Linear response of the Lorenz system

Anomalous transport of a classical wave-particle entity in a tilted potential

A classical wave-particle entity in the form of a millimetric walking droplet can emerge on the free surface of a vertically vibrating liquid bath. Such wave-particle entities have been shown to exhibit hydrodynamic analogs of quantum systems. Using an idealized theoretical model of this wave-particle entity in a tilted potential, we explore its transport behavior. The integro-differential equation of motion governing the dynamics of the wave-particle entity transforms to a Lorenz-like system of ordinary differential equations that drives the particle's velocity. Several anomalous transport regimes such as absolute negative mobility, differential negative mobility, and lock-in regions corresponding to force-independent mobility are observed. These observations motivate experiments in the hydrodynamic walking-droplet system for the experimental realizations of anomalous transport phenomena.

Handy fluctuation-dissipation relation to approach generic noisy systems and chaotic dynamics

We introduce a general formulation of the fluctuation-dissipation relations (FDRs) holding also in far-from-equilibrium stochastic dynamics. A great advantage of this version of the FDR is that it does not require explicit knowledge of the stationary probability density function. Our formula applies to Markov stochastic systems with generic noise distributions: When the noise is additive and Gaussian, the relation reduces to those known in the literature; for multiplicative and non-Gaussian distributions (e.g., Cauchy noise) it provides exact results in agreement with numerical simulations. Our formula allows us to reproduce, in a suitable small-noise limit, the response functions of deterministic, strongly nonlinear dynamical models, even in the presence of chaotic behavior: This could have important practical applications in several contexts, including geophysics and climate. As a case of study, we consider the Lorenz '63 model, which is paradigmatic for the chaotic properties of deterministic dynamical systems.

Adiabatic dynamic causal modelling

This technical note introduces adiabatic dynamic causal modelling, a method for inferring slow changes in biophysical parameters that control fluctuations of fast neuronal states. The application domain we have in mind is inferring slow changes in variables (e.g., extracellular ion concentrations or synaptic efficacy) that underlie phase transitions in brain activity (e.g., paroxysmal seizure activity). The scheme is efficient and yet retains a biophysical interpretation, in virtue of being based on established neural mass models that are equipped with a slow dynamic on the parameters (such as synaptic rate constants or effective connectivity). In brief, we use an adiabatic approximation to summarise fast fluctuations in hidden neuronal states (and their expression in sensors) in terms of their second order statistics; namely, their complex cross spectra. This allows one to specify and compare models of slowly changing parameters (using Bayesian model reduction) that generate a sequence of empirical cross spectra of electrophysiological recordings. Crucially, we use the slow fluctuations in the spectral power of neuronal activity as empirical priors on changes in synaptic parameters. This introduces a circular causality, in which synaptic parameters underwrite fast neuronal activity that, in turn, induces activity-dependent plasticity in synaptic parameters. In this foundational paper, we describe the underlying model, establish its face validity using simulations and provide an illustrative application to a chemoconvulsant animal model of seizure activity.

Sensitivity of long periodic orbits of chaotic systems

The properties of long, numerically determined periodic orbits of two low-dimensional chaotic systems, the Lorenz equations and the Kuramoto-Sivashinsky system in a minimal-domain configuration, are examined. The primary question is to establish whether the sensitivity of period averaged quantities with respect to parameter perturbations computed over long orbits can be used as a sufficiently good proxy for the response of the chaotic state to finite-amplitude parameter perturbations. To address this question, an inventory of thousands of orbits at least two orders of magnitude longer than the shortest admissible cycles is constructed. The expectation of period averages, Floquet exponents, and sensitivities over such set is then obtained. It is shown that all these quantities converge to a limiting value as the orbit period is increased. However, while period averages and Floquet exponents appear to converge to analogous quantities computed from chaotic trajectories, the limiting value of the sensitivity is not necessarily consistent with the response of the chaotic state, similar to observations made with other shadowing algorithms.

