Perturbation theory for the Fokker–Planck operator in chaos

Heninger, Jeffrey; Lippolis, Domenico; Cvitanović, Predrag

doi:10.1016/j.cnsns.2017.06.025

Cited by 9 publications

(8 citation statements)

References 38 publications

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“…For the Lorenz equations, the distributions of the gradients converge to a delta function, centered around 0.0219. This value is close to a finite-difference approximation of the gradient of the leading Lyapunov exponent, which, being a long-time average, is not in general a continuous function of the parameters 28,79 and the comparison suffers from the same problems highlighted for the sensitivity of the period average in section VI D.…”

Section: E Sensitivity Of Floquet Exponentsmentioning

confidence: 62%

“…However, systems of practical interest rarely satisfy hyperbolicity and the literature is rich of examples where infinite-time averaged quantities exhibit complex, non smooth behavior when parameters are varied [22][23][24] . Such systems are infinitesimally close to internal structural bifurcations 25 and parameter perturbations might have a catastrophic impact on the invariant measure that supports ergodic averages, unless stochastic components are introduced [26][27][28] or in the "thermodynamic" limit of high-dimensional systems 12,29,30 .…”

Section: Introductionmentioning

confidence: 99%

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Sensitivity of long periodic orbits of chaotic systems

Lasagna

2019

Preprint

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The relationship between the stability of unstable periodic orbits of dissipative chaotic systems and the sensitivity of period averages to parameter perturbations is clarified. Using Floquet theory, we show that period averages vary most rapidly when structural perturbations of the equations of motion are well expressed by adjoint Floquet eigenfunctions associated to Floquet exponents of small magnitude, along invariant subspaces closest to marginality. Building on this analysis, we then construct a large inventory of periodic orbits of the Lorenz equations and of the Kuramoto-Sivashinsky system in a minimal-domain configuration, focusing on long periodic orbits spanning a large fraction of the chaotic attractor. We examine how the statistical distributions of Floquet exponents, period averages and their sensitivities to parameter perturbations vary as the period T increases. We show that, at least for these two systems, the distribution of these quantities converges to a Dirac delta function as T → ∞. We observe that the limiting value of period averages and Floquet exponents tend to the same asymptotic value obtained on long chaotic trajectories. Conversely, the limiting value of the sensitivity is not consistent with the response of long-time averages to finiteamplitude parameter perturbations, due to the lack of a linear response for non-hyperbolic dynamics.

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Section: E Sensitivity Of Floquet Exponentsmentioning

confidence: 62%

Section: Introductionmentioning

confidence: 99%

Sensitivity of long periodic orbits of chaotic systems

Lasagna

2019

Preprint

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“…Figures 1, 3 and 5 show that even small stochastic forcing smoothes out the fine structure of the strange attractor [16]; in particular the ring-like steps in the PDF disappear.…”

Section: A Direct Numerical Simulationmentioning

confidence: 95%

Statistics of the stochastically forced Lorenz attractor by the Fokker-Planck equation and cumulant expansions

Allawala

Marston

2016

Phys. Rev. E

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We investigate the Fokker-Planck description of the equal-time statistics of the three-dimensional Lorenz-63 attractor with additive white noise. The invariant measure is found by computing the zero (or null) mode of the linear Fokker-Planck operator as a problem of sparse linear algebra. Two variants are studied: A self-adjoint construction of the linear operator, and the replacement of diffusion with hyperdiffusion. We also access the low-order statistics of the system by a perturbative expansion in equal-time cumulants. Comparison is made to statistics obtained by the standard approach of accumulation via direct numerical simulation. Theoretical and computational aspects of the Fokker-Planck and cumulant expansion methods are discussed.

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“…In the real world, the evolution of a nonlinear system is often contaminated by the omnipresent noise. Heninger proposed the finest state-space resolution that can be achieved in a physical dynamical system is limited by noise [35], which in fact revives a form of perturbation theory in [36]. The finite approximation of the Koopman operator presented above of course brings error in the computation but at the same time grants robustness even in the presence of noise.…”

Section: Robustness Of the Partition In The Presence Of Noisementioning

confidence: 99%

Koopman Operator and Phase Space Partition of Chaotic Maps

Zhang,

Lan

2020

Preprint

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Koopman operator describes evolution of observables in the phase space, which could be used to extract characteristic dynamical features of a nonlinear system. Here, we show that it is possible to carry out interesting symbolic partitions based on properly constructed eigenfunctions of the operator for chaotic maps. The partition boundaries are the extrema of these eigenfunctions, the accuracy of which is improved by including more basis functions in the numerical computation. The validity of this scheme is demonstrated in well-known 1-d and 2-d maps. It seems no obstacle to extend the computation to nonlinear systems of high dimensions, which provides a possible way of dissecting complex dynamics.

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Perturbation theory for the Fokker–Planck operator in chaos

Cited by 9 publications

References 38 publications

Sensitivity of long periodic orbits of chaotic systems

Sensitivity of long periodic orbits of chaotic systems

Statistics of the stochastically forced Lorenz attractor by the Fokker-Planck equation and cumulant expansions

Koopman Operator and Phase Space Partition of Chaotic Maps

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