Search

We study a system of coupled phase oscillators near a saddle-node on an invariant circle bifurcation and driven by random intrinsic frequencies. Under the variation of control parameters, the system undergoes a phase transition changing the qualitative properties of collective dynamics. Using the Ott-Antonsen reduction and geometric techniques for ordinary differential equations, we identify a heteroclinic bifurcation in a family of vector fields on a cylinder, which explains the change in collective dynamics. Specifically, we show that the heteroclinic bifurcation separates two topologically distinct families of limit cycles: contractible limit cycles before the bifurcation from noncontractibile ones after the bifurcation. Both families are stable for the model at hand., 22 pages, 12 figures

In his classical work on synchronization, Kuramoto derived the formula for the critical value of the coupling strength corresponding to the transition to synchrony in large ensembles of all-to-all coupled phase oscillators with randomly distributed intrinsic frequencies. We extend this result to a large class of coupled systems on convergent families of deterministic and random graphs. Specifically, we identify the critical values of the coupling strength (transition points), between which the incoherent state is linearly stable and is unstable otherwise. We show that the transition points depend on the largest positive or/and smallest negative eigenvalue(s) of the kernel operator defined by the graph limit. This reveals the precise mechanism, by which the network topology controls transition to synchrony in the Kuramoto model on graphs. To illustrate the analysis with concrete examples, we derive the transition point formula for the coupled systems on Erdős-Renyi, small-world, and \begin{document}$ k$\end{document} -nearest-neighbor families of graphs. As a result of independent interest, we provide a rigorous justification for the mean field limit for the Kuramoto model on graphs. The latter is used in the derivation of the transition point formulas. In the second part of this work [ 8 ], we study the bifurcation corresponding to the onset of synchronization in the Kuramoto model on convergent graph sequences.

In our previous work [ 3 ], we initiated a mathematical investigation of the onset of synchronization in the Kuramoto model (KM) of coupled phase oscillators on convergent graph sequences. There, we derived and rigorously justified the mean field limit for the KM on graphs. Using linear stability analysis, we identified the critical values of the coupling strength, at which the incoherent state looses stability, thus, determining the onset of synchronization in this model. In the present paper, we study the corresponding bifurcations. Specifically, we show that similar to the original KM with all-to-all coupling, the onset of synchronization in the KM on graphs is realized via a pitchfork bifurcation. The formula for the stable branch of the bifurcating equilibria involves the principal eigenvalue and the corresponding eigenfunctions of the kernel operator defined by the limit of the graph sequence used in the model. This establishes an explicit link between the network structure and the onset of synchronization in the KM on graphs. The results of this work are illustrated with the bifurcation analysis of the KM on Erdős-Renyi, small-world, as well as certain weighted graphs on a circle.

The instability of mixing in the Kuramoto model of coupled phase oscillators is the key to understanding a range of spatiotemporal patterns, which feature prominently in collective dynamics of systems ranging from neuronal networks, to coupled lasers, to power grids. In this paper, we describe a codimension-2 bifurcation of mixing whose unfolding, in addition to the classical scenario of the onset of synchronization, also explains the formation of clusters and chimeras. We use a combination of linear stability analysis and Penrose diagrams to identify and analyze a variety of spatiotemporal patterns including stationary and traveling coherent clusters and twisted states, as well as their combinations with regions of incoherent behavior called chimera states. The linear stability analysis is used to estimate of the velocity distribution within these structures. Penrose diagrams, on the other hand, predict accurately the basins of their existence. Furthermore, we show that network topology can endow chimera states with nontrivial spatial organization. In particular, we present twisted chimera states, whose coherent regions are organized as stationary or traveling twisted states. The analytical results are illustrated with numerical bifurcation diagrams computed for the Kuramoto model with uni-, bi-, and tri-modal frequency distributions and all-to-all and nonlocal nearest-neighbor connectivity., 18 pages, 7 figures