Towards a General Theory of Extremes for Observables of Chaotic Dynamical Systems

In this paper we provide a connection between the geometrical properties of the attractor of a chaotic dynamical system and the distribution of extreme values. We show that the extremes of so-called physical observables are distributed according to the classical generalised Pareto distribution and derive explicit expressions for the scaling and the shape parameter. In particular, we derive that the shape parameter does not depend on the chosen observables, but only on the partial dimensions of the invariant measure on the stable, unstable, and neutral manifolds. The shape parameter is negative and is close to zero when high-dimensional systems are considered. This result agrees with what was derived recently using the generalized extreme value approach. Combining the results obtained using such physical observables and the properties of the extremes of distance observables, it is possible to derive estimates of the partial dimensions of the attractor along the stable and the unstable directions of the flow. Moreover, by writing the shape parameter in terms of moments of the extremes of the considered observable and by using linear response theory, we relate the sensitivity to perturbations of the shape parameter to the sensitivity of the moments, of the partial dimensions, and of the Kaplan-Yorke dimension of the attractor. Preliminary numerical investigations provide encouraging results on the applicability of the theory presented here. The results presented here do not apply for all combinations of Axiom A systems and observables, but the breakdown seems to be related to very special geometrical configurations.

Dynamic structures of the time correlation functions of chaotic nonequilibrium fluctuations

Using the projection operator formalism we explore the decay form of the time correlation function U_(n)(t) identical with < û_(n)(t)û*_(n)(0)> of the state variable û_(n)(t) in the chaotic Kuramoto-Sivashinsky equation. The decay form turns out to be the algebraic decay 1/[1+(gamma_(na)(t)2] in the initial regime t<1/gamma_(ne) and the exponential decay exp(-gamma_(ne)t) in the final regime t>1/gamma_(ne) . The memory function Gamma_(n)(t) that represents the chaos-induced transport is found to obey the Gaussian decay exp[-(beta_(ng)t)2] in the case of large wave numbers, but the 3/2 power decay exp[-(beta_(n3)t)3/2] in the case of small wave numbers. The power spectrum of û_(n)(t) is given by the real part U'_(n)(omega) of the Fourier-Laplace transform of U_(n)(t) and has a dominant peak at omega=0 . This peak within the linewidth (-)gamma_(ne) (approximately equal to gamma_(ne)) is given by the Lorentzian spectrum (-)gamma2_(ne)/(omega2+(-)gamma2_(ne)) . However, the wings of the peak outside the width (-) gamma_(ne) turn out to take the exponential spectrum exp(-omega/gamma_(na)) . Thus it is found that the exponential decay exp(-gamma_(ne)t) appears to lead to the universal Lorentzian peak, while the algebraic decay 1/[1+(gamma_(na)t)2] arises to bring about the exponential wing.

Gravitational dynamics of an infinite shuffled lattice of particles

We study, using numerical simulations, the dynamical evolution of self-gravitating point particles in static Euclidean space, starting from a simple class of infinite "shuffled lattice" initial conditions. These are obtained by applying independently to each particle on an infinite perfect lattice a small random displacement, and are characterized by a power spectrum (structure factor) of density fluctuations which is quadratic in the wave number k, at small k. For a specified form of the probability distribution function of the "shuffling" applied to each particle, and zero initial velocities, these initial configurations are characterized by a single relevant parameter: the variance delta(2) of the "shuffling" normalized in units of the lattice spacing l. The clustering, which develops in time starting from scales around l, is qualitatively very similar to that seen in cosmological simulations, which begin from lattices with applied correlated displacements and incorporate an expanding spatial background. From very soon after the formation of the first nonlinear structures, a spatiotemporal scaling relation describes well the evolution of the two-point correlations. At larger times the dynamics of these correlations converges to what is termed "self-similar" evolution in cosmology, in which the time dependence in the scaling relation is specified entirely by that of the linearized fluid theory. Comparing simulations with different delta, different resolution, but identical large scale fluctuations, we are able to identify and study features of the dynamics of the system in the transient phase leading to this behavior. In this phase, the discrete nature of the system explicitly plays an essential role.

Time-ordered fluctuation-dissipation relation for incompressible isotropic turbulence

The Kraichnan-Wyld perturbation expansion is used to justify the introduction of a renormalized response function connecting two-point covariances at different times. The resulting relationship was specialized by a suitable choice of initial conditions to the form of a fluctuation-dissipation relation (FDR). This was further developed to reconcile the time symmetry of the covariance with the causality of the response by the introduction of time ordering along with a counterterm. This formulation provides a solution to an old problem, that of representing the time dependence of the covariance and response by exponential forms. We show that the derivative (with respect to difference time) of the covariance now vanishes at the origin. This allows one to study the relationships between two-time spectral closures and time-independent theories such as the self-consistent field theory of Edwards or the more recent renormalization group approaches. We also show that the renormalized response function is transitive with respect to intermediate times and report a different Langevin equation model for turbulence. We note the potential value of this time-ordering procedure in all applications of the FDR.