractors via easily measurable quantities. We provide a simple formula via the Lyapunov exponents, holding in quadratic approximation, and describe the construction of the quartic approximation. A different approximation scheme for finding the correlation dimension from expansion in the flow compressibility is also provided.

Reynolds number dependence of Lyapunov exponents of turbulence and fluid particles

“…(We remark that truncation of the series for γ(k) does not contradict Pawula's theorem [14].) This assumption is made stronger by the observations of [5] for the Lyapunov exponent of turbulence λ v 1 , considered in more detail later.…”

“…Cumulant series and realization independence of γ(k)-A representation of γ(k) as series in cumulants was introduced in [14]. Using the cumulant expansion theorem [28] we obtain:…”

“…( 18) is well-defined. In particular, we mention its salient properties (see [14] and references therein): This function is convex, has two zeros at k = −3 and k = 0, is negative for −3 < k < 0 and positive elsewhere. Note also γ ′ (0) = λ 1 > 0 and γ ′ (−3) = λ 3 < 0.…”

“…This system is characterized by a positive Lyapunov exponent λ 1 that describes exponential growth of distance r between two trajectories below the viscous scale [9], the phenomenon often referred to as the Lagrangian chaos, see e.g. [10][11][12][13][14].…”

“…The first type of intermittency is well-known, see e.g. the detailed discussion in [14]. It does not have anything to do with the intermittency of turbulence.…”