Search

In this paper we address two questions about the synchronization of coupled oscillators in the Kuramoto model with all-to-all coupling. In the first part we use some classical results in convex geometry to prove bounds on the size of the frequency set supporting the existence of stable, phase locked solutions and show that the set of such frequencies can be expressed by a seminorm which we call the Kuramoto norm. In the second part we use some ideas from extreme order statistics to compute upper and lower bounds on the probability of synchronization for very general frequency distributions. We do so by computing exactly the limiting extreme value distribution of a quantity that is equivalent to the Kuramoto norm., Keywords: Kuramoto model, convex analysis, permutahedron, extreme-value statistics

Given an admissible map γ f for a homogeneous network N , it is known that the Jacobian D γ f ( x ) around a fully synchronous point x = ( x 0 , … , x 0 ) is again an admissible map for N . Motivated by this, we study the spectra of linear admissible maps for homogeneous networks. In particular, we define so-called network multipliers. These are (relatively small) matrices that depend linearly on the coefficients of the response function, and whose eigenvalues together make up the spectrum of the corresponding admissible map. More precisely, given a network N , we define a finite set of network multipliers ( Λ l ) l = 1 k and a class of networks C containing N . This class is furthermore closed under taking quotient networks, subnetworks and disjoint unions. We then show that the eigenvalues of an admissible map for any network in C are given by those of (a subset of) the network multipliers, with fixed multiplicities ( m l ) l = 1 k and independently of the given (finite dimensional) phase space of a node. The coefficients of all the network multipliers of C are furthermore linearly independent, which implies that one may find the multiplicities ( m l ) l = 1 k by simply expressing the trace of an admissible map as a linear combination of the traces of the multipliers. In particular, we will give examples of networks where the network multipliers need not be constructed, but can be determined by looking at small networks in C . We also show that network multipliers are multiplicative with respect to composition of linear maps.

We present a variety of results analyzing the behavior of a class of stochastic processes---referred to as piecewise deterministic Markov processes (PDMPs)---for the infinite-time interval and determine general conditions on when the moments of such processes will, or will not, be well-behaved. We also characterize the types of finite-time blowups that are possible for these processes, and obtain bounds on their probabilities.

The Kuramoto–Sakaguchi model is a generalization of the well-known Kuramoto model that adds a phase-lag paramater or “frustration” to a network of phase-coupled oscillators. The Kuramoto model is a flow of gradient type, but adding a phase-lag breaks the gradient structure, significantly complicating the analysis of the model. We present several results determining the stability of phase-locked configurations: the first of these gives a sufficient condition for stability, and the second a sufficient condition for instability. In fact, the instability criterion gives a count, modulo 2, of the dimension of the unstable manifold to a fixed point and having an odd count is a sufficient condition for instability of the fixed point. We also present numerical results for both small ( N≤10) and large ( N=50) collections of Kuramoto–Sakaguchi oscillators.

The generalized distance matrix of a graph is the matrix whose entries depend only on the pairwise distances between vertices, and the generalized distance spectrum is the set of eigenvalues of this matrix. This framework generalizes many of the commonly studied spectra of graphs. We show that for a large class of graphs these eigenvalues can be computed explicitly. We also present the applications of our results to competition models in ecology and rapidly mixing Markov Chains.

We present and analyze a model of opinion formation on an arbitrary network whose dynamics comes from a global energy function. We study the global and local minimizers of this energy, which we call stable opinion configurations, and describe the global minimizers under certain assumptions on the friendship graph. We show a surprising result that the number of stable configurations is not necessarily monotone in the strength of connection in the social network, i.e. the model sometimes supports more stable configurations when the interpersonal connections are made stronger.

Many applied problems can be posed as a dynamical system defined on a network with attractive and repulsive interactions. Examples include synchronization of nonlinear oscillator networks; the behavior of groups, or cliques, in social networks; and the study of optimal convergence for consensus algorithm. It is important to determine the index of a matrix, i.e., the number of positive and negative eigenvalues, and the dimension of the kernel. In this paper we consider the common examples where the matrix takes the form of a signed graph Laplacian. We show that the there are topological constraints on the index of the Laplacian matrix related to the dimension of a certain homology group. When the homology group is trivial, the index of the operator is determined only by the topology of the network and is independent of the strengths of the interactions. In general, these constraints give bounds on the number of positive and negative eigenvalues, with the dimension of the homology group counting the number ...

