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.

The chaotic milling behaviors of interacting swarms after collision.

We consider the problem of characterizing the dynamics of interacting swarms after they collide and form a stationary center of mass. Modeling efforts have shown that the collision of near head-on interacting swarms can produce a variety of post-collision dynamics including coherent milling, coherent flocking, and scattering behaviors. In particular, recent analysis of the transient dynamics of two colliding swarms has revealed the existence of a critical transition whereby the collision results in a combined milling state about a stationary center of mass. In the present work, we show that the collision dynamics of two swarms that form a milling state transitions from periodic to chaotic motion as a function of the repulsive force strength and its length scale. We used two existing methods as well as one new technique: Karhunen-Loeve decomposition to show the effective modal dimension chaos lives in, the 0-1 test to identify chaos, and then constrained correlation embedding to show how each swarm is embedded in the other when both swarms combine to form a single milling state after collision. We expect our analysis to impact new swarm experiments which examine the interaction of multiple swarms.

Outbreak Size Distribution in Stochastic Epidemic Models.

Motivated by recent epidemic outbreaks, including those of COVID-19, we solve the canonical problem of calculating the dynamics and likelihood of extensive outbreaks in a population within a large class of stochastic epidemic models with demographic noise, including the susceptible-infected-recovered (SIR) model and its general extensions. In the limit of large populations, we compute the probability distribution for all extensive outbreaks, including those that entail unusually large or small (extreme) proportions of the population infected. Our approach reveals that, unlike other well-known examples of rare events occurring in discrete-state stochastic systems, the statistics of extreme outbreaks emanate from a full continuum of Hamiltonian paths, each satisfying unique boundary conditions with a conserved probability flux.

Delay induced swarm pattern bifurcations in mixed reality experiments.

Swarms of coupled mobile agents subject to inter-agent wireless communication delays are known to exhibit multiple dynamic patterns in space that depend on the strength of the interactions and the magnitude of the communication delays. We experimentally demonstrate communication delay-induced bifurcations in the spatiotemporal patterns of robot swarms using two distinct hardware platforms in a mixed reality framework. Additionally, we make steps toward experimentally validating theoretically predicted parameter regions where transitions between swarm patterns occur. We show that multiple rotation patterns persist even when collision avoidance strategies are incorporated, and we show the existence of multi-stable, co-existing rotational patterns not predicted by usual mean field dynamics. Our experiments are the first significant steps toward validating existing theory and the existence and robustness of the delay-induced patterns in real robotic swarms.

Torus bifurcations of large-scale swarms having range dependent communication delay.

Dynamical emergent patterns of swarms are now fairly well established in nature and include flocking and rotational states. Recently, there has been great interest in engineering and physics to create artificial self-propelled agents that communicate over a network and operate with simple rules, with the goal of creating emergent self-organizing swarm patterns. In this paper, we show that when communicating networks have range dependent delays, rotational states, which are typically periodic, undergo a bifurcation and create swarm dynamics on a torus. The observed bifurcation yields additional frequencies into the dynamics, which may lead to quasi-periodic behavior of the swarm.

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.

Epidemic Extinction and Control in Heterogeneous Networks.

We consider epidemic extinction in finite networks with a broad variation in local connectivity. Generalizing the theory of large fluctuations to random networks with a given degree distribution, we are able to predict the most probable, or optimal, paths to extinction in various configurations, including truncated power laws. We find that paths for heterogeneous networks follow a limiting form in which infection first decreases in low-degree nodes, which triggers a rapid extinction in high-degree nodes, and finishes with a residual low-degree extinction. The usefulness of our approach is further demonstrated through optimal control strategies that leverage the dependence of finite-size fluctuations on network topology. Interestingly, we find that the optimal control is a mix of treating both high- and low-degree nodes based on theoretical predictions, in contrast to methods that ignore dynamical fluctuations.

A Framework for Inferring Unobserved Multistrain Epidemic Subpopulations Using Synchronization Dynamics.

A new method is proposed to infer unobserved epidemic subpopulations by exploiting the synchronization properties of multistrain epidemic models. A model for dengue fever is driven by simulated data from secondary infective populations. Primary infective populations in the driven system synchronize to the correct values from the driver system. Most hospital cases of dengue are secondary infections, so this method provides a way to deduce unobserved primary infection levels. We derive center manifold equations that relate the driven system to the driver system and thus motivate the use of synchronization to predict unobserved primary infectives. Synchronization stability between primary and secondary infections is demonstrated through numerical measurements of conditional Lyapunov exponents and through time series simulations.

Breaking the symmetry: immune enhancement increases persistence of dengue viruses in the presence of asymmetric transmission rates.

