arXiv subjects

Wolfram Just

Publications and source records attributed to Wolfram Just.

At least 19 recordsLinked to original sources

arXiv2025

The stochastic discrete nonlinear Schr\"odinger equation: microscopic derivation and finite-temperature phase transition

We study a stochastic version of the one-dimensional discrete nonlinear Schr{\"o}dinger equation (DNSE), which is derived from first principles, and thus possesses all the properties required by statistical mechanics, such as detailed balance and the H-theorem. The stochastic version shows disordered and localised dynamics, and displays a corresponding phase transition at a finite temperature value. The phase transition can be captured in a quantitative way by a mean-field type approach. The corresponding coarsening dynamics shows an unexpected dependence on the noise strength, which is reminiscent of stochastic resonance. The phase transition is linked with negative temperature phase transitions, which have been reported recently for the Hamiltonian dynamics of the DNSE. Our approach gives a clue to how these negative temperature phase transitions can be implemented in experimental setups, which are inevitably coupled to a positive temperature heat bath.

cond-mat.stat-mech↗

arXiv2025

Long-term behavior of the master equation on a countable network and approximation methods of the (stationary) solutions via finite subsystems in the thermodynamic limit

The Master equation on directed networks - also called the differential Chapman-Kolmogorov equation - is a linear differential equation, which describes the probability evolution in a discrete system. While this is well understood, if the underlying graph is finite, the mathematics required for the treatment of a network with countable many nodes is way more complicated and advanced. In this paper we provide criteria for the rates of the system, which makes it possible to approximate the solution by finite subsystems in the thermodynamic limit. By writing the phase space as a direct sum of stationary states and states which vanish in the time limit, we give a new proof of when the time limit for an countable, infinite dimensional system exists and when it can be interchanged with the limit of large systems. We give sufficient criteria, when these two limits commute and demonstrate on various examples, what happens when these criteria are violated and only one of these limits exists.

math-ph↗

arXiv2025

Dynamics of localized states in the stochastic discrete nonlinear Schr\"odinger equation

We revisit aspects of dynamics and stability of localized states in the deterministic and stochastic discrete nonlinear Schr\"odinger equation. By a combination of analytic and numerical techniques, we show that localized initial conditions disperse if the strength of the nonlinear part drops below a threshold and that localized states are unstable in a noisy environment. As expected, the constants of motion in the nonlinear Schr\"odinger equation play a crucial role. An infinite temperature state emerges when multiplicative noise is applied, while additive noise yields unbounded dynamics since conservation of normalization is violated.

nlin.PS↗

arXiv2024

Non-periodic not everywhere dense trajectories in triangular billiards

Building on tools that have been successfully used in the study of rational billiards, such as induced maps and interval exchange transformations, we provide a construction of a one-parameter family of isosceles triangles exhibiting non-periodic trajectories that are not everywhere dense. This provides, by elementary means, a definitive negative answer to a long-standing open question on the density of non-periodic trajectories in triangular billiards.

math.DS↗

arXiv2024

Convergence properties of dynamic mode decomposition for analytic interval maps

Extended dynamic mode decomposition (EDMD) is a data-driven algorithm for approximating spectral data of the Koopman operator associated to a dynamical system, combining a Galerkin method of order N and collocation method of order M. Spectral convergence of this method subtly depends on appropriate choice of the space of observables. For chaotic analytic full branch maps of the interval, we derive a constraint between M and N guaranteeing spectral convergence of EDMD.

math.DS↗

arXiv2023

EDMD for expanding circle maps and their complex perturbations

We show that spectral data of the Koopman operator arising from an analytic expanding circle map $τ$ can be effectively calculated using an EDMD-type algorithm combining a collocation method of order m with a Galerkin method of order n. The main result is that if $m \geq δn$, where $δ$ is an explicitly given positive number quantifying by how much $τ$ expands concentric annuli containing the unit circle, then the method converges and approximates the spectrum of the Koopman operator, taken to be acting on a space of analytic hyperfunctions, exponentially fast in n. Additionally, these results extend to more general expansive maps on suitable annuli containing the unit circle.

