arXiv subjects

Hildegard Meyer-Ortmanns

Publications and source records attributed to Hildegard Meyer-Ortmanns.

At least 19 recordsLinked to original sources

arXiv2026

Metastability induced by non-reciprocal adaptive couplings in Kuramoto models

Non-reciprocal couplings are frequently found in systems out-of-equilibrium such as neuronal networks. We consider generalized Kuramoto models with non-reciprocal adaptive couplings. The non-reciprocity refers to the type of couplings according to Hebbian or anti-Hebbian rules and to different time scales on which the couplings evolve. The main effect of this specific combination of deterministic dynamics is an induced metastability of anti-phase synchronized clusters of oscillators. Metastable switching is typical for neuronal networks and a characteristic of brain dynamics. We analyze the metastability as a function of the system parameters, in particular of the size and the network connectivity. The mechanism behind sudden changes in the order parameters is individual oscillators which change their cluster affiliation from time to time, providing ``weak ties" between clusters of synchronized oscillators, where an individual oscillator may represent an entire brain area. The time series exhibit random features but arise from deterministic dynamics.

nlin.AO↗

arXiv2025

Topologically protected edge oscillations in nonlinear dynamical units

Many examples from quantum and classical physics are known where topological protection is responsible for the robustness of the dynamics. Less explored is the role of topological protection in the context of classical oscillatory systems. As on-site dynamics, we consider prototypical oscillator models with possible applications in biochemical systems. However, our choice of coupling geometry is inspired by models from condensed matter physics which -- in isolation -- guarantee non-trivial topology in momentum space. We choose directed couplings between units on a two-dimensional grid, alternating between weak and strong values, such that oscillations become localized at the edges of the grid while bulk units transition to oscillation-death states, resulting in a frequency chimera-like state. These patterns are resilient to parameter mismatches, additive noise, and structural defects. To explain the robustness of these edge oscillations we use topological characteristics and calculate Zak phases. As it turns out, the edge-localized oscillations result from a bulk-boundary correspondence that applies to our system even though our derived effective Hamiltonian is non-Hermitian. By appropriately tuning the system parameters, it is possible to control which regions of the two-dimensional grid are in an oscillatory state and which settle to oscillation-death states.

nlin.AO↗

arXiv2022

Rare desynchronization events in power grids: On data implementation and dimensional reductions

We discuss the frequency of desynchronization events in power grids for realistic data input. We focus on the role of time correlations in the fluctuating power production and propose a new method for implementing colored noise that reproduces non-Gaussian data by means of cumulants of data increment distributions. Our desynchronization events are caused by overloads. We extend known and propose different methods of dimensional reduction to considerably reduce the high-dimensional phase space and to predict the rare desynchronization events with reasonable computational costs. The first method splits the system into two areas, connected by heavily loaded lines, and treats each area as a single node. The second method considers a separation of the timescales of power fluctuations and phase angle dynamics and completely disregards the latter. The fact that this separation turns out to be justified, albeit only to exponential accuracy in the strength of fluctuations, means that the number of rare events does not sensitively depend on inertia or damping for realistic heterogeneous parameters and long correlation times. Neither does the number of desynchronization events automatically increase with non-Gaussian fluctuations in the power production as one might have expected. On the other hand, the analytical expressions for the average time to desynchronization depend sensitively on the finite correlation time of the fluctuating power input.

nlin.AO↗

arXiv2022

Heteroclinic units acting as pacemakers: Entrained dynamics for cognitive processes

Heteroclinic dynamics is a suitable framework for describing transient and reproducible dynamics such as cognitive processes in the brain. We demonstrate how heteroclinic units can act as pacemakers to entrain larger sets of units from a resting state to hierarchical heteroclinic motion that is able to describe fast oscillations modulated by slow oscillations. Such features are observed in brain dynamics. The entrainment range depends on the type of coupling, the spatial location of the pacemaker and the individual bifurcation parameters of the pacemaker and the driven units. Noise as well as a small back-coupling to the pacemaker facilitate synchronization. Units can be synchronously entrained to different temporal patterns encoding transiently excited neural populations, depending on the selected path in the heteroclinic network. Via entrainment, these temporal patterns, locally generated by the pacemakers, can be communicated to the resting units in target waves over a spatial grid. For getting entrained there is no need of fine-tuning the parameters of the resting units. Thus, entrainment provides one way of processing information over the grid, when information is encoded in the generated spatiotemporal patterns.

math.DS↗

arXiv2022

Belief propagation for supply networks: Efficient clustering of their factor graphs