The dengue viruses exist as four antigenically distinct serotypes. These four serotypes co-circulate and interact with each other through multiple immune-mediated mechanisms. Though the majority of previous efforts to understand the transmission dynamics of dengue have assumed identical characteristics for these four serotypes, empirical data suggests that they differ from one another in important ways. Here, we examine dynamics and persistence in models that do not assume symmetry between the dengue viruses. We find that for serotype transmission rates that are only slightly asymmetric, increased transmissibility of secondary infections through immune enhancement increases the persistence of all dengue viruses in opposition to findings in symmetric models. We identify an optimal magnitude of immune enhancement that maximizes the probability of persistence of all four serotypes. In contrast to other pathogen systems where heterogeneity between serotypes in transmissibility facilitates competitive exclusion (Bremmermann and Thieme, 1989), here we find that in the presence of Antibody Dependent Enhancement (ADE) heterogeneity can increase the persistence of multiple serotypes of dengue.

Predicting unobserved exposures from seasonal epidemic data.

We consider a stochastic Susceptible-Exposed-Infected-Recovered (SEIR) epidemiological model with a contact rate that fluctuates seasonally. Through the use of a nonlinear, stochastic projection, we are able to analytically determine the lower dimensional manifold on which the deterministic and stochastic dynamics correctly interact. Our method produces a low dimensional stochastic model that captures the same timing of disease outbreak and the same amplitude and phase of recurrent behavior seen in the high dimensional model. Given seasonal epidemic data consisting of the number of infectious individuals, our method enables a data-based model prediction of the number of unobserved exposed individuals over very long times.

Coherent Pattern Prediction in Swarms of Delay-Coupled Agents.

We consider a general swarm model of self-propelling agents interacting through a pairwise potential in the presence of noise and communication time delay. Previous work has shown that a communication time delay in the swarm induces a pattern bifurcation that depends on the size of the coupling amplitude. We extend these results by completely unfolding the bifurcation structure of the mean field approximation. Our analysis reveals a direct correspondence between the different dynamical behaviors found in different regions of the coupling-time delay plane with the different classes of simulated coherent swarm patterns. We derive the spatiotemporal scales of the swarm structures, as well as demonstrate how the complicated interplay of coupling strength, time delay, noise intensity, and choice of initial conditions can affect the swarm. In particular, our studies show that for sufficiently large values of the coupling strength and/or the time delay, there is a noise intensity threshold that forces a transition of the swarm from a misaligned state into an aligned state. We show that this alignment transition exhibits hysteresis when the noise intensity is taken to be time dependent.

Modeling synchronization in networks of delay-coupled fiber ring lasers.

We study the onset of synchronization in a network of N delay-coupled stochastic fiber ring lasers with respect to various parameters when the coupling power is weak. In particular, for groups of three or more ring lasers mutually coupled to a central hub laser, we demonstrate a robust tendency toward out-of-phase (achronal) synchronization between the N-1 outer lasers and the single inner laser. In contrast to the achronal synchronization, we find the outer lasers synchronize with zero-lag (isochronal) with respect to each other, thus forming a set of N-1 coherent fiber lasers.

Maximal sensitive dependence and the optimal path to epidemic extinction.

Extinction of an epidemic or a species is a rare event that occurs due to a large, rare stochastic fluctuation. Although the extinction process is dynamically unstable, it follows an optimal path that maximizes the probability of extinction. We show that the optimal path is also directly related to the finite-time Lyapunov exponents of the underlying dynamical system in that the optimal path displays maximum sensitivity to initial conditions. We consider several stochastic epidemic models, and examine the extinction process in a dynamical systems framework. Using the dynamics of the finite-time Lyapunov exponents as a constructive tool, we demonstrate that the dynamical systems viewpoint of extinction evolves naturally toward the optimal path.

Set-based corral control in stochastic dynamical systems: making almost invariant sets more invariant.

We consider the problem of stochastic prediction and control in a time-dependent stochastic environment, such as the ocean, where escape from an almost invariant region occurs due to random fluctuations. We determine high-probability control-actuation sets by computing regions of uncertainty, almost invariant sets, and Lagrangian coherent structures. The combination of geometric and probabilistic methods allows us to design regions of control, which provide an increase in loitering time while minimizing the amount of control actuation. We show how the loitering time in almost invariant sets scales exponentially with respect to the control actuation, causing an exponential increase in loitering times with only small changes in actuation force. The result is that the control actuation makes almost invariant sets more invariant.

Optimal periodic closure for minimizing risk in emerging disease outbreaks.