The Kuramoto model of coupled second order damped oscillators on convergent sequences of graphs is analyzed in this work. The oscillators in this model have random intrinsic frequencies and interact with each other via nonlinear coupling. The connectivity of the coupled system is assigned by a graph which may be random as well. In the thermodynamic limit the behavior of the system is captured by the Vlasov equation, a hyperbolic partial differential equation for the probability distribution of the oscillators in the phase space. We study stability of mixing, a steady state solution of the Vlasov equation, corresponding to the uniform distribution of phases. Specifically, we identify a critical value of the strength of coupling, at which the system undergoes a pitchfork bifurcation. It corresponds to the loss of stability of mixing and marks the onset of synchronization. As for the classical Kuramoto model, the presence of the continuous spectrum on the imaginary axis poses the main difficulty for the stability analysis. To overcome this problem, we use the methods from the generalized spectral theory developed for the original Kuramoto model. The analytical results are illustrated with numerical bifurcation diagrams computed for the Kuramoto model on Erd\H{o}s--R\'enyi and small-world graphs. Applications of the second-order Kuramoto model include power networks, coupled pendula, and various biological networks. The analysis in this paper provides a mathematical description of the onset of synchronization in these systems.

The Kuramoto model of coupled phase oscillators with inertia on Erdos-Renyi graphs is analyzed in this work. For a system with intrinsic frequencies sampled from a bimodal distribution we identify a variety of two cluster patterns and study their stability. To this end, we decompose the description of the cluster dynamics into two systems: one governing the (macro) dynamics of the centers of mass of the two clusters and the second governing the (micro) dynamics of individual oscillators inside each cluster. The former is a low-dimensional ODE whereas the latter is a system of two coupled Vlasov PDEs. Stability of the cluster dynamics depends on the stability of the low-dimensional group motion and on coherence of the oscillators in each group. We show that the loss of coherence in one of the clusters leads to the loss of stability of a two-cluster state and to formation of chimera states. The analysis of this paper can be generalized to cover states with more than two clusters and to coupled systems on W-random graphs. Our results apply to a model of a power grid with fluctuating sources., 22 pages, 11 figures

We correct and improve the main result in Medvedev, “The nonlinear heat equation on W-random graphs”, Arch. Rational Mech. Anal., 212(3), pp. 781–803.

The relation between the structural properties of the network and its dynamics is a central question in the analysis of dynamical networks. It is especially relevant for complex networks found in real-world applications. This work presents mathematically rigorous analysis of coupled dynamical systems on power law graphs. Specifically, we study large systems of coupled Kuramoto phase oscillators. In the limit as the size of the network tends to infinity, we derive analytically tractable mean field partial differential equation for the probability density function describing the state of the coupled system. The mean field limit is used to establish an explicit formula for the synchronization threshold for coupled phase oscillators with randomly distributed intrinsic frequencies. Furthermore, we study stable spatial patterns generated by the Kuramoto model with repulsive coupling. In particular, we identify a family of stable steady-state solutions having multiple regions with distinct statistical properties. We call these solutions contrast states. Like chimera states, contrast states exhibit coexisting regions of highly localized (coherent) behavior and highly irregular (incoherent) distribution of phases. We provide a detailed mathematical analysis of contrast states in the KM using the Ott–Antonsen ansatz. The analysis of synchronization and contrast states provides new insights into the role of power law connectivity in shaping dynamics of coupled dynamical systems. In particular, we show that despite sparse connectivity, power law networks possess remarkable synchronizability: the synchronization threshold can be made arbitrarily low by varying the parameter of the power law distribution.

We derive a sufficient condition for stability in probability of an equilibrium of a randomly perturbed map in \begin{document}$\mathbb{R}^d$\end{document} . This condition can be used to stabilize unstable equilibria by random forcing. Analytical results on stabilization are illustrated with numerical examples of randomly perturbed nonlinear maps in one-and two-dimensional spaces.

In this paper, we study convergence of coupled dynamical systems on convergent sequences of graphs to a continuum limit. We show that the solutions of the initial value problem for the dynamical system on a convergent graph sequence tend to that for the nonlocal diffusion equation on a unit interval, as the graph size tends to infinity. We improve our earlier results in [Arch. Ration. Mech. Anal., 21 (2014), pp. 781--803] and extend them to a larger class of graphs, which includes directed and undirected, sparse and dense, random and deterministic graphs. There are three main ingredients of our approach. First, we employ a flexible framework for incorporating random graphs into the models of interacting dynamical systems, which fits seamlessly with the derivation of the continuum limit. Next, we prove the averaging principle for approximating a dynamical system on a random graph by its deterministic (averaged) counterpart. The proof covers systems on sparse graphs and yields almost sure convergence on time intervals of order $\log n,$ where $n$ is the number of vertices. Finally, a Galerkin scheme is developed to show convergence of the averaged model to the continuum limit. The analysis of this paper covers the Kuramoto model of coupled phase oscillators on a variety of graphs including sparse Erd\H{o}s-R{\' e}nyi, small-world, and power law graphs., Comment: To appear in Communications in Mathematical Sciences

