Driven synchronization in random networks of oscillators

Resilience of the slow component in timescale-separated synchronized oscillators

Physiological networks are usually made of a large number of biological oscillators evolving on a multitude of different timescales. Phase oscillators are particularly useful in the modelling of the synchronization dynamics of such systems. If the coupling is strong enough compared to the heterogeneity of the internal parameters, synchronized states might emerge where phase oscillators start to behave coherently. Here, we focus on the case where synchronized oscillators are divided into a fast and a slow component so that the two subsets evolve on separated timescales. We assess the resilience of the slow component by, first, reducing the dynamics of the fast one using Mori-Zwanzig formalism. Second, we evaluate the variance of the phase deviations when the oscillators in the two components are subject to noise with possibly distinct correlation times. From the general expression for the variance, we consider specific network structures and show how the noise transmission between the fast and slow components is affected. Interestingly, we find that oscillators that are among the most robust when there is only a single timescale, might become the most vulnerable when the system undergoes a timescale separation. We also find that layered networks seem to be insensitive to such timescale separations.

Stability of Kuramoto networks subject to large and small fluctuations from heterogeneous and spatially correlated noise

Oscillatory networks subjected to noise are broadly used to model physical and technological systems. Due to their nonlinear coupling, such networks typically have multiple stable and unstable states that a network might visit due to noise. In this article, we focus on the assessment of fluctuations resulting from heterogeneous and spatially correlated noise inputs on Kuramoto model networks. We evaluate the typical, small fluctuations near synchronized states and connect the network variance to the overlap between stable modes of synchronization and the input noise covariance. Going beyond small to large fluctuations, we introduce the indicator mode approximation that projects the dynamics onto a single amplitude dimension. Such an approximation allows for estimating rates of fluctuations to saddle instabilities, resulting in phase slips between connected oscillators. Statistics for both regimes are quantified in terms of effective noise amplitudes that are compared and contrasted for several noise models. Bridging the gap between small and large fluctuations, we show that a larger network variance does not necessarily lead to higher rates of large fluctuations.

Dynamics of periodically forced finite N-oscillators, with implications for the social synchronization of animal rest-activity rhythms

The possible mechanisms for the synchronization of rest-activity rhythms of individual animals living in groups is a relatively understudied question; synchronized rhythms could occur by entrainment of individuals to a common external force and/or by social synchronization between individuals. To gain insight into this question, we explored the synchronization dynamics of populations of globally coupled Kuramoto oscillators and analyzed the effects of a finite oscillator number (N) and the variable strengths of their periodic forcing (F) and mutual coupling (K). We found that increasing N promotes entrainment to a decreasing value of F, but that F could not be reduced below a certain level determined by the number of oscillators and the distribution width of their intrinsic frequencies. Our analysis prompts some simple predictions of ecologically optimal animal group sizes under differing natural conditions.

Unstable modes and bistability in delay-coupled swarms

It is known that introducing time delays into the communication network of mobile-agent swarms produces coherent rotational patterns, from both theory and experiments. Often such spatiotemporal rotations can be bistable with other swarming patterns, such as milling and flocking. Yet, most known bifurcation results related to delay-coupled swarms rely on inaccurate mean-field techniques. As a consequence, the utility of applying macroscopic theory as a guide for predicting and controlling swarms of mobile robots has been limited. To overcome this limitation, we perform an exact stability analysis of two primary swarming patterns in a general model with time-delayed interactions. By correctly identifying the relevant spatiotemporal modes, we are able to accurately predict unstable oscillations beyond the mean-field dynamics and bistability in large swarms-laying the groundwork for comparisons to robotics experiments.

Network desynchronization by non-Gaussian fluctuations

Many networks must maintain synchrony despite the fact that they operate in noisy environments. Important examples are stochastic inertial oscillators, which are known to exhibit fluctuations with broad tails in many applications, including electric power networks with renewable energy sources. Such non-Gaussian fluctuations can result in rare network desynchronization. Here we build a general theory for inertial oscillator network desynchronization by non-Gaussian noise. We compute the rate of desynchronization and show that higher moments of noise enter at specific powers of coupling: either speeding up or slowing down the rate exponentially depending on how noise statistics match the statistics of a network's slowest mode. Finally, we use our theory to introduce a technique that drastically reduces the effective description of network desynchronization. Most interestingly, when instability is associated with a single edge, the reduction is to one stochastic oscillator.

Homoclinic-doubling and homoclinic-gluing bifurcations in the Takens-Bogdanov normal form with D 4 symmetry

In this paper, we investigate the dynamics of a fourth-order normal form near a double Takens-Bogdanov bifurcation. The reduced system of this normal form possesses eight pairs of homoclinic orbits for certain parameter values. The nonlinear time transformation method is applied to obtain an analytical approximation of the homoclinic orbit in the perturbed system and to construct the homoclinic bifurcation curve as well. Using numerical continuation, period-doubling and homoclinic-doubling cascades emanating from a codimension-2 bifurcation point are found. A codimension-2 homoclinic-gluing bifurcation point at which several homoclinic orbits concerning the origin glue together to form a new homoclinic orbit is also obtained. It is shown that in the vicinity of these bifurcation points, the system may exhibit chaos and chaotic attractors.

Rare slips in fluctuating synchronized oscillator networks

We study rare phase slips due to noise in synchronized Kuramoto oscillator networks. In the small-noise limit, we demonstrate that slips occur via large fluctuations to saddle phase-locked states. For tree topologies, slips appear between subgraphs that become disconnected at a saddle-node bifurcation, where phase-locked states lose stability generically. This pattern is demonstrated for sparse networks with several examples. Scaling laws are derived and compared for different tree topologies. On the other hand, for dense networks slips occur between oscillators on the edges of the frequency distribution. If the distribution is discrete, the probability-exponent for large fluctuations to occur scales linearly with the system size. However, if the distribution is continuous, the probability is a constant in the large network limit, as individual oscillators fluctuate to saddles while all others remain fixed. In the latter case, the network's coherence is approximately preserved.

Self-similarity in explosive synchronization of complex networks

We report that explosive synchronization of networked oscillators (a process through which the transition to coherence occurs without intermediate stages but is rather characterized by a sudden and abrupt jump from the network's asynchronous to synchronous motion) is related to self-similarity of synchronous clusters of different size. Self-similarity is revealed by destructing the network synchronous state during the backward transition and observed with the decrease of the coupling strength between the nodes of the network. As illustrative examples, networks of Kuramoto oscillators with different topologies of links have been considered. For each one of such topologies, the destruction of the synchronous state goes step by step with self-similar configurations of interacting oscillators. At the critical point, the invariance of the phase distribution in the synchronized cluster with respect to the cluster size is reported.

Hybrid dynamics in delay-coupled swarms with mothership networks

Swarming behavior continues to be a subject of immense interest because of its centrality in many naturally occurring systems in physics and biology, as well as its importance in applications such as robotics. Here we examine the effects on swarm pattern formation from delayed communication and topological heterogeneity, and in particular, the inclusion of a relatively small number of highly connected nodes, or "motherships," in a swarm's communication network. We find generalized forms of basic patterns for networks with general degree distributions, and a variety of dynamic behaviors including parameter regions with multistability and hybrid motions in bimodal networks. The latter is an interesting example of how heterogeneous networks can have dynamics that is a mix of different states in homogeneous networks, where high- and low-degree nodes have distinct behavior simultaneously.