math.DS↗

arXiv2022

Resonances for rational Anosov maps on the torus

A complete description of resonances for rational toral Anosov diffeomorphisms preserving certain Reinhardt domains is presented. As a consequence it is shown that every homotopy class of two-dimensional Anosov diffeomorphisms contains maps with the sequence of resonances decaying stretched-exponentially. This is achieved by introducing a certain group of rational toral diffeomorphisms and computing the resonances of the respective composition operator considered on suitable anisotropic spaces of hyperfunctions. The class of examples is sufficiently rich to also include non-linear Anosov maps with trivial resonances, or resonances decaying exponentially, as well as with or without area-preservation or reversing symmetries.

math.DS↗

arXiv2022

Continuation with Non-invasive Control Schemes: Revealing Unstable States in a Pedestrian Evacuation Scenario

This paper presents a framework to perform bifurcation analysis in laboratory experiments or simulations. We employ control-based continuation to study the dynamics of a macroscopic variable of a microscopically defined model, exploring the potential viability of the underlying feedback control techniques in an experiment. In contrast to previous experimental studies that used iterative root-finding methods on the feedback control targets, we propose a feedback control law that is inherently non-invasive. That is, the control discovers the location of equilibria and stabilizes them simultaneously. We call the proposed control zero-in-equilibrium feedback control and we prove that it is able to stabilize branches of equilibria, except at singularities of codimension n+1, where n is the number of state space dimensions the feedback can depend on. We apply the method to a simulated evacuation scenario were pedestrians have to reach an exit after maneuvering left or right around an obstacle. The scenario shows a hysteresis phenomenon with bistability and tipping between two possible steady pedestrian flows in microscopic simulations. We demonstrate for the evacuation scenario that the proposed control law is able to uniformly discover and stabilize steady flows along the entire branch, including points where other non-invasive approaches to feedback control become singular.

math.DS↗

arXiv2019

Dynamic mode decomposition for analytic maps

Extended dynamic mode decomposition (EDMD) provides a class of algorithms to identify patterns and effective degrees of freedom in complex dynamical systems. We show that the modes identified by EDMD correspond to those of compact Perron-Frobenius and Koopman operators defined on suitable Hardy-Hilbert spaces when the method is applied to classes of analytic maps. Our findings elucidate the interpretation of the spectra obtained by EDMD for complex dynamical systems. We illustrate our results by numerical simulations for analytic maps.

math.DS↗

arXiv2017

Nonequilibrium dynamics of a pure dry friction model subjected to coloured noise

We investigate the impact of noise on a two-dimensional simple paradigmatic piecewise-smooth dynamical system. For that purpose we consider the motion of a particle subjected to dry friction and coloured noise. The finite correlation time of the noise provides an additional dimension in phase space, a nontrivial probability current, and thus establishes a proper nonequilibrium regime. Furthermore, the finite noise correlation time allows for the study of stick-slip phenomena which show up as a singular component in the stationary probability density. Analytic insight can be provided by application of the unified coloured noise approximation, developed by Jung and Hänggi (Phys. Rev. A 35, 4464 (R) (1987)). The analysis of probability currents and a closer look at power spectral densities underpin the observed stick-slip transitions which are related with a critical value of the noise correlation time.

cond-mat.stat-mech↗

arXiv2017

Visibility graphs and symbolic dynamics

Visibility algorithms are a family of geometric and ordering criteria by which a real-valued time series of N data is mapped into a graph of N nodes. This graph has been shown to often inherit in its topology non-trivial properties of the series structure, and can thus be seen as a combinatorial representation of a dynamical system. Here we explore in some detail the relation between visibility graphs and symbolic dynamics. To do that, we consider the degree sequence of horizontal visibility graphs generated by the one-parameter logistic map, for a range of values of the parameter for which the map shows chaotic behaviour. Numerically, we observe that in the chaotic region the block entropies of these sequences systematically converge to the Lyapunov exponent of the system. Via Pesin identity, this in turn suggests that these block entropies are converging to the Kolmogorov- Sinai entropy of the map, which ultimately suggests that the algorithm is implicitly and adaptively constructing phase space partitions which might have the generating property. To give analytical insight, we explore the relation k(x), x \in[0,1] that, for a given datum with value x, assigns in graph space a node with degree k. In the case of the out-degree sequence, such relation is indeed a piece-wise constant function. By making use of explicit methods and tools from symbolic dynamics we are able to analytically show that the algorithm indeed performs an effective partition of the phase space and that such partition is naturally expressed as a countable union of subintervals, where the endpoints of each subinterval are related to the fixed point structure of the iterates of the map and the subinterval enumeration is associated with particular ordering structures that we called motifs.