The Kuramoto model of coupled phase oscillators on small-world (SW) graphs is analyzed in this work. When the number of oscillators in the network goes to infinity, the model acquires a family of steady state solutions of degree q , called q -twisted states. We show that this class of solutions plays an important role in the formation of spatial patterns in the Kuramoto model on SW graphs. In particular, the analysis of q -twisted states elucidates the role of long-range random connections in shaping the attractors in this model. We develop two complementary approaches for studying q -twisted states in the coupled oscillator model on SW graphs: linear stability analysis and numerical continuation. The former approach shows that long-range random connections in the SW graphs promote synchronization and yields the estimate of the synchronization rate as a function of the SW randomization parameter. The continuation shows that the increase of the long-range connections results in patterns consisting of one or several plateaus separated by sharp interfaces. These results elucidate the pattern formation mechanisms in nonlocally coupled dynamical systems on random graphs.

We study a nonlocal diffusion equation approximating the dynamics of coupled phase oscillators on large graphs. Under appropriate assumptions, the model has a family of steady state solutions called twisted states. We prove a sufficient condition for stability of twisted states with respect to perturbations in the Sobolev and BV spaces. As an application, we study the stability of twisted states in the Kuramoto model on small-world graphs.

Using the theory of $L^p$-graphons [C. Borgs et al., preprint, arXiv:1401.2906, 2014; C. Borgs et al., preprint, arXiv:1408.0744, 2014], we derive and rigorously justify the continuum limit for systems of differential equations on sparse random graphs. Specifically, we show that the solutions of the initial value problems for the discrete models can be approximated by those of an appropriate nonlocal diffusion equation. Our results apply to a range of spatially extended dynamical models of different physical, biological, social, and economic networks. Importantly, our assumptions cover network topologies featured in many important real-world networks. In particular, we derive the continuum limit for coupled dynamical systems on power law graphs. The latter is the main motivation for this work.

In his classical work, Kuramoto analytically described the onset of synchronization in all-to-all coupled networks of phase oscillators with random intrinsic frequencies. Specifically, he identified a critical value of the coupling strength, at which the incoherent state loses stability and a gradual build-up of coherence begins. Recently, Kuramoto's scenario was shown to hold for a large class of coupled systems on convergent families of deterministic and random graphs. Guided by these results, in the present work, we study several model problems illustrating the link between network topology and synchronization in coupled dynamical systems. First, we identify several families of graphs, for which the transition to synchronization in the Kuramoto model starts at the same critical value of the coupling strength and proceeds in practically the same way. These examples include Erd\H{o}s-R\'enyi random graphs, Paley graphs, complete bipartite graphs, and certain stochastic block graphs. These examples illustrate that some rather simple structural properties such as the volume of the graph may determine the onset of synchronization, while finer structural features may affect only higher order statistics of the transition to synchronization. Further, we study the transition to synchronization in the Kuramoto model on power law and small-world random graphs. The former family of graphs endows the Kuramoto model with very good synchronizability: the synchronization threshold can be made arbitrarily low by varying the parameter of the power law degree distribution. For the Kuramoto model on small-world graphs, in addition to the transition to synchronization, we identify a new bifurcation leading to stable random twisted states. The examples analyzed in this work complement the results in [Chiba, Medvedev, The mean field analysis for the Kuramoto model on graphs (parts I and II), arxiv]., Comment: 18 pages, 12 figures

For a class of coupled limit cycle oscillators, we give a condition on a linear coupling operator that is necessary and sufficient for exponential stability of the synchronous solution. We show that with certain modifications our method of analysis applies to networks with partial, time-dependent, and nonlinear coupling schemes, as well as to ensembles of local systems with nonperiodic attractors. We also study robustness of synchrony to noise. To this end, we analytically estimate the degree of coherence of the network oscillations in the presence of noise. Our estimate of coherence highlights the main ingredients of stochastic stability of the synchronous regime. In particular, it quantifies the contribution of the network topology. The estimate of coherence for the randomly perturbed network can be used as means for analytic inference of degree of stability of the synchronous solution of the unperturbed deterministic network. Furthermore, we show that in large networks, the effects of noise on the dynamics of each oscillator can be effectively controlled by varying the strength of coupling, which provides a powerful mechanism of denoising. This suggests that the organization of oscillators in a coupled network may play an important role in maintaining robust oscillations in random environment. The analysis is complemented with the results of numerical simulations of a neuronal network. PACS: 05.45.Xt, 05.40.Ca Keywords: synchronization, coupled oscillators, denoising, robustness to noise, compartmental model, major revisions; two new sections