We consider a cascading model of excitable neural dynamics and show that over a wide variety of parameter regimes, these systems admit unique attractors. For large coupling strengths, this attractor is a limit cycle, and for small coupling strengths, it is a fixed point. We also show that the cascading model considered here is a mean-field limit of an existing stochastic model.

We consider the existence of non-synchronized fixed points to the Kuramoto model defined on sparse networks: specifically, networks where each vertex has degree exactly three. We show that "most" such networks support multiple attracting phase-locked solutions that are not synchronized, and study the depth and width of the basins of attraction of these phase-locked solutions. We also show that it is common in "large enough" graphs to find phase-locked solutions where one or more of the links has angle difference greater than $\pi/2$.

The Gershgorin Circle Theorem is a well-known and efficient method for bounding the eigenvalues of a matrix in terms of its entries. If $A$ is a symmetric matrix, by writing $A = B + x{\bf 1}$, where ${\bf 1}$ is the matrix with unit entries, we consider the problem of choosing $x$ to give the optimal Gershgorin bound on the eigenvalues of $B$, which then leads to one-sided bounds on the eigenvalues of $A$. We show that this $x$ can be found by an efficient linear program (whose solution can in may cases be written in closed form), and we show that for large classes of matrices, this shifting method beats all existing piecewise linear or quadratic bounds on the eigenvalues. We also apply this shifting paradigm to some nonlinear estimators and show that under certain symmetries this also gives rise to a tractable linear program. As an application, we give a novel bound on the lowest eigenvalue of a adjacency matrix in terms of the "top two" or "bottom two" degrees of the corresponding graph, and study the efficacy of this method in obtaining sharp eigenvalue estimates for certain classes of matrices., Comment: 18 pages, 7 figures

We consider the Kuramoto model of coupled oscillators with nearest-neighbour coupling and additive white noise. We show that synchronous solutions which are stable without the addition of noise become metastable and that we have transitions amongst synchronous solutions on long timescales. We compute these timescales and, moreover, compute the most likely path in phase space that transitions will follow. We show that these transition timescales do not increase as the number of oscillators in the system increases, and are roughly constant in the system size. Finally, we show that the transitions correspond to a splitting of one synchronous solution into two communities which move independently for some time and which rejoin to form a different synchronous solution.

We study the top Lyapunov exponent of the response of a two-dimensional non-Hamiltonian system driven by additive white noise. The origin is not a fixed point for the system; however, there is an invariant measure for the one-point motion of the system. In this paper we consider the stability of the two-point motion by Khasminskii's method of linearization along trajectories. The specific system we consider is the third-order truncated normal form of the unfolding of a Hopf bifurcation. We show that in the small noise limit the top Lyapunov exponent always approaches zero from below (and is thus negative for noise sufficiently small); we also show that there exist open sets of parameters for which the top Lyapunov exponent is positive. Thus the two-point motion can be either stable or unstable, while the stationary density that describes the one-point motion always exists.

We analyze a stochastic neuronal network model which corresponds to an all-to-all net- work of discretized integrate-and-fire neurons where the sy napses are failure-prone. This network exhibits different phases of behavior corresponding to syn chrony and asynchrony, and we show that this is due to the limiting mean-field system possessing multiple attractors. We also show that this mean-field limit exhibits a first-order phase transitio n as a function of the connection strength — as the synapses are made more reliable, there is a sudden onset of synchronous behavior. A detailed understanding of the dynamics involves both a characterization of the size of the giant component in a certain random graph process, and control of the pathwise dynamics of the system by obtaining exponential bounds for the probabilities of events far from the mean.