We consider belief propagation (BP) as an efficient and scalable tool for state estimation and optimization problems in supply networks such as power grids. BP algorithms make use of factor graph representations, whose assignment to the problem of interest is not unique. It depends on the state variables and their mutual interdependencies. Many short loops in factor graphs may impede the accuracy of BP. We propose a systematic way to cluster loops of naively assigned factor graphs such that the resulting transformed factor graphs have no additional loops as compared to the original network. They guarantee an accurate performance of BP with only slightly increased computational effort, as we demonstrate by a concrete and realistic implementation for power grids. The method outperforms existing alternatives to handle the loops. We point to other applications to supply networks such as gas-pipeline or other flow networks that share the structure of constraints in the form of analogues to Kirchhoff's laws. Whenever small and abundant loops in factor graphs are systematically generated by constraints between variables in the original network, our factor-graph assignment in BP complements other approaches. It provides a fast and reliable algorithm to perform marginalization in tasks like state determination, estimation, or optimization issues in supply networks.

cs.AI↗

arXiv2022

The cavity method for minority games between arbitrageurs on financial markets

We use the cavity method from statistical physics for analyzing the transient and stationary dynamics of a minority game that is played by agents performing market arbitrage. On the level of linear response the method allows to include the reaction of the market to individual actions of the agents as well as the reaction of the agents to individual information items of the market. This way we derive a self-consistent solution to the minority game. In particular we analyze the impact of general nonlinear price functions on the amount of arbitrage if noise from external fluctuations is present. We identify the conditions under which arbitrage gets reduced due to the presence of noise. When the cavity method is extended to time dependent response of the market price to previous actions of the agents, the individual contributions of noise can be pursued over different time scales in the transient dynamics until a stationary state is reached and when the stationary state is reached. The contributions are from external fluctuations in price and information and from noise due to the choice of strategies. The dynamics explains the time evolution of scores of the agents' strategies: it changes from initially a random walk to non-Markovian dynamics and bounded excursions on an intermediate time scale to effectively random switching in the choice between strategies on long time scales. In contrast to the Curie-Weiss level of a mean-field approach, the market response included by the cavity method captures the realistic feature that the agents can have a preference for a certain choice of strategies without getting stuck to a single choice. The breakdown of the method in the phase transition region indicates possible market mechanisms leading to critical volatility and a possible regime shift.

q-fin.TR↗

arXiv2021

Rare extinction events in cyclic predator-prey games

In the May-Leonard model of three cyclically competing species, we analyze the statistics of rare events in which all three species go extinct due to strong but rare fluctuations. These fluctuations are from the tails of the probability distribution of species concentrations. They render a coexistence of three populations unstable even if the coexistence is stable in the deterministic limit. We determine the mean time to extinction (MTE) by using a WKB-ansatz in the master equation that represents the stochastic description of this model. This way, the calculation is reduced to a problem of classical mechanics and amounts to solving a Hamilton-Jacobi equation with zero-energy Hamiltonian. We solve the corresponding Hamilton's equations of motion in six-dimensional phase space numerically by using the Iterative Action Minimization Method. This allows to project on the optimal path to extinction, starting from a parameter choice where the three-species coexistence-fixed point undergoes a Hopf bifurcation and becomes stable. Specifically for our system of three species, extinction events can be triggered along various paths to extinction, differing in their intermediate steps. We compare our analytical predictions with results from Gillespie simulations for two-species extinctions, complemented by an analytical calculation of the MTE in which the remaining third species goes extinct. From Gillespie simulations we also analyze how the distributions of times to extinction change upon varying the bifurcation parameter. Even within the same model and the same dynamical regime, the MTE depends on the distance from the bifurcation point in a way that contains the system size dependence in the exponent. It is challenging and worthwhile to quantify how rare the rare events of extinction are.

physics.bio-ph↗

arXiv2021

Minority games played by arbitrageurs on the energy market

Along with the energy transition, the energy markets change their organization toward more decentralized and self-organized structures, striving for locally optimal profits. These tendencies may endanger the physical grid stability. One realistic option is the exhaustion of reserve energy due to an abuse by arbitrageurs. We map the energy market to different versions of a minority game and determine the expected amount of arbitrage as well as its fluctuations as a function of the model parameters. Of particular interest are the impact of heterogeneous contributions of arbitrageurs, the interplay between external stochastic events and nonlinear price functions of reserve power, and the effect of risk aversion due to suspected penalties. The non-monotonic dependence of arbitrage on the control parameters reveals an underlying phase transition that is the counterpart to replica symmetry breaking in spin glasses. As conclusions from our results we propose economic and statutory measures to counteract a detrimental effect of arbitrage.