We prove a sufficient condition for synchronization for coupled one-dimensional maps and estimate the size of the window of parameters where synchronization takes place. It is shown that coupled systems on graphs with positive eigenvalues (EVs) of the normalized graph Laplacian concentrated around 1 are more amenable for synchronization. In the light of this condition, we review spectral properties of Cayley, quasirandom, power-law graphs, and expanders and relate them to synchronization of the corresponding networks. The analysis of synchronization on these graphs is illustrated with numerical experiments. The results of this paper highlight the advantages of random connectivity for synchronization of coupled chaotic dynamical systems., Comment: Keywords: synchronization, chaos, Cayley graph, quasirandom graph, power-law graph, expander

We use qualitative methods for singularly perturbed systems of differential equations and the principle of averaging to compute the first return map for the dynamics of a slow variable (calcium concentration) in the model of an excitable cell. The bifurcation structure of the system with continuous time endows the map with distinct features: it is a unimodal map with a boundary layer corresponding to the homoclinic bifurcation in the original model. This structure accounts for different periodic and aperiodic regimes and transitions between them. All parameters in the discrete system have biophysical meaning, which allows for precise interpretation of various dynamical patterns. Our results provide analytical explanation for the numerical studies reported previously.

We study chains of strongly electrically coupled relaxation oscillators modeling dopamine neurons. When individual oscillators are in the regime close to an Andronov–Hopf bifurcation (AHB), the coupled system exhibits a variety of oscillatory behavior. We show that the proximity of individual oscillators to the AHB has a significant impact on the system dynamics in a wide range of parameters. It manifests itself through a family of stable multimodal periodic solutions that are composed out of large-amplitude relaxation oscillations and small-amplitude oscillations. This family of solutions has a rich bifurcation structure. The waveform and the period vary greatly across the family. The structure and bifurcations of the stable periodic solutions of the coupled system are investigated using numerical and analytic techniques.

Transient increases in spontaneous firing rate of mesencephalic dopaminergic neurons have been suggested to act as a reward prediction error signal. A mechanism previously proposed involves subthreshold calcium-dependent oscillations in all parts of the neuron. In that mechanism, the natural frequency of oscillation varies with diameter of cell processes, so there is a wide variation of natural frequencies on the cell, but strong voltage coupling enforces a single frequency of oscillation under resting conditions. In previous work, mathematical analysis of a simpler system of oscillators showed that the chain of oscillators could produce transient dynamics in which the frequency of the coupled system increased temporarily, as seen in a biophysical model of the dopaminergic neuron. The transient dynamics was shown to be consequence of a slow drift along an invariant subset of phase space, with rate of drift given by a Lyapunov function. In this paper, we show that the same mathematical structure exists for the full biophysical model, giving physiological meaning to the slow drift and the Lyapunov function, which is shown to describe differences in intracellular calcium concentration in different parts of the cell. The duration of transients was long, being comparable to the time constant of calcium disposition. These results indicate that brief changes in input to the dopaminergic neuron can produce long lasting firing rate transients whose form is determined by intrinsic cell properties.

The Kuramoto model (KM) of coupled phase oscillators on complete, Paley, and Erdos-Renyi (ER) graphs is analyzed in this work. As quasirandom graphs, the complete, Paley, and ER graphs share many structural properties. For instance, they exhibit the same asymptotics of the edge distributions, homomorphism densities, graph spectra, and have constant graph limits. Nonetheless, we show that the asymptotic behavior of solutions in the KM on these graphs can be qualitatively different. Specifically, we identify twisted states, steady state solutions of the KM on complete and Paley graphs, which are stable for one family of graphs but not for the other. On the other hand, we show that the solutions of the IVPs for the KM on complete and random graphs remain close on finite time intervals, provided they start from close initial conditions and the graphs are sufficiently large. Therefore, the results of this paper elucidate the relation between the network structure and dynamics in coupled nonlinear dynamical systems. Furthermore, we present new results on synchronization and stability of twisted states for the KM on Cayley and random graphs., Journal of Nonlinear Science, 2015

The leading term of the error of eigenvalues of a discrete analog of the eigenvalue problem for an elliptic operator with variable coefficients is obtained. A method for refining eigenvalues by evaluating a correction with the help of a discrete problem of second-order accuracy is proposed.

