Persistent stability of a chaotic system

Huber, Greg; Pradas, Marc; Pumir, Alain; Wilkinson, M.

doi:10.1016/j.physa.2017.10.042

Cited by 5 publications

(10 citation statements)

References 25 publications

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“…On the other hand, our intermediate result (50) for the finite w rate, r w (f ), requires L/L w ≫ 1. From the definition (42)…”

Section: Elastic Line (D = 1) -Continuous Modelmentioning

confidence: 81%

“…. On the other hand, our intermediate result (50) for the finite w rate, r w (f ), requires L/L w ≫ 1. From the definition (42) of L w , that is equivalent to w ≪ v p L 1/2 c (L/L c ) 2 .…”

Section: Elastic Line (D = 1) -Continuous Modelmentioning

confidence: 81%

“…For the case of periodic depinning studied numerically in the main text, ζ = 3/2 (the roughness of the Larkin model, see Appendix C). In that case W ∼ L ζ is equivalent to L ∼ L w and one cannot use the second term in the asymptotic rate formula (50). Instead one must replace it by − 1 L ln det L −2 w − ∂ 2 τ .…”

Section: Elastic Line (D = 1) -Continuous Modelmentioning

confidence: 99%

“…. For example the problem of calculating large deviation function addressed here is closely related to recent work in chaos theory[50], to be explored.…”

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confidence: 99%

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Exponential number of equilibria and depinning threshold for a directed polymer in a random potential

et al. 2018

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By extending the Kac-Rice approach to manifolds of finite internal dimension, we show that the mean number N tot of all possible equilibria (i.e. force-free configurations, a.k.a. equilibrium points) of an elastic line (directed polymer), confined in a harmonic well and submitted to a quenched random Gaussian potential in dimension d = 1 + 1, grows exponentially N tot ∼ exp (r L) with its length L. The growth rate r is found to be directly related to the generalized Lyapunov exponent (GLE) which is a momentgenerating function characterizing the large-deviation type fluctuations of the solution to the initial value problem associated with the random Schrödinger operator of the 1D Anderson localization problem. For strong confinement, the rate r is small and given by a non-perturbative (instanton, Lifshitz tail-like) contribution to GLE. For weak confinement, the rate r is found to be proportional to the inverse Larkin length of the pinning theory. As an application, identifying the depinning with a landscape "topology trivialization" phenomenon, we obtain an upper bound for the depinning threshold f c , in the presence of an applied force, for elastic lines and d-dimensional manifolds, expressed through the mean modulus of the spectral determinant of the Laplace operators with a random potential. We also discuss the question of counting of stable equilibria. Finally, we extend the method to calculate the asymptotic number of equilibria at fixed energy (elastic, potential and total), and obtain the (annealed) distribution of the energy density over these equilibria (i.e. force-free configurations). Some connections with the Larkin model are also established.

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“…On the other hand, our intermediate result (50) for the finite w rate, r w (f ), requires L/L w ≫ 1. From the definition (42)…”

Section: Elastic Line (D = 1) -Continuous Modelmentioning

confidence: 81%

Section: Elastic Line (D = 1) -Continuous Modelmentioning

confidence: 81%

Section: Elastic Line (D = 1) -Continuous Modelmentioning

confidence: 99%

“…. For example the problem of calculating large deviation function addressed here is closely related to recent work in chaos theory[50], to be explored.…”

mentioning

confidence: 99%

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Exponential number of equilibria and depinning threshold for a directed polymer in a random potential

et al. 2018

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“…In particular, there is no linear part in the corresponding rate functions. We note that the authors of [96] computedˆ( ) L k from the leading eigenvalue of an operator L k similar to L k , but associated with the spatial FTLEŝ t , with equation of motion (28). Our expression for the tilted generator L k in phase space shows that L k and L k have the same leading eigenvalue.…”

Section: Fluctuation Relation and Spatial Rate Functionmentioning

confidence: 77%

Fractal catastrophes

et al. 2020

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We analyse the spatial inhomogeneities ('spatial clustering') in the distribution of particles accelerated by a force that changes randomly in space and time. To quantify spatial clustering, the phase-space dynamics of the particles must be projected to configuration space. Folds of a smooth phase-space manifold give rise to catastrophes ('caustics') in this projection. When the inertial particle dynamics is damped by friction, however, the phase-space manifold converges towards a fractal attractor. It is believed that caustics increase spatial clustering also in this case, but a quantitative theory is missing. We solve this problem by determining how projection affects the distribution of finite-time Lyapunov exponents (FTLEs). Applying our method in one spatial dimension we find that caustics arising from the projection of a dynamical fractal attractor ('fractal catastrophes') make a distinct and universal contribution to the distribution of spatial FTLEs. Our results explain a projection formula for the spatial fractal correlation dimension, and how a fluctuation relation for the distribution of FTLEs for white-in-time Gaussian force fields breaks upon projection. We explore the implications of our results for heavy particles in turbulence, and for wave propagation in random media. in configuration space. These singular points are cusp or fold catastrophes, also called 'caustics' [46,47], due to their similarities with the random focusing of light in geometrical optics [49,50]. For smooth phase-space manifolds, catastrophes are known to lead to finite-time singularities in the spatial particle density ñ t (x) ( figure 1(b)), suggesting that caustics may increase spatial clustering [46]. These consideration are, however, too imprecise to quantify spatial clustering. More importantly, these arguments rest on the notion of a smooth manifold, and it is unclear to which extent they apply to fractal phase-space attractors(figure 1(c)). In other words, it is not understood how caustic folds affect the spatial fractal dimensions, although it is generally assumed that they do. Computing the effect of caustics in the long-time limit is challenging because they give rise to non-perturbative effects [51], and are therefore thought to cause perturbation expansions for spatial fractal dimensions [6, 7, 51-55] to fail.Here we show how to describe the effect of caustics on spatial clustering from first principles by projecting the phase-space FTLEs to configuration space. We demonstrate our method by deriving the spatial FTLE distribution for inertial particles accelerated by a spatially smooth but random force field in one spatial dimension. Our main result is that caustics give rise to a distinct and universal contribution to the spatial FTLE distribution, independent of the details of the force field. This caustic contribution results in an exponentially increased probability of observing dense clusters of particles. Furthermore, we demonstrate how caustics affect the distribution of spatial separations, and we show that ...

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