A global bifurcation organizing rhythmic activity in a coupled network.

We study a system of coupled phase oscillators near a saddle-node on 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 Ott-Antonsen reduction and geometric techniques for ordinary differential equations, we identify heteroclinic bifurcation in a family of vector fields on a cylinder, which explains the change in collective dynamics. Specifically, we show that heteroclinic bifurcation separates two topologically distinct families of limit cycles: contractible limit cycles before bifurcation from noncontractibile ones after bifurcation. Both families are stable for the model at hand.

Stability of twisted states on lattices of Kuramoto oscillators.

Real world systems comprised of coupled oscillators have the ability to exhibit spontaneous synchronization and other complex behaviors. The interplay between the underlying network topology and the emergent dynamics remains a rich area of investigation for both theory and experiment. In this work, we study lattices of coupled Kuramoto oscillators with non-local interactions. Our focus is on the stability of twisted states. These are equilibrium solutions with constant phase shifts between oscillators resulting in spatially linear profiles. Linear stability analysis follows from studying the quadratic form associated with the Jacobian matrix. Novel estimates on both stable and unstable regimes of twisted states are obtained in several cases. Moreover, exploiting the "almost circulant" nature of the Jacobian obtains a surprisingly accurate numerical test for stability. While our focus is on 2D square lattices, we show how our results can be extended to higher dimensions.

Dynamics of a cell motility model near the sharp interface limit.

Phase-field models have recently had great success in describing the dynamic morphologies and motility of eukaryotic cells. In this work we investigate the minimal phase-field model introduced in Berlyand et al. (2017). Rigorous analysis of its sharp interface limit dynamics was completed in Mizuhara et al. (2016) and Mizuhara et al. (2019), where it was observed that persistent cell motion was not stable. In this work we numerically study the pre-limiting phase-field model near the sharp interface limit, to better understand this lack of persistent motion. We find that immobile, persistent, and rotating states are all exhibited in this minimal model, and investigate the loss of persistent motion in the sharp interface limit.

Bifurcations in the Kuramoto model on graphs.

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 [Chiba and Medvedev, "The mean field analysis of the Kuramoto model on graphs. I. The mean field equation and the transition point formulas," Discrete and Continuous Dynamical Systems-Series A (to be published); "The mean field analysis of the Kuramoto model on graphs. II. Asymptotic stability of the incoherent state, center manifold reduction, and bifurcations," Discrete and Continuous Dynamical Systems-Series A (submitted).]. 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 a similar manner. These examples include Erdos-Rényi 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. Furthermore, 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 and Medvedev, "The mean field analysis of the Kuramoto model on graphs. I. The mean field equation and the transition point formulas," Discrete and Continuous Dynamical Systems-Series A (to be published); "The mean field analysis of the Kuramoto model on graphs. II. Asymptotic stability of the incoherent state, center manifold reduction, and bifurcations," Discrete and Continuous Dynamical Systems-Series A (submitted).

Minimal model of directed cell motility on patterned substrates.

Crawling cell motility is vital to many biological processes such as wound healing and the immune response. Using a minimal model we investigate the effects of patterned substrate adhesiveness and biophysical cell parameters on the direction of cell motion. We show that cells with low adhesion site formation rates may move perpendicular to adhesive stