Alignment of Lyapunov Vectors: A Quantitative Criterion to Predict Catastrophes?

Beims, Marcus W.; Gallas, Jason A. C.

doi:10.1038/srep37102

Cited by 26 publications

(25 citation statements)

References 32 publications

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“…In addition, since a break of hyperbolicity may be due to a homoclinic tangency where the stable and unstable manifolds are tangent, the angle between the CLVs has been suggested in [69] and [70] as an indicator of the degree of hyperbolicity of the dynamics. Moreover, it has been conjectured in [61] that the alignment of the CLVs could provide a criterion to predict crises, although the latter are understood there as chaotic bursts (in other words, an extreme fluctuation in an otherwise weakly chaotic trajectory) rather than as an attractor crisis. However, due to the singularity in the Lorenz attractor, the angle between the unstable and the central CLVs can be arbitrarily small even away from the crisis, so that the loss of hyperbolicity should not provide a precursor of the crisis.…”

Section: Lyapunov Exponents and Covariant Lyapunov Vectorsmentioning

confidence: 99%

Resonances in a Chaotic Attractor Crisis of the Lorenz Flow

2017

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Local bifurcations of stationary points and limit cycles have successfully been characterized in terms of the critical exponents of these solutions. Lyapunov exponents and their associated covariant Lyapunov vectors have been proposed as tools for supporting the understanding of critical transitions in chaotic dynamical systems. However, it is in general not clear how the statistical properties of dynamical systems change across a boundary crisis during which a chaotic attractor collides with a saddle.This behavior is investigated here for a boundary crisis in the Lorenz flow, for which neither the Lyapunov exponents nor the covariant Lyapunov vectors provide a criterion for the crisis. Instead, the convergence of the time evolution of probability densities to the invariant measure, governed by the semigroup of transfer operators, is expected to slow down at the approach of the crisis. Such convergence is described by the eigenvalues of the generator of this semigroup, which can be divided into two families, referred to as the stable and unstable Ruelle-Pollicott resonances, respectively. The former describes the convergence of densities to the attractor (or escape from a repeller) and is estimated from many short time series sampling the phase space. The latter is responsible for the decay of correlations, or mixing, and can be estimated from a long times series, invoking ergodicity.It is found numerically for the Lorenz flow that the stable resonances do approach the imaginary axis during the crisis, as is indicative of the loss of global A. Tantet · V. Lucarini 2 Alexis Tantet et al. stability of the attractor. On the other hand, the unstable resonances, and a fortiori the decay of correlations, do not flag the proximity of the crisis, thus questioning the usual design of early warning indicators of boundary crises of chaotic attractors and the applicability of response theory close to such crises.It is a problem of fundamental relevance in mathematical, natural, and applied sciences to understand under which conditions a system may undergo abrupt changes under perturbation and, if so, predict when these changes will occur. Much of our understanding of such transitions comes from the bifurcation theory of autonomous dynamical systems [1,2,3], with extensions to nonautonomous [4] and random [5] dynamical systems. In particular, local bifurcations, taking place for example when a stationary point or a limit cycle loses stability, are characterized by the critical exponents of these invariant sets. They yield a local measure of the relaxation rate of trajectories to these sets. As the latter become less stable, these exponents approach zero, resulting in the slowing down of the convergence of trajectories to the attractor. This critical slowing down has allowed to design early-warning signals of critical transitions by monitoring the rate of decay of correlations [6], peaks in power spectra [7], or of recovery from perturbations [8]. See [9], for a review, and [10] for applications to climate science.Most physica...

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Section: Lyapunov Exponents and Covariant Lyapunov Vectorsmentioning

confidence: 99%

Resonances in a Chaotic Attractor Crisis of the Lorenz Flow

2017

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“…However the main objective of this work is to show the appearance of optical rogue waves and to identify the physical mechanism at their origin. Special interest is put also in establishing our ability to predict them [21][22][23][24][25]. We establish clearly the relevance of the existence of generalized multistability and the role played by an external crisis of the chaotic attractor in order to generate optical rogue waves.…”

Section: Introductionmentioning

confidence: 82%

Extreme Events in Lasers with Modulation of the Field Polarization

Gomel

Boyer

Métayer

et al. 2019

Advances in Condensed Matter Physics

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We develop a theoretical model for a unidirectional ring laser consisting of an isotropic active medium inside a cavity containing a birefringent Kerr cell. We analyze the dynamical behavior of the system as we modulate the voltage applied to the Kerr cell. We discuss the bifurcation diagram and we study the regions of control parameter space where it becomes possible to observe and predict extreme events.

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“…They present slow, long-wavelength behavior in the tangent space dynamics. Besides, the properties of the covariant Lyapunov vectors [38][39][40][41] at the full SS might also be promis-ing from the theoretical point of view and applications.…”

Section: Discussionmentioning

confidence: 99%

Intermittent stickiness synchronization

2019

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This work uses the statistical properties of Finite-Time Lyapunov exponents (FTLEs) to investigate the Intermittent Stickiness Synchronization (ISS) observed in the mixed phase space of high-dimensional Hamiltonian systems. Full Stickiness Synchronization (SS) occurs when all FTLEs from a chaotic trajectory tend to zero for arbitrary long time-windows. This behavior is a consequence of the sticky motion close to regular structures which live in the high-dimensional phase space and affects all unstable directions proportionally by the same amount, generating a kind of collective motion. Partial SS occurs when at least one FTLE approaches to zero. Thus, distinct degrees of partial SS may occur, depending on the values of nonlinearity and coupling parameters, on the dimension of the phase space and on the number of positive FTLEs. Through filtering procedures used to precisely characterize the sticky motion, we are able to compute the algebraic decay exponents of the ISS and to obtain remarkable evidence about the existence of a universal behavior related to the decay of time correlations encoded in such exponents. In addition, we show that even though the probability to find full SS is small compared to partial SSs, the full SS may appear for very long times due to the slow algebraic decay of time correlations in mixed phase space. In this sense, observations of very late intermittence between chaotic motion and full SSs become rare events.

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Alignment of Lyapunov Vectors: A Quantitative Criterion to Predict Catastrophes?

Cited by 26 publications

References 32 publications

Resonances in a Chaotic Attractor Crisis of the Lorenz Flow

Resonances in a Chaotic Attractor Crisis of the Lorenz Flow

Extreme Events in Lasers with Modulation of the Field Polarization

Intermittent stickiness synchronization

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