For systems of coupled differential equations on a sequence of W-random graphs, we derive the continuum limit in the form of an evolution integral equation. We prove that solutions of the initial value problems (IVPs) for the discrete model converge to the solution of the IVP for its continuum limit. These results combined with the analysis of nonlocally coupled deterministic networks in Medvedev (The nonlinear heat equation on dense graphs and graph limits. ArXiv e-prints, 2013) justify the continuum (thermodynamic) limit for a large class of coupled dynamical systems on convergent families of graphs.

We use the combination of ideas and results from the theory of graph limits and nonlinear evolution equations to provide a rigorous mathematical justification for taking continuum limit for certain nonlocally coupled networks and to extend this method to cover many complex networks, for which it has not been applied before. Specifically, for dynamical networks on convergent sequences of simple and weighted graphs, we prove convergence of solutions of the initial-value problems for discrete models to those of the limiting continuous equations. In addition, for sequences of simple graphs converging to {0, 1}-valued graphons, it is shown that the convergence rate depends on the fractal dimension of the boundary of the support of the graph limit. These results are then used to study the regions of continuity of chimera states and the attractors of the nonlocal Kuramoto equation on certain multipartite graphs. Furthermore, the analytical tools developed in this work are used in the rigorous justification of the continuum limit for networks on random graphs that we undertake in a companion paper (Medvedev, 2013). As a by-product of the analysis of the continuum limit on deterministic and random graphs, we identify the link between this problem and the convergence analysis of several classical numerical schemes: the collocation, Galerkin, and Monte-Carlo methods. Therefore, our results can be used to characterize convergence of these approximate methods of solving initial-value problems for nonlinear evolution equations with nonlocal interactions.

A system of autonomous differential equations with a stable limit cycle and perturbed by small white noise is analyzed in this work. In the vicinity of the limit cycle of the unperturbed deterministic system, we define, construct, and analyze the Poincare map of the randomly perturbed periodic motion. We show that the time of the first exit from a small neighborhood of the fixed point of the map, which corresponds to the unperturbed periodic orbit, is well approximated by the geometric distribution. The parameter of the geometric distribution tends to zero together with the noise intensity. Therefore, our result can be interpreted as an estimate of the stability of periodic motion to random perturbations. In addition, we show that the geometric distribution of the first exit times translates into statistical properties of solutions of important differential equation models in applications. To this end, we demonstrate three distinct examples from mathematical neuroscience featuring complex oscillatory patterns characterized by the geometric distribution. We show that in each of these models the statistical properties of emerging oscillations are fully explained by the general properties of randomly perturbed periodic motions identified in this paper.

The mathematical theory of pattern formation in electrically coupled networks of excitable neurons forced by small noise is presented in this work. Using the Freidlin–Wentzell large-deviation theory for randomly perturbed dynamical systems and the elements of the algebraic graph theory, we identify and analyze the main regimes in the network dynamics in terms of the key control parameters: excitability, coupling strength, and network topology. The analysis reveals the geometry of spontaneous dynamics in electrically coupled network. Specifically, we show that the location of the minima of a certain continuous function on the surface of the unit n-cube encodes the most likely activity patterns generated by the network. By studying how the minima of this function evolve under the variation of the coupling strength, we describe the principal transformations in the network dynamics. The minimization problem is also used for the quantitative description of the main dynamical regimes and transitions between them. In particular, for the weak and strong coupling regimes, we present asymptotic formulas for the network activity rate as a function of the coupling strength and the degree of the network. The variational analysis is complemented by the stability analysis of the synchronous state in the strong coupling regime. The stability estimates reveal the contribution of the network connectivity and the properties of the cycle subspace associated with the graph of the network to its synchronization properties. This work is motivated by the experimental and modeling studies of the ensemble of neurons in the Locus Coeruleus, a nucleus in the brainstem involved in the regulation of cognitive performance and behavior.