arXiv subjects

Hildegard Meyer-Ortmanns

Publications and source records attributed to Hildegard Meyer-Ortmanns.

At least 37 records · Page 2Linked to original sources

arXiv2015

Curing critical links in oscillator networks as power grid models

Modern societies crucially depend on the robust supply with electric energy. Blackouts of power grids can thus have far reaching consequences. During a blackout, often the failure of a single infrastructure, such as a critical transmission line, results in several subsequent failures that spread across large parts of the network. Preventing such large-scale outages is thus key for assuring a reliable power supply. Here we present a non-local curing strategy for oscillatory power grid networks based on the global collective redistribution of loads. We first identify critical links and compute residual capacities on alternative paths on the remaining network from the original flows. For each critical link, we upgrade lines that constitute bottlenecks on such paths. We demonstrate the viability of this strategy for random ensembles of network topologies as well as topologies derived from real transmission grids and compare the nonlocal strategy against local back-ups of critical links. These strategies are independent of the detailed grid dynamics and combined may serve as an effective guideline to reduce outages in power grid networks by intentionally strengthen optimally selected links.

nlin.AO↗

arXiv2015

Boundary drive induced formation of aggregate condensates in stochastic transport with short-range interactions

We discuss the effects of particle exchange through open boundaries and the induced drive on the phase structure and condensation phenomena of a stochastic transport process with tunable short-range interactions featuring pair-factorized steady states (PFSS) in the closed system. In this model, the steady state of the particle hopping process can be tuned to yield properties from the zero-range process (ZRP) condensation model to those of models with spa- tially extended condensates. By varying the particle exchange rates as well as the presence of a global drift, we observe a phase transition from a free particle gas to a phase with condensates aggregated to the boundaries. While this transition is similar to previous results for the ZRP, we find that the mechanism is different as the presence of the boundary actually influences the interaction due to the non-zero interaction range.

cond-mat.stat-mech↗

arXiv2015

Open boundary conditions in stochastic transport processes with pair-factorized steady states

Using numerical methods we discuss the effects of open boundary conditions on condensation phenomena in the zero-range process (ZRP) and transport processes with pair-factorized steady states (PFSS), an extended model of the ZRP with nearest-neighbor interaction. For the zero-range process we compare to analytical results in the literature with respect to criticality and condensation. For the extended model we find a similar phase structure, but observe supercritical phases with droplet formation for strong boundary drives.

cond-mat.stat-mech↗

arXiv2014

Networks of coupled circuits: From a versatile toggle switch to collective coherent behavior

We study the versatile performance of networks of coupled circuits. Each of these circuits is composed of a positive and a negative feedback loop in a motif that is frequently found in genetic and neural networks. When two of these circuits are coupled with mutual repression, the system can function as a toggle switch. The variety of its states can be controlled by two parameters as we demonstrate by a detailed bifurcation analysis. In the bistable regimes switches between the coexisting attractors can be induced by noise. When we couple larger sets of these units, we numerically observe collective coherent modes of individual fixed-point and limit-cycle behavior. It is there the monotonic change of a single bifurcation parameter that allows to control the onset and arrest of the synchronized oscillations. This mechanism may play a role in biological applications, in particular in connection with the segmentation clock. While tuning the bifurcation parameter, also a variety of transient patterns emerges upon approaching the stationary states, in particular a self-organized pacemaker in a completely uniformly equipped ensemble, so that the symmetry breaking happens dynamically.

nlin.AO↗

arXiv2014

Long-Range Response to Transmission Line Disturbances in DC Electricity Grids

We consider a DC electricity grid composed of transmission lines connecting power generators and consumers at its nodes. The DC grid is described by nonlinear equations derived from Kirchhoff's law. For an initial distribution of consumed and generated power, and given transmission line conductances, we determine the geographical distribution of voltages at the nodes. Adjusting the generated power for the Joule heating losses, we then calculate the electrical power flow through the transmission lines. Next, we study the response of the grid to an additional transmission line between two sites of the grid and calculate the resulting change in the power flow distribution. This change is found to decay slowly in space, with a power of the distance from the additional line. We find the geographical distribution of the power transmission, when a link is added. With a finite probability the maximal load in the grid becomes larger when a transmission line is added, a phenomenon that is known as Braess' paradox. We find that this phenomenon is more pronounced in a DC grid described by the nonlinear equations derived from Kirchhoff's law than in a linearised flow model studied previously in Ref. \cite{witthaut2013}. We observe furthermore that the increase in the load of the transmission lines due to an added line is of the same order of magnitude as Joule heating. Interestingly, for a fixed system size the load of the lines increases with the degree of disorder in the geographical distribution of consumers and producers.