Without vaccines and treatments, societies must rely on non-pharmaceutical intervention strategies to control the spread of emerging diseases such as COVID-19. Though complete lockdown is epidemiologically effective, because it eliminates infectious contacts, it comes with significant costs. Several recent studies have suggested that a plausible compromise strategy for minimizing epidemic risk is periodic closure, in which populations oscillate between wide-spread social restrictions and relaxation. However, no underlying theory has been proposed to predict and explain optimal closure periods as a function of epidemiological and social parameters. In this work we develop such an analytical theory for SEIR-like model diseases, showing how characteristic closure periods emerge that minimize the total outbreak, and increase predictably with the reproductive number and incubation periods of a disease- as long as both are within predictable limits. Using our approach we demonstrate a sweet-spot effect in which optimal periodic closure is maximally effective for diseases with similar incubation and recovery periods. Our results compare well to numerical simulations, including in COVID-19 models where infectivity and recovery show significant variation.

Critical transition for colliding swarms.

Swarming patterns that emerge from the interaction of many mobile agents are a subject of great interest in fields ranging from biology to physics and robotics. In some application areas, multiple swarms effectively interact and collide, producing complex spatiotemporal patterns. Recent studies have begun to address swarm-on-swarm dynamics, and in particular the scattering of two large, colliding swarms with nonlinear interactions. To build on early numerical insights, we develop a self-propelled, rigid-body approximation that can be used to predict the parameters under which colliding swarms are expected to form a milling state. Our analytical method relies on the assumption that, upon collision, two swarms oscillate near a limit cycle, where each swarm rotates around the other while maintaining an approximately constant and uniform density. Using this approach we are able to predict the critical swarm-on-swarm interaction coupling, below which two colliding swarms merely scatter, as a function of physical swarm parameters. We show that the critical coupling gives a lower bound for all impact parameters, including head-on collision, and corresponds to a saddle-node bifurcation of a stable limit cycle in the uniform, constant density approximation. Our results are tested and found to agree with both small and large multiagent simulations.

Swarm shedding in networks of self-propelled agents.

Understanding swarm pattern formation is of great interest because it occurs naturally in many physical and biological systems, and has artificial applications in robotics. In both natural and engineered swarms, agent communication is typically local and sparse. This is because, over a limited sensing or communication range, the number of interactions an agent has is much smaller than the total possible number. A central question for self-organizing swarms interacting through sparse networks is whether or not collective motion states can emerge where all agents have coherent and stable dynamics. In this work we introduce the phenomenon of swarm shedding in which weakly-connected agents are ejected from stable milling patterns in self-propelled swarming networks with finite-range interactions. We show that swarm shedding can be localized around a few agents, or delocalized, and entail a simultaneous ejection of all agents in a network. Despite the complexity of milling motion in complex networks, we successfully build mean-field theory that accurately predicts both milling state dynamics and shedding transitions. The latter are described in terms of saddle-node bifurcations that depend on the range of communication, the inter-agent interaction strength, and the network topology.

Switching exponent scaling near bifurcation points for non-Gaussian noise.

We study noise-induced switching of a system close to bifurcation parameter values where the number of stable states changes. For non-Gaussian noise, the switching exponent, which gives the logarithm of the switching rate, displays a non-power-law dependence on the distance to the bifurcation point. This dependence is found for Poisson noise. Even weak additional Gaussian noise dominates switching sufficiently close to the bifurcation point, leading to a crossover in the behavior of the switching exponent.

Stability of milling patterns in self-propelled swarms on surfaces.

In some physical and biological swarms, agents effectively move and interact along curved surfaces. The associated constraints and symmetries can affect collective-motion patterns, but little is known about pattern stability in the presence of surface curvature. To make progress, we construct a general model for self-propelled swarms moving on surfaces using Lagrangian mechanics. We find that the combination of self-propulsion, friction, mutual attraction, and surface curvature produce milling patterns where each agent in a swarm oscillates on a limit cycle with different agents splayed along the cycle such that the swarm's center-of-mass remains stationary. In general, such patterns loose stability when mutual attraction is insufficient to overcome the constraint of curvature, and we uncover two broad classes of stationary milling-state bifurcations. In the first, a spatially periodic mode undergoes a Hopf bifurcation as curvature is increased, which results in unstable spatiotemporal oscillations. This generic bifurcation is analyzed for the sphere and demonstrated numerically for several surfaces. In the second, a saddle-node-of-periodic orbits occurs in which stable and unstable milling states collide and annihilate. The latter is analyzed for milling states on cylindrical surfaces. Our results contribute to the general understanding of swarm pattern formation and stability in the presence of surface curvature and may aid in designing robotic swarms that can be controlled to move over complex surfaces and terrains.

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.