For singular perturbation problems, the renormalization group (RG) method of Chen, Goldenfeld, and Oono [Phys. Rev. E. 49 (1994) 4502–4511] has been shown to be an effective general approach for deriving reduced or amplitude equations that govern the long time dynamics of the system. It has been applied to a variety of problems traditionally analyzed using disparate methods, including the method of multiple scales, boundary layer theory, the WKBJ method, the Poincare–Lindstedt method, the method of averaging, and others. In this article, we show how the RG method may be used to generate normal forms for large classes of ordinary differential equations. First, we apply the RG method to systems with autonomous perturbations, and we show that the reduced or amplitude equations generated by the RG method are equivalent to the classical Poincare–Birkhoff normal forms for these systems up to and including terms of O ( ϵ 2 ) , where ϵ is the perturbation parameter. This analysis establishes our approach and generalizes to higher order. Second, we apply the RG method to systems with nonautonomous perturbations, and we show that the reduced or amplitude equations so generated constitute time-asymptotic normal forms, which are based on KBM averages. Moreover, for both classes of problems, we show that the main coordinate changes are equivalent, up to translations between the spaces in which they are defined. In this manner, our results show that the RG method offers a new approach for deriving normal forms for nonautonomous systems, and it offers advantages since one can typically more readily identify resonant terms from naive perturbation expansions than from the nonautonomous vector fields themselves. Finally, we establish how well the solution to the RG equations approximates the solution of the original equations on time scales of O ( 1 / ϵ ) .

We consider the Moran process with two populations competing under an iterated Prisoner’s Dilemma in the presence of mutation, and concentrate on the case where there are multiple evolutionarily stable strategies. We perform a complete bifurcation analysis of the deterministic system which arises in the infinite population size. We also study the Master equation and obtain asymptotics for the invariant distribution and metastable switching times for the stochastic process in the case of large but finite population. We also show that the stochastic system has asymmetries in the form of a skew for parameter values where the deterministic limit is symmetric.

There are a number of models of coupled oscillator networks where the question of the stability of fixed points reduces to calculating the index of a graph Laplacian. Some examples of such models include the Kuramoto and Kuramoto--Sakaguchi equations as well as the swing equations, which govern the behavior of generators coupled in an electrical network. We show that the index calculation can be related to a dual calculation which is done on the first homology group of the graph, rather than the vertex space. We also show that this representation is computationally attractive for relatively sparse graphs, where the dimension of the first homology group is low, as is true in many applications. We also give explicit formulae for the dimension of the unstable manifold to a phase-locked solution for graphs containing one or two loops. As an application, we present some novel results for the Kuramoto model defined on a ring and compute the longest possible edge length for a stable solution., 18 pages, 2 figures, 1 table

We propose a precise definition of a continuous time dynamical system made up of interacting open subsystems. The interconnections of subsystems are coded by directed graphs. We prove that the appropriate maps of graphs called graph fibrations give rise to maps of dynamical systems. Consequently surjective graph fibrations give rise to invariant subsystems and injective graph fibrations give rise to projections of dynamical systems.

We consider Markov chains with fast and slow variables and show that in a suitable scaling limit, the dynamics becomes deterministic, yet is far away from the standard mean field approximation. This new limit is an instance of self-induced stochastic resonance which arises due to matching between a rare event timescale on the one hand and the natural timescale separation in the underlying problem on the other. Here it is illustrated on a model of a molecular motor, where it is shown to explain the regularity of the motor gait observed in some experiments.

We consider the effect of stochastic forcing on a one-dimensional excitable reaction–diffusion system, where the noise only acts on a small subdomain of the medium. We show that there are distinguished scaling limits in which the stochastic forcing gives rise to a mean response of a periodic wavetrain of pulses which are deterministic in the limit. Thus, the system responds coherently and periodically to stochastic forcing. This is an instance of self-induced stochastic resonance, which has previously only been analysed for finite-dimensional systems.

We study energy transfer in a “resonant duet” — a resonant quartet where symmetries support a reduced subsystem with only two degrees of freedom — where one mode is forced by white noise and the other is damped. We consider a physically motivated family of nonlinear damping forms, and investigate their effect on the dynamics of the system. A variety of statistical steady-states arise in different parameter regimes, including intermittent bursting phases, non–equilibrium states highly constrained by slaving among amplitudes and phases, and Gaussian and non-Gaussian quasi–equilibrium regimes. All of this can be understood analytically using asymptotic techniques for stochastic differential equations.

