Quasipotentials for simple noisy maps with complicated dynamics

Hamm, Andreas; Graham, R.

doi:10.1007/bf01055697

Cited by 30 publications

(22 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Without going into the details of Lagrangian manifolds (see e.g. [36]) we mention that generically these invariant manifolds have tangling bends which lead to accumulating Maxwell points at which the eigenfunctions are not differentiable (see [9]; many interesting aspects of this phenomenon have been studied in the case of quasipotentials for continuous time systems [34,37,38,39]). …”

Section: Proofmentioning

confidence: 99%

“…The discrete time version [8,9,26] has been used successfully to investigate the influence of noise on renormalisation schemes in the context of transitions from regular to chaotic behaviour [27], and other universal aspects of the influence of noise on bifurcations [28].…”

Section: Proofmentioning

confidence: 99%

“…We mention already here that the most interesting link between probability theory and possibility theory is established by the much used estimates of large deviation type [4,5]. In the context of dynamical systems this means that the so called quasipotentials or nonequilibrium potentials [6,7,8,9] -a standard tool for studying stochastic perturbations -have a natural interpretation in the context of possibility theory.…”

Section: Introductionmentioning

confidence: 99%

“…[34,9]) a connection to Hamiltonian systems has been established and exploited. In the same spirit we formulate a connection between symplectic maps (see e.g.…”

mentioning

confidence: 99%

See 3 more Smart Citations

Uncertain dynamical systems defined by pseudomeasures

Hamm

1997

Journal of Mathematical Physics

Self Cite

View full text Add to dashboard Cite

This paper deals with uncertain dynamical systems in which predictions about the future state of a system are assessed by so called pseudomeasures. Two special cases are stochastic dynamical systems, where the pseudomeasure is the conventional probability measure, and fuzzy dynamical systems in which the pseudomeasure is a so called possibility measure.New results about possibilistic systems and their relation to deterministic and to stochastic systems are derived by using idempotent pseudolinear algebra.By expressing large deviation estimates for stochastic perturbations in terms of possibility measures, we obtain a new interpretation of the Freidlin-Wentzell quasipotentials for stochastic perturbations of dynamical systems as invariant possibility densities.

show abstract

Section: Proofmentioning

confidence: 99%

Section: Proofmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…[34,9]) a connection to Hamiltonian systems has been established and exploited. In the same spirit we formulate a connection between symplectic maps (see e.g.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Uncertain dynamical systems defined by pseudomeasures

Hamm

1997

Journal of Mathematical Physics

Self Cite

View full text Add to dashboard Cite

show abstract

“…The theory of quasipotentials, providing a formal link between equilibruim and nonequilibrium physics, has been applied to many systems [50,51]. In particular, the theory of quasipotentials has been used to estimate escape times from the attractors of noisy period-doubling maps [52][53][54] and to estimate the invariant probability distributions of their strange attractors in the chaotic regime [53,55,56].Here, we investigate the invariant probability distributions that characterize critical fluctuations in the pitchfork bifurcations of period-doubling maps far from the chaotic threshold. In the limit of weak noise, we find an exact correspondence between the static behavior of fluctuations at a pitchfork bifurcation and the critical behavior of finite-size mean field theory [57].…”

mentioning

confidence: 99%

Critical fluctuations of noisy period-doubling maps

et al. 2017

View full text Add to dashboard Cite

PACS 64.60.-i -First pacs description PACS 02.50.-r -Second pacs description PACS 05.45.-a -Third pacs description Abstract -We extend the theory of quasipotentials in dynamical systems by calculating, within a broad class of period-doubling maps, an exact potential for the critical fluctuations of pitchfork bifurcations in the weak noise limit. These far-from-equilibrium fluctuations are described by finitesize mean field theory, placing their static properties in the same universality class as the Ising model on a complete graph. We demonstrate that the effective system size of noisy period-doubling bifurcations exhibits universal scaling behavior along period-doubling routes to chaos.Period-doubling bifurcations have been observed in a wide variety of natural systems spanning many areas of science [1]. Univariate, discrete-time maps are the simplest dynamical systems to exhibit a period-doubling route to chaos [2,3]. Applications range from the population dynamics of species with non-overlapping generations [4,5] to the oscillations of rf-driven Josephson junctions [6,7]. The impact of noise on period-doubling maps has been extensively studied in both ecology [8][9][10][11] and physics [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30], including the universal scaling of the Lyapunov exponent along period-doubling routes to chaos [31][32][33][34][35][36]. Connections between such noisy dynamical systems and the universality classes of equilibrium statistical physics has been a subject of great fascination [37][38][39][40][41][42][43][44][45][46][47][48][49]. The theory of quasipotentials, providing a formal link between equilibruim and nonequilibrium physics, has been applied to many systems [50,51]. In particular, the theory of quasipotentials has been used to estimate escape times from the attractors of noisy period-doubling maps [52][53][54] and to estimate the invariant probability distributions of their strange attractors in the chaotic regime [53,55,56].Here, we investigate the invariant probability distributions that characterize critical fluctuations in the pitchfork bifurcations of period-doubling maps far from the chaotic threshold. In the limit of weak noise, we find an exact correspondence between the static behavior of fluctuations at a pitchfork bifurcation and the critical behavior of finite-size mean field theory [57]. This correspondence places pitchfork bifurcations in the same universality class as the Ising model on a complete graph [58][59][60][61][62]. Analytical estimates of critical exponents and amplitudes agree well with the results of numerical simulations. We conclude with evidence of universal scaling behavior in the effective system size of critical fluctuations along period-doubling routes to chaos. p-1 arXiv:1502.04074v3 [cond-mat.stat-mech]

show abstract

Fluctuations in the steady state

Graham¹

25 Years of Non-Equilibrium Statistical Mechanics

View full text Add to dashboard Cite

Quasipotentials for simple noisy maps with complicated dynamics

Cited by 30 publications

References 27 publications

Uncertain dynamical systems defined by pseudomeasures

Uncertain dynamical systems defined by pseudomeasures

Critical fluctuations of noisy period-doubling maps

Fluctuations in the steady state

Contact Info

Product

Resources

About