Lyapunov Modes in Hard-Disk Systems

Eckmann, Jean‐Pierre; Forster, Christina; Posch, Harald A.; Zabey, Emmanuel

doi:10.1007/s10955-004-2687-4

Cited by 49 publications

(119 citation statements)

References 25 publications

Supporting

Mentioning

118

Contrasting

Order By: Relevance

“…However, only the first two of these (1) are observed (as the two-point steps or transverse modes) [6,8]. The second two modes are longi-tudinal and we notice that the last two basis vectors (3) have time dependent normalisation coefficients.…”

mentioning

confidence: 81%

“…These steps in the spectrum are accompanied by global wavelike structures in the corresponding Lyapunov vectors, or Lyapunov modes [5,6,7,8]. The significance of this phenomenon is that this structure appears in the vectors associated with the Lyapunov exponents that are closest to zero, therefore it is connected with the slow macroscopic behavior of the system.…”

mentioning

confidence: 99%

“…It follows from these equations and the assumption that ψ 2 (t) exp{iω lya t} ∼ ψ 1 (t) * ψ 1 (0) φ(t) exp{i(ω acf − ω lya )t}, that ω lya ∼ ω acf /2. The frequency is inversely proportional to the oscillation period, so we obtain relation (8).…”

mentioning

confidence: 99%

See 2 more Smart Citations

Time-Oscillating Lyapunov Modes and the Momentum Autocorrelation Function

Taniguchi

Morriss

2005

Phys. Rev. Lett.

View full text Add to dashboard Cite

The time-dependent structure of the Lyapunov vectors corresponding to the steps of Lyapunov spectra and their basis set representation are discussed for a quasi-one-dimensional many-hard-disk systems. Time-oscillating behavior is observed in two types of Lyapunov modes, one associated with the time translational invariance and another with the spatial translational invariance, and their phase relation is specified. It is shown that the longest period of the Lyapunov modes is twice as long as the period of the longitudinal momentum auto-correlation function. A simple explanation for this relation is proposed. This result gives the first quantitative connection between the Lyapunov modes and an experimentally accessible quantity.

show abstract

mentioning

confidence: 81%

mentioning

confidence: 99%

See 1 more Smart Citation

Time-Oscillating Lyapunov Modes and the Momentum Autocorrelation Function

Taniguchi

Morriss

2005

Phys. Rev. Lett.

View full text Add to dashboard Cite

show abstract

“…, t. If t 0 ≫ 1 we can assume these CLVs to be perfectly converged. Now repeat the backward procedure starting for some randomly chosen vector at time (44) and evolve backward up to time n. This procedure, allows to study the convergence towards v (i) n of the "approximate" CLV u (i) n (k) as a function of the finite backward evolution time k on which the vector depends. Let us define δu…”

Section: Convergence Towards the Covariant Lyapunov Vectorsmentioning

confidence: 99%

Covariant Lyapunov vectors

Ginelli¹,

Chaté²,

Livi³

et al. 2013

J. Phys. A: Math. Theor.

140

View full text Add to dashboard Cite

Abstract. The recent years have witnessed a growing interest for covariant Lyapunov vectors (CLVs) which span local intrinsic directions in the phase space of chaotic systems. Here we review the basic results of ergodic theory, with a specific reference to the implications of Oseledets' theorem for the properties of the CLVs. We then present a detailed description of a "dynamical" algorithm to compute the CLVs and show that it generically converges exponentially in time. We also discuss its numerical performance and compare it with other algorithms presented in literature. We finally illustrate how CLVs can be used to quantify deviations from hyperbolicity with reference to a dissipative system (a chain of Hénon maps) and a Hamiltonian model (a Fermi-Pasta-Ulam chain).

show abstract

“…From this point of view, Lyapunov exponents with small absolute values are connected to large and macroscopic timescale behavior of many-body systems, and in this region the wave-like structure of Lyapunov vectors, known as the Lyapunov modes, is observed [14,15,16,17,18]. The Lyapunov mode is a reflection of a collective movement (phonon mode) of many-body systems, and comes from dynamical conservation laws and translational invariance [18,19,20,21]. On the other hand, large Lyapunov exponents are dominated by small and microscopic time-scale movement, and in this region the spatially localized behavior of Lyapunov vector, the so-called Lyapunov localization, appears [22,23,24,25,26,27,28].…”

Section: Introductionmentioning

confidence: 99%

Dynamics of strongly localized Lyapunov vectors in many-hard-disk systems

Taniguchi

Morriss

2006

Phys. Rev. E

View full text Add to dashboard Cite

The dynamics of the localized region of the Lyapunov vector for the largest Lyapunov exponent is discussed in quasi-one-dimensional hard-disk systems at low density. We introduce a hopping rate to quantitatively describe the movement of the localized region of this Lyapunov vector, and show that it is a decreasing function of hopping distance, implying spatial correlation of the localized regions. This behavior is explained quantitatively by a brick accumulation model derived from hard-disk dynamics in the low density limit, in which hopping of the localized Lyapunov vector is represented as the movement of the highest brick position. We also give an analytical expression for the hopping rate, which is obtained us a sum of probability distributions for brick height configurations between two separated highest brick sites. The results of these simple models are in good agreement with the simulation results for hard-disk systems.

show abstract

Lyapunov Modes in Hard-Disk Systems

Cited by 49 publications

References 25 publications

Time-Oscillating Lyapunov Modes and the Momentum Autocorrelation Function

Time-Oscillating Lyapunov Modes and the Momentum Autocorrelation Function

Covariant Lyapunov vectors

Dynamics of strongly localized Lyapunov vectors in many-hard-disk systems

Contact Info

Product

Resources

About