econ.TH↗

arXiv2020

State Estimation of Power Flows for Smart Grids via Belief Propagation

Belief propagation is an algorithm that is known from statistical physics and computer science. It provides an efficient way of calculating marginals that involve large sums of products which are efficiently rearranged into nested products of sums to approximate the marginals. It allows a reliable estimation of the state and its variance of power grids that is needed for the control and forecast of power grid management. At prototypical examples of IEEE-grids we show that belief propagation not only scales linearly with the grid size for the state estimation itself, but also facilitates and accelerates the retrieval of missing data and allows an optimized positioning of measurement units. Based on belief propagation, we give a criterion for how to assess whether other algorithms, using only local information, are adequate for state estimation for a given grid. We also demonstrate how belief propagation can be utilized for coarse-graining power grids towards representations that reduce the computational effort when the coarse-grained version is integrated into a larger grid. It provides a criterion for partitioning power grids into areas in order to minimize the error of flow estimates between different areas.

cs.AI↗

arXiv2019

Dynamically generated hierarchies in games of competition

Spatial many-species predator-prey systems have been shown to yield very rich space-time patterns. This observation begs the question whether there exist universal mechanisms for generating this type of emerging complex patterns in non-equilibrium systems. In this work we investigate the possibility of dynamically generated hierarchies in predator-prey systems. We analyze a nine-species model with competing interactions and show that the studied situation results in the spontaneous formation of spirals within spirals. The parameter dependence of these intriguing nested spirals is elucidated. This is achieved through the numerical investigation of various quantities (correlation lengths, densities of empty sites, Fourier analysis of species densities, interface fluctuations) that allows to gain a rather complete understanding of the spatial arrangements and the temporal evolution of the system. A possible generalization of the interaction scheme yielding dynamically generated hierarchies is discussed. As cyclic interactions occur spontaneously in systems with competing strategies, the mechanism discussed in this work should contribute to our understanding of various social and biological systems.

cond-mat.stat-mech↗

arXiv2018

On the fate of dynamical systems under a trade-off between cost and precision

We analyze the fate of dynamical systems that consist of two kind of processes. The first type is supposed to perform a certain function by processing information at a required high accuracy, which is, however, limited to less than 100 percent, while the second process serves to maintain the required precision. Both processes are assumed to be subject to a trade-off between cost and precision, where the cost have to be paid from renewable but limited resources. In a discrete map we pursue the time evolution of errors and determine the conditions under which the fate of the system is either a stable performance at the desired accuracy, or a deterioration. Deterioration may be realized either as an accumulation of errors or a decline of resources when they are all absorbed for maintenance. We point to possible implications for living organisms and their perspectives to avoid an accumulation of errors in the course of time.

physics.bio-ph↗

arXiv2018

A hierarchical heteroclinic network: Controlling the time evolution along its paths

We consider a heteroclinic network in the framework of winnerless competition of species. It consists of two levels of heteroclinic cycles. On the lower level, the heteroclinic cycle connects three saddles, each representing the survival of a single species; on the higher level, the cycle connects three such heteroclinic cycles, in which nine species are involved. We show how to tune the predation rates in order to generate the long time scales on the higher level from the shorter time scales on the lower level. Moreover, when we tune a single bifurcation parameter, first the motion along the lower and next along the higher-level heteroclinic cycles are replaced by a heteroclinic cycle between 3-species coexistence-fixed points and by a 9-species coexistence-fixed point, respectively. We also observe a similar impact of additive noise. Beyond its usual role of preventing the slowing-down of heteroclinic trajectories at small noise level, its increasing strength can replace the lower-level heteroclinic cycle by 3-species coexistence fixed-points, connected by an effective limit cycle, and for even stronger noise the trajectories converge to the 9-species coexistence-fixed point. The model has applications to systems in which slow oscillations modulate fast oscillations with sudden transitions between the temporary winners.

nlin.AO↗

arXiv2017

Temporal self-similar synchronization patterns and scaling in repulsively coupled oscillators

We study synchronization patterns in repulsively coupled Kuramoto oscillators and focus on the impact of disorder in the natural frequencies. Among other choices we select the grid size and topology in a way that we observe a dynamically induced dimensional reduction with a continuum of attractors as long as the natural frequencies are uniformly chosen. When we introduce disorder in these frequencies, we find limit cycles with periods that are orders of magnitude longer than the natural frequencies of individual oscillators. Moreover we identify sequences of temporary patterns of phase-locked motion, which are self-similar in time and whose periods scale with a power of the inverse width about a uniform frequency distribution. This behavior provides challenges for future research.

nlin.AO↗

arXiv2017

Comparison of electricity market designs for stable decentralized power grids

In this study, we develop a theoretical model of strategic equilibrium bidding and price-setting behaviour by heterogeneous and boundedly rational electricity producers and a grid operator in a single electricity market under uncertain information about production capabilities and electricity demand. We compare eight different market design variants and several levels of centralized electricity production that influence the spatial distribution of producers in the grid, their unit production and curtailment costs, and the mean and standard deviation of their production capabilities. Our market design variants differ in three aspects. Producers are either paid their individual bid price ("pay as bid") or the (higher) market price set by the grid operator ("uniform pricing"). They are either paid for their bid quantity ("pay requested") or for their actual supply ("pay supplied") which may differ due to production uncertainty. Finally, excess production is either required to be curtailed or may be supplied to the grid. Overall, we find the combination of uniform pricing, paying for requested amounts, and required curtailment to perform best or second best in many respects, and to provide the best compromise between the goals of low economic costs, low consumer costs, positive profits, low balancing, low workload, and honest bidding behaviour.