nlin.AO↗

arXiv2014

A simple mechanism for controlling the onset and arrest of collective oscillations in genetic circuits

We study a system of dynamical units, each of which shows excitable or oscillatory behavior, depending on the choice of parameters. When we couple these units with repressive bonds, we can control the duration of collective oscillations for an intermediate period between collective fixed-point behavior. The control mechanism works by monotonically increasing a single bifurcation parameter. Both the onset and arrest of oscillations are due to bifurcations. Depending on the coupling strength, the network topology and the tuning speed, our numerical simulations reveal a rich dynamics out-of-equilibrium with multiple inherent time scales, long transients towards the stationary states and interesting transient patterns like self-organized pacemakers. Zooming into the transition regime, we pursue the arrest of oscillations along the arms of spirals. We point out possible relations to the genetic network of segmentation clocks.

q-bio.MN↗

arXiv2012

A network of phase oscillators as a device for sequential pattern generation

We design a system of phase oscillators that is able to produce temporally periodic sequences of patterns. Patterns are cluster partitions which encode information as phase differences between phase oscillators. The architecture of our system consists of a retrieval network with N globally coupled phase oscillators, a pacemaker that controls the sequence retrieval, and a set of patterns stored in the couplings between the pacemaker and the retrieval network. The system performs in analogy to a central pattern generator of neural networks and is very robust against perturbations in the retrieval process.

nlin.CD↗

arXiv2012

Stochastic Description of a Bistable Frustrated Unit

Mixed positive and negative feedback loops are often found in biological systems which support oscillations. In this work we consider a prototype of such systems, which has been recently found at the core of many genetic circuits showing oscillatory behaviour. Our model consists of two interacting species A and B, where A activates not only its own production, but also that of its repressor B. While the self-activation of A leads already to a bistable unit, the coupling with a negative feedback loop via B makes the unit frustrated. In the deterministic limit of infinitely many molecules, such a bistable frustrated unit is known to show excitable and oscillatory dynamics, depending on the maximum production rate of A which acts as a control parameter. We study this model in its fully stochastic version and we find oscillations even for parameters which in the deterministic limit are deeply in the fixed-point regime. The deeper we go into this regime, the more irregular these oscillations are, becoming finally random excitations whenever fluctuations allow the system to overcome the barrier for a large excursion in phase space. The fluctuations can no longer be fully treated as a perturbation. The smaller the system size (the number of molecules), the more frequent are these excitations. Therefore, stochasticity caused by demographic noise makes this unit even more flexible with respect to its oscillatory behaviour.

cond-mat.stat-mech↗

arXiv2009

Preferential attachment renders an evolving network of populations robust against crashes

We study a model for the evolution of chemical species under a combination of population dynamics on a short time scale and a selection mechanism on a longer time scale. Least fit nodes are replaced by new nodes whose links are attached to the nodes of the given network via preferential attachment. In contrast to a random attachment of newly incoming nodes that was used in previous work, this preferential attachment mechanism accelerates the generation of a so-called autocatalytic set after a start from a random geometry and the growth of this structure until it saturates in a stationary phase in which the whole system is an autocatalytic set. Moreover, the system in the stationary phase becomes much more stable against crashes in the population size as compared to random attachment. We explain in detail in terms of graph theoretical notions which structure of the resulting network is responsible for this stability. Essentially it is a very dense core with many loops and less nodes playing the role of a keystone that prevents the system against crashes almost completely.