We show that the set of itineraries for the complex exponential family λez is closely related, through the continued fraction expansion, to the set of Brjuno numbers.

We study the problem that arises in a class of stochastic processes referred to as Stochastic Hybrid Systems (SHS) when computing the moments of the states using the generator of the process and Dynkin's formula. We focus on the case when the SHS is at equilibrium or approaching equilibrium. We present a family of such processes for which infinite-dimensional linear-system analysis tools are ineffective, and discuss a few differing perspectives on how to tackle such problems by assuming that the SHS state distribution is such that its entropy is maximum. We also provide a numerical algorithm that allows us to efficiently compute maximum entropy solutions, and compare results with Monte Carlo simulations for some illustrative SHS.

we use symbolic dynamics to describe the set of allowable itineraries of orbits of the complex exponential family λexp(z). First, we show that the set of allowable itineraries is the same for every member of this family. We then show that the set of itineraries is the same for every map of finite exponential order. In addition, we study other transcendental entire functions, and show that they also have the same set of itineraries. Finally, we give an example of a class of functions with a different set of itineraries.

In this paper we consider the question of accessibility of points in the Julia sets of complex exponential functions in the case where the exponential admits an attracting cycle. In the case of an attracting fixed point it is known that the Julia set is a Cantor bouquet and that the only points accessible from the basin are the endpoints of the bouquet. In case the cycle has period two or greater, there are many more restrictions on which points in the Julia set are accessible. In this paper we give precise conditions for a point to be accessible in the periodic point case in terms of the kneading sequence for the cycle.

We propose a new framework for the study of continuous time dynamical systems on networks. We view such dynamical systems as collections of interacting control systems. We show that a class of maps between graphs called graph fibrations give rise to maps between dynamical systems on networks. This allows us to produce conjugacy between dynamical systems out of combinatorial data. In particular we show that surjective graph fibrations lead to synchrony subspaces in networks. The injective graph fibrations, on the other hand, give rise to surjective maps from large dynamical systems to smaller ones. One can view these surjections as a kind of "fast/slow" variable decompositions or as "abstractions" in the computer science sense of the word., 37 pages. Major revision of arXiv:1008.5359 [math.DS]. Following referees' suggestions we made the paper more accessible for applied dynamicists

We carefully examine two mechanisms--coherence resonance and self-induced stochastic resonance--by which small random perturbations of excitable systems with large time scale separation may lead to the emergence of new coherent behaviors in the form of limit cycles. We analyze what controls the degree of coherence in these two mechanisms and classify their very different properties. In particular we show that coherence resonance arises only at the onset of bifurcation and is rather insensitive against variations in the noise amplitude and the time scale separation ratio. In contrast, self-induced stochastic resonance may arise away from bifurcations and the properties of the limit cycle it induces are controlled by both the noise amplitude and the time scale separation ratio.

Dept. Mathematics, University of Illinois at Urbana-Champaign, M/C 382, Urbana, IL 61801Some confusion appears in the literature regarding the extension of the method of multiple scales to three or more timescales. While the work of Murdock and Wang [1] correctly indicates some obstructions to such an extension in some typesof problems, other work suggests that the extension should almost never be possible [2]. These pessimistic results generallyfollow from the imposition of additional restrictions in the calculation, which really are not necessary [3]. We will showon some simple ODE models how a systematic implementation of the method of multiple scales can succeed in correctlycapturing three active time scales, though more calculation is required than one might expect from naive considerations. Onthe other hand, an extension of the method of averaging to three time scales produces an incorrect approximation in theseproblems [4].

We present a detailed analysis of the stability of synchronized solutions to the Kuramoto system of oscillators. We derive an analytical expression counting the dimension of the unstable manifold associated to a given stationary solution. From this we are able to derive a number of consequences, including: analytic expressions for the first and last frequency vectors to synchronize, upper and lower bounds on the probability that a randomly chosen frequency vector will synchronize, and very sharp results on the large $N$ limit of this model. One of the surprises in this calculation is that for frequencies that are Gaussian distributed the correct scaling for full synchrony is not the one commonly studied in the literature---rather, there is a logarithmic correction to the scaling which is related to the extremal value statistics of the random frequency vector., 25 Pages, 4 Figures