physics.soc-ph↗

arXiv2017

Emerging long orbits and self-similar temporal sequences in classical oscillators

We analyze repulsively coupled Kuramoto oscillators, which are exposed to a distribution of natural frequencies. This source of disorder leads to closed orbits with a variety of different periods, which can be orders of magnitude longer than periods of individual oscillators. By construction the attractor space is quite rich. This may cause long transients until the deterministic trajectories find their stationary orbits. The smaller the width of the distribution about the common natural frequency is, the longer are the emerging time scales on average. Among the long-period orbits we find self-similar sequences of temporary phase-locked motion on different time scales. The ratio of time scales is determined by the ratio of widths of the distributions.

nlin.AO↗

arXiv2016

Extreme prices in electricity balancing markets from an approach of statistical physics

An increase in energy production from renewable energy sources is viewed as a crucial achievement in most industrialized countries. The higher variability of power production via renewables leads to a rise in ancillary service costs over the power system, in particular costs within the electricity balancing markets, mainly due to an increased number of extreme price spikes. This study focuses on forecasting the behavior of price and volumes of the Italian balancing market in the presence of an increased share of renewable energy sources. Starting from configurations of load and power production, which guarantee a stable performance, we implement fluctuations in the load and in renewables; in particular we artificially increase the contribution of renewables as compared to conventional power sources to cover the total load. We then forecast the amount of provided energy in the balancing market and its fluctuations, which are induced by production and consumption. Within an approach of agent based modeling we estimate the resulting energy prices and costs. While their average values turn out to be only slightly affected by an increased contribution from renewables, the probability for extreme price events is shown to increase along with undesired peaks in the costs.

q-fin.GN↗

arXiv2016

Rock-paper-scissors played within competing domains in predator-prey games

We consider (N,r) games of predation with N species and r < N prey and predators, acting in a cyclic way. Further basic reactions include reproduction, decay and diffusion over a regular grid, without a hard constraint on the occupation number per site, so in a "bosonic" implementation. For special combinations of N and r and appropriate parameter choices we observe games within games, that is, different coexisting games, depending on the spatial resolution. As concrete and simplest example we analyze the (6,3) game. Once the players segregate from a random initial distribution, domains emerge, which effectively play a (2,1)-game on the coarse scale of domain diameters, while agents inside the domains play (3,1) (rock-paper-scissors), leading to spiral formation with species chasing each other. As the (2,1)-game has a winner in the end, the coexistence of domains is transient, while agents inside the remaining domain coexist, until demographic fluctuations lead to extinction of all but one species in the very end. This means that we observe a dynamical generation of multiple space and time scales with emerging re-organization of players upon segregation, starting from a simple set of rules on the smallest scale and changed rules from the coarser perspective. These observations are based on Gillespie simulations. We discuss the deterministic limit derived from a van Kampen expansion. The linear stability analysis predicts the number of forming domains, their composition in terms of species; it also explains the instability of interfaces between domains, which drives their extinction; spiral patterns are identified as motion along heteroclinic cycles. The numerical solutions reproduce the observed patterns of the Gillespie simulations including even extinction events, so that the mean-field analysis here is very conclusive, which is due to the specific implementation of rules.

nlin.PS↗

arXiv2016

Coarse-Grained Modeling of Genetic Circuits as a Function of the Inherent Time Scales

From a coarse-grained perspective the motif of a self-activating species, activating a second species which acts as its own repressor, is widely found in biological systems, in particular in genetic systems with inherent oscillatory behavior. Here we consider a specific realization of this motif as a genetic circuit, in which genes are described as directly producing proteins, leaving out the intermediate step of mRNA production. We focus on the effect that inherent time scales on the underlying fine-grained scale can have on the bifurcation patterns on a coarser scale in time. Time scales are set by the binding and unbinding rates of the transcription factors to the promoter regions of the genes. Depending on the ratio of these rates to the decay times of the proteins, the appropriate averaging procedure for obtaining a coarse-grained description changes and leads to sets of deterministic equations, which differ in their bifurcation structure. In particular the desired intermediate range of regular limit cycles fades away when the binding rates of genes are of the same order or less than the decay time of at least one of the proteins. Our analysis illustrates that the common topology of the widely found motif alone does not necessarily imply universal features in the dynamics.

physics.bio-ph↗