Long-Time Correlations and Expansion-Rate Spectra of Chaos in Hamiltonian Systems

Horita, Takehiko; Hata, Hideaki; Ishizaki, Ryuji; Mori, H.

doi:10.1143/ptp.83.1065

Cited by 47 publications

(23 citation statements)

References 4 publications

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“…The trajectories that at time t > 0 are still travelling in the same direction as at time t = 0 produce two peaks located at the propagation fronts, x = W t and x = −W t, and their contribution to p(x, t) is given by the second term on the right hand side of Eq. (26). The number of trajectories that contribute to the peaks is given by the function I p that has been evaluated in detail by the authors of Ref.…”

Section: Lévy Walkmentioning

confidence: 99%

Lévy scaling: The diffusion entropy analysis applied to DNA sequences

2002

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We address the problem of the statistical analysis of a time series generated by complex dynamics with a new method: the Diffusion Entropy Analysis (DEA) (Fractals, 9, 193 (2001)). This method is based on the evaluation of the Shannon entropy of the diffusion process generated by the time series imagined as a physical source of fluctuations, rather than on the measurement of the variance of this diffusion process, as done with the traditional methods. We compare the DEA to the traditional methods of scaling detection and we prove that the DEA is the only method that always yields the correct scaling value, if the scaling condition applies. Furthermore, DEA detects the real scaling of a time series without requiring any form of de-trending. We show that the joint use of DEA and variance method allows to assess whether a time series is characterized by Lévy or Gauss statistics. We apply the DEA to the study of DNA sequences, and we prove that their large-time scales are characterized by Lévy statistics, regardless of whether they are coding or non-coding sequences. We show that the DEA is a reliable technique and, at the same time, we use it to confirm the validity of the dynamic approach to the DNA sequences, proposed in earlier work. 03.65.Bz,

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Section: Lévy Walkmentioning

confidence: 99%

Lévy scaling: The diffusion entropy analysis applied to DNA sequences

2002

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“…ref. [8]. Conventional wisdom has it that such tori are uncharacteristic of real fluid turbulence, where the aperiodic flows make it more difficult for invariant structures to exist.…”

Section: Description Of the Mapmentioning

confidence: 99%

“…In two or more dimensions, it is similar to, but not identical to, a structure function. The difference arises because the mean gradient g in (8) results from integrating the gradient along a line, whereas the 2D smoothing operator in (9) implies w --fFc(r)(-dy + ;dx),…”

Section: Zt(q) = (Ivci Q ) (9)mentioning

confidence: 99%

Tracer microstructure in the large-eddy dominated regime

Pierrehumbert

1994

Chaos, Solitons & Fractals

255

301

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Abstract--We have re-examined the inertial range behavior of a passive scalar which is advected by a large-scale velocity field causing a cascade of tracer variance to small scales, where it is dissipated by diffusion. This has been done within the context of an idealized model based on mixing by a 2D area-preserving map alternating with a weak diffusion step; the model is a special case of the general advection-diffusion problem. Both freely decaying and forced equilibrium systems were considered. Our main interest in this concerns the validity of Batchelor's theory predicting a k -I tracer variance spectrum, but the tracer microstructure has been diagnosed in terms of concentration probability distribution functions, generalized dimensions of the dissipation field, structure functions, and cancellation exponents. 2D simulations carried out at 10242 resolution show that i;, the decaying case the evolution settles into a 'fractal eigenmode' in which the variance decays exponentially with time at a rate dependent on the Lyapunov exponent but independent of the diffusion coefficient. Although the concentration pattern is self-similar with time, the power spectrum is not algebraic. Concentration PDFs have exponential tails. The dissipation field is not multifractal, and formally has Dq = 2 for all q. The convergence of the squared-gradient PDFs under coarse-graining indicates some underlying fractal behavior, however, and we have introduced the notion of 'fractal degree of freedom' systems to describe such entities. Cancellation exponents and structure functions were also considered, and have a self-similarity which is compatible with a non-intermittent behavior of the dissipation field. These matters have also been addressed for the equilibrium case. The main difference is that the power spectrum of concentration variance in equilibrium exhibits a power-law inertial range, though it is steeper than k -1, but not as steep as k -2. Other features are similar to the decaying case. Very high resolution simulations of the undiffused problem indicate that a k -1 spectrum is approached asymptotically, but only at resolutions corresponding to 106 x 106. The passive scalar behavior is compared and contrasted with the behavior of scalar pseudo-vorticity (an 'active scalar') in the family of generalized 2D turbulence models introduced elsewhere in this issue.

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“…On the other hand, the running diffusion coefficient, defined as in equation (30) with an upper integration limit equal to t, rather than ∞, appears to be identical to the result derived from equation (24) for all times, and reduces to (27) …”

Section: The Hybrid Kinetic Equationmentioning

confidence: 52%

“…Orbits starting in the neighbourhood of such a mode are carried away very far before decorrelating. It has been shown in detail that such a behaviour leads to a superdiffusive regime with D → ∞ for certain ranges of K (corresponding to the stability domains of the accelerator modes) [26], [27]. This explains quite well the presence of the peaks (singularities) in figure 5.…”

Section: Chaotic Hamiltonian Dynamical Systemsmentioning

confidence: 63%

Strange Diffusion

Balescu¹

1998

Condens. Matter Phys.

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Strange diffusion is defined as a process in which the mean square displacement of a randomly walking test particle behaves asymptotically as x 2 (t) ∼ t α , with α =1 . A brief review of the properties of such processes is presented, stressing their deep difference from the corresponding normal diffusive evolution. A number of examples are discussed, including diffusion of charged particles in a fluctuating magnetic field, continuous time random walks and transport in chaotic Hamiltonian systems.

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Long-Time Correlations and Expansion-Rate Spectra of Chaos in Hamiltonian Systems

Abstract: 1065The repeated sticking of a chaotic orbit to critical tori with an inverse· power distribution of sticking times is shown to produce a universal spectrum of expansion rates A of nearby orbits with a linearity with zero slope for O

Cited by 47 publications

References 4 publications

Lévy scaling: The diffusion entropy analysis applied to DNA sequences

Lévy scaling: The diffusion entropy analysis applied to DNA sequences

Tracer microstructure in the large-eddy dominated regime

Strange Diffusion

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