nlin.CD↗

arXiv2016

Complete spectral data for analytic Anosov maps of the torus

Using analytic properties of Blaschke factors we construct a family of analytic hyperbolic diffeomorphisms of the torus for which the spectral properties of the associated transfer operator acting on a suitable Hilbert space can be computed explicitly. As a result, we obtain explicit expressions for the decay of correlations of analytic observables without resorting to any kind of perturbation argument.

nlin.CD↗

arXiv2014

Modulating coherence resonance in non-excitable systems by time-delayed feedback

We propose a paradigmatic model system, a subcritical Hopf normal form subjected to noise and time-delayed feedback, to investigate the impact of time delay on coherence resonance in non-excitable systems. We develop analytical tools to estimate the stationary distribution and the time correlations in nonlinear stochastic delay differential equations. These tools are applied to our model to propose a novel quantity to measure coherence resonance induced by a saddle-node bifurcation of periodic orbits.

nlin.AO↗

arXiv2014

On large deviation properties of Brownian motion with dry friction

We investigate piecewise-linear stochastic models as with regards to the probability distribution of functionals of the stochastic processes, a question which occurs frequently in large deviation theory. The functionals that we are looking into in detail are related to the time a stochastic process spends at a phase space point or in a phase space region, as well as to the motion with inertia. For a Langevin equation with discontinuous drift, we extend the so-called backward Fokker-Planck technique for nonnegative support functionals to arbitrary support functionals, to derive explicit expressions for the moments of the functional. Explicit solutions for the moments and for the distribution of the so-called local time, the occupation time and the displacement are derived for the Brownian motion with dry friction, including quantitative measures to characterize deviation from Gaussian behaviour in the asymptotic long time limit.

cond-mat.stat-mech↗

arXiv2014

First-passage time of Brownian motion with dry friction

We provide an analytic solution to the first-passage time (FPT) problem of a piecewise-smooth stochastic model, namely Brownian motion with dry friction, using two different but closely related approaches which are based on eigenfunction decompositions on the one hand and on the backward Kolmogorov equation on the other. For the simple case containing only dry friction, a phase transition phenomenon in the spectrum is found which relates to the position of the exit point, and which affects the tail of the FPT distribution. For the model containing as well a driving force and viscous friction the impact of the corresponding stick-slip transition and of the transition to ballistic exit is evaluated quantitatively. The proposed model is one of the very few cases where FPT properties are accessible by analytical means.

cond-mat.stat-mech↗

arXiv2013

Spectral structure of transfer operators for expanding circle maps

We explicitly determine the spectrum of transfer operators (acting on spaces of holomorphic functions) associated to analytic expanding circle maps arising from finite Blaschke products. This is achieved by deriving a convenient natural representation of the respective adjoint operators.

math.DS↗

arXiv2013

Weak-noise limit of a piecewise-smooth stochastic differential equation

We investigate the validity and accuracy of weak-noise (saddle-point or instanton) approximations for piecewise-smooth stochastic differential equations (SDEs), taking as an illustrative example a piecewise-constant SDE, which serves as a simple model of Brownian motion with solid friction. For this model, we show that the weak-noise approximation of the path integral correctly reproduces the known propagator of the SDE at lowest order in the noise power, as well as the main features of the exact propagator with higher-order corrections, provided that the singularity of the path integral associated with the non-smooth SDE is treated with some heuristics. We also show that, as in the case of smooth SDEs, the deterministic paths of the noiseless system correctly describe the behaviour of the non-smooth SDE in the low-noise limit. Finally, we consider a smooth regularisation of the piecewise-constant SDE and study to which extent this regularisation can rectify some of the problems encountered when dealing with discontinuous drifts and singularities in SDEs.

cond-mat.stat-mech↗

arXiv2013

Analytic expanding circle maps with explicit spectra

We show that for any $λ\in \mathbb{C}$ with $|λ|<1$ there exists an analytic expanding circle map such that the eigenvalues of the associated transfer operator (acting on holomorphic functions) are precisely the nonnegative powers of $λ$ and $\barλ$. As a consequence we obtain a counterexample to a variant of a conjecture of Mayer on the reality of spectra of transfer operators.

math.DS↗