Structure of characteristic Lyapunov vectors in anharmonic Hamiltonian lattices

Romero-Bastida, M.; Pazó, Diego; López, Juan M.; Rodríguez, Miguel A.

doi:10.1103/physreve.82.036205

Cited by 22 publications

(27 citation statements)

References 48 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This does not occur all the time but in an intermittent manner. Interestingly, these structural features-namely, replication and clustering-have recently been reported to occur generically for LVs in chaotic spatially extended systems [23][24][25]. This deepens in the analogy between DDSs and systems with extensive chaos in one dimension, which happens to hold even at the level of non-leading LVs.…”

Section: Long-range Correlations Of Lyapunov Vectorsmentioning

confidence: 98%

“…The fact that the LV surfaces obey scaling laws and that systems with spatiotemporal chaos could be divided into a few universality classes, according to the scaling of the associated surfaces, has implications in our understanding of chaos in extended systems from a statistical physics point of view. Moreover, the generic replication and clustering of characteristic LVs along the main vector direction in extended systems [23][24][25] (also observed here for time-delay systems, see previous section) makes it even more interesting to determine the universality class of the main LV, since this is expected to provide a great deal of information about the structure of space and time correlations of the LV corresponding to leading as well as sub-leading unstable directions.…”

Section: Main Lyapunov Vector and The Universality Class Questionmentioning

confidence: 99%

See 1 more Smart Citation

Characteristic Lyapunov vectors in chaotic time-delayed systems

Pazó

López

2010

Phys. Rev. E

Self Cite

View full text Add to dashboard Cite

We compute Lyapunov vectors (LVs) corresponding to the largest Lyapunov exponents in delay-differential equations with large time delay. We find that characteristic LVs, and backward (Gram-Schmidt) LVs, exhibit long-range correlations, identical to those already observed in dissipative extended systems. In addition we give numerical and theoretical support to the hypothesis that the main LV belongs, under a suitable transformation, to the universality class of the Kardar-Parisi-Zhang equation. These facts indicate that in the large delay limit (an important class of) delayed equations behave exactly as dissipative systems with spatiotemporal chaos.

show abstract

Section: Long-range Correlations Of Lyapunov Vectorsmentioning

confidence: 98%

Section: Main Lyapunov Vector and The Universality Class Questionmentioning

confidence: 99%

Characteristic Lyapunov vectors in chaotic time-delayed systems

Pazó

López

2010

Phys. Rev. E

Self Cite

View full text Add to dashboard Cite

show abstract

“…Two identical chaotic systems, one described with MKS units and the other with cgs units exhibit the same (time-averaged) rates of divergence even though the mass and length scales differ. It is also possible, usual, and useful to define "local" or "instantaneous" Lyapunov exponents by following two or more constrained trajectories and measuring their tendencies to separate or approach each other as a function of the time of measurement [1][2][3][4][5][6][7][8][9]15]. The MKS and cgs values of these local exponents differ.…”

Section: Local and Global Gram-schmidt Covariant Vectors And Exponmentioning

confidence: 99%

“…We reïterate that this symmetry can be violated, for short times, in response to inhomogeneities or to "external perturbations". [15] By now many groups [1][2][3][4][5][6] have illustrated the algebraic steps necessary to map the "covariant" exponents from one coordinate system to another. A careless reader of some of this work might well conclude (as we did) that "covariant" vectors and exponents are somehow coordinate-frame independent.…”

Section: Local and Global Gram-schmidt Covariant Vectors And Exponmentioning

confidence: 99%

Why Instantaneous Values of the “Covariant” Lyapunov Exponents Depend upon the Chosen State-Space Scale

Hoover¹,

Hoover²

2014

CMST

View full text Add to dashboard Cite

Abstract:We explore a simple example of a chaotic thermostated harmonic-oscillator system which exhibits qualitatively different local Lyapunov exponents for simple scale-model constant-volume transformations of its coordinate q and momentum p: {q, p} → {(Q/s), (sP )}. The time-dependent thermostat variable ζ(t) is unchanged by such scaling. The original (qpζ) motion and the scale-model (QP ζ) version of the motion are physically identical. But both the local Gram-Schmidt Lyapunov exponents and the related local "covariant" exponents change with the change of scale. Thus this model furnishes a clearcut chaotic time-reversible example showing how and why both the local Lyapunov exponents and covariant exponents vary with the scale factor s. Key words: Lyapunov Instability, Thermostats, Chaotic Dynamics I. LOCAL AND GLOBAL GRAM-SCHMIDT COVARIANT VECTORS AND EXPONENTSThe popularity of the time-dependent (or "local", or "instantaneous") covariant Lyapunov vectors and their associated exponents as descriptions of chaotic motion seems to us to be linked to a (false) impression extracted from the literature. Some of the literature implies that these descriptors have a special significance independent of such details as the coordinate system used to describe them. A selected literature, some of it quite clear, can be found in References 1-7. If the chosen coordinate system were really insignificant it would be hard to understand a simple, but nonchaotic, counterexample: the one-dimensional harmonic oscillator, which exhibits a strong dependence of its largest local Lyapunov exponent λ 1 (t) on the chosen Cartesian coordinate system [1,8,9].We remind the reader that this local instantaneous Lyapunov exponent λ 1 (t) (the largest of them when time averaged) measures the local rate of divergence of two nearby trajectories. Think of them as a reference trajectory and a satellite trajectory, with the satellite constrained to remain near the reference. It is unnecessary to consider exponents beyond the first to understand why it is that the local Lyapunov exponents, covariant or not, are in fact not scale-independent and do indeed depend upon the chosen coordinate system or set of measurement units. The oftrepeated statement that the local covariant exponents are "norm-independent" should not be misunderstood (as we did) to mean that the exponents are independent of a scale factor, as in a change of units from cgs to MKS.Here we focus on a simple chaotic continuous-flow example [10], the thermostated three-dimensional flow of a harmonic oscillator with coordinate q, momentum p, and friction coefficient ζ(t) in the unscaled (q, p, ζ) phase space:The variation of temperature with coordinate T (q) makes possible dissipation, and phase-volume shrinkage,⊗ < 0, onto a torus, or a strange attractor with fractional dimensionality, or a one-dimensional limit cycle. For the evolution of this model see References 11-14.

show abstract

“…Recently, reasonably efficient algorithms for the computation of covariant vectors have become available [4,5], which were applied to a variety of systems [1,6,7]. In the panel on the left of the figure we demonstrate the time-reversal symmetry displayed by the local covariant exponents for a one-dimensional harmonic oscillator coupled to a position-dependent temperature T (q) = 1 + ε tanh(q) with a two-stage Nosé-Hoover thermostat, which makes use of two thermostat variables [1].…”

mentioning

confidence: 99%