nlin.AO↗

arXiv2006

Phase transition between synchronous and asynchronous updating algorithms

We update a one-dimensional chain of Ising spins of length $L$ with algorithms which are parameterized by the probability $p$ for a certain site to get updated in one time step. The result of the update event itself is determined by the energy change due to the local change in the configuration. In this way we interpolate between the Metropolis algorithm at zero temperature for $p$ of the order of 1/L and for large $L$, and a synchronous deterministic updating procedure for $p=1$. As function of $p$ we observe a phase transition between the stationary states to which the algorithm drives the system. These are non-absorbing stationary states with antiferromagnetic domains for $p>p_c$, and absorbing states with ferromagnetic domains for $p\leq p_c$. This means that above this transition the stationary states have lost any remnants to the ferromagnetic Ising interaction. A measurement of the critical exponents shows that this transition belongs to the universality class of parity conservation.

cond-mat.other↗

arXiv2006

Universality class of triad dynamics on a triangular lattice

We consider triad dynamics as it was recently considered by Antal \emph{et al.} [T. Antal, P. L. Krapivsky, and S. Redner, Phys. Rev. E {\bf 72}, 036121 (2005)] as an approach to social balance. Here we generalize the topology from all-to-all to a regular one of a two-dimensional triangular lattice. The driving force in this dynamics is the reduction of frustrated triads in order to reach a balanced state. The dynamics is parameterized by a so-called propensity parameter $p$ that determines the tendency of negative links to become positive. As a function of $p$ we find a phase transition between different kind of absorbing states. The phases differ by the existence of an infinitely connected (percolated) cluster of negative links that forms whenever $p\leq p_c$. Moreover, for $p\leq p_c$, the time to reach the absorbing state grows powerlike with the system size $L$, while it increases logarithmically with $L$ for $p > p_c$. From a finite-size scaling analysis we numerically determine the critical exponents $β$ and $ν$ together with $γ$, $τ$, $σ$. The exponents satisfy the hyperscaling relations. We also determine the fractal dimension $d_f$ that fulfills a hyperscaling relation as well. The transition of triad dynamics between different absorbing states belongs to a universality class with new critical exponents. We generalize the triad dynamics to four-cycle dynamics on a square lattice. In this case, again there is a transition between different absorbing states, going along with the formation of an infinite cluster of negative links, but the usual scaling and hyperscaling relations are violated.

cond-mat.stat-mech↗

arXiv2006

Reducing Frustration in Spin Systems: Social Balance as an XOR-SAT problem

Reduction of frustration was the driving force in an approach to social balance as it was recently considered by Antal \emph{et al.} [ T. Antal, P. L. Krapivsky, and S. Redner, Phys. Rev. E {\bf 72}, 036121 (2005). ]. We generalize their triad dynamics to $k$-cycle dynamics for arbitrary integer $k$. We derive the phase structure, determine the stationary solutions and calculate the time it takes to reach a frozen state. The main difference in the phase structure as a function of $k$ is related to $k$ being even or odd. As a second generalization we dilute the all-to-all coupling as considered by Antal \emph{et al.} to a random network with connection probability $w<1$. ...

physics.soc-ph↗

arXiv2006

Reentrant Synchronization and Pattern Formation in Pacemaker-Entrained Kuramoto Oscillators

We study phase entrainment of Kuramoto oscillators under different conditions on the interaction range and the natural frequencies. In the first part the oscillators are entrained by a pacemaker acting like an impurity or a defect. We analytically derive the entrainment frequency for arbitrary interaction range and the entrainment threshold for all-to-all couplings. For intermediate couplings our numerical results show a reentrance of the synchronization transition as a function of the coupling range. The origin of this reentrance can be traced back to the normalization of the coupling strength. In the second part we consider a system of oscillators with an initial gradient in their natural frequencies, extended over a one-dimensional chain or a two-dimensional lattice. Here it is the oscillator with the highest natural frequency that becomes the pacemaker of the ensemble, sending out circular waves in oscillator-phase space. No asymmetric coupling between the oscillators is needed for this dynamical induction of the pacemaker property nor need it be distinguished by a gap in the natural frequency.

cond-mat.other↗

arXiv2006

Entrainment of Coupled Oscillators on Regular Networks by Pacemakers

We study Kuramoto oscillators, driven by one pacemaker, on $d$-dimensional regular topologies with nearest neighbor interactions. We derive the analytical expressions for the common frequency in the case of phase-locked motion and for the critical frequency of the pacemaker, placed at an arbitrary position in the lattice, so that above the critical frequency no phase-locked motion is possible. We show that the mere change in topology from an open chain to a ring induces synchronization for a certain range of pacemaker frequencies and couplings, while keeping the other parameters fixed. Moreover we demonstrate numerically that the critical frequency of the pacemaker decreases as a power of the linear size of the lattice with an exponent equal to the dimension of the system. This leads in particular to the conclusion that for infinite-dimensional topologies the critical frequency for having entrainment decreases exponentially with the size of the system, or, more generally, with the depth of the network, that is the average distance of the oscillators from the pacemaker.

cond-mat.other↗

arXiv2005

Competition of Languages and their Hamming Distance

We consider the spreading and competition of languages that are spoken by a population of individuals. The individuals can change their mother tongue during their lifespan, pass on their language to their offspring and finally die. The languages are described by bitstrings, their mutual difference is expressed in terms of their Hamming distance. Language evolution is determined by mutation and adaptation rates. In particular we consider the case where the replacement of a language by another one is determined by their mutual Hamming distance. As a function of the mutation rate we find a sharp transition between a scenario with one dominant language and fragmentation into many language clusters. The transition is also reflected in the Hamming distance between the two languages with the largest and second to largest number of speakers. We also consider the case where the population is localized on a square lattice and the interaction of individuals is restricted to a certain geometrical domain. Here it is again the Hamming distance that plays an essential role in the final fate of a language of either surviving or being extinct.

physics.soc-ph↗

arXiv2005

Kuramoto Oscillators on Chains, Rings and Cayley-trees

We study systems of Kuramoto oscillators, driven by one pacemaker, on $d$-dimensional regular topologies like linear chains, rings, hypercubic lattices and Cayley-trees. For the special cases of next-neighbor and infinite-range interactions, we derive the analytical expressions for the common frequency in the case of phase-locked motion and for the critical frequency of the pacemaker, placed at an arbitrary position on the lattice, so that above the critical frequency no phase-locked motion is possible. These expressions depend on the number of oscillators, the type of coupling, the coupling strength, and the range of interactions. In particular we show that the mere change in topology from an open chain with free boundary conditions to a ring induces synchronization for a certain range of pacemaker frequencies and couplings, keeping the other parameters fixed. We also study numerically the phase evolution above the critical eigenfrequency of the pacemaker for arbitrary interaction ranges and find some interesting remnants to phase-locked motion below the critical frequency.

cond-mat.stat-mech↗

arXiv2005

Synchronization of R{ö}ssler Oscillators on Scale-free Topologies

We study the synchronization of R{ö}ssler oscillators as prototype of chaotic systems, when they are coupled on scale-free complex networks. We find that the underlying topology crucially affects the global synchronization properties. Especially, we show that the existence of loops facilitates the synchronizability of the system, whereas Rössler oscillators do not synchronize on tree-like topologies beyond a certain size. By considering Cayley trees, modified by various shortcuts, we find that also the distribution of shortest path lengths between two oscillators plays an important role for the global synchronization.

cond-mat.stat-mech↗

arXiv2005

Self-similar Scale-free Networks and Disassortativity

Self-similar networks with scale-free degree distribution have recently attracted much attention, since these apparently incompatible properties were reconciled in a paper by Song et al. by an appropriate box-counting method that enters the measurement of the fractal dimension. We study two genetic regulatory networks ({\it Saccharomyces cerevisiae} and {\it Escherichai coli} and show their self-similar and scale-free features, in extension to the datasets studied by Song et al. Moreover, by a number of numerical results we support the conjecture that self-similar scale-free networks are not assortative. From our simulations so far these networks seem to be disassortative instead. We also find that the qualitative feature of disassortativity is scale-invariant under renormalization, but it appears as an intrinsic feature of the renormalization prescription, as even assortative networks become disassortative after a sufficient number of renormalization steps.

cond-mat.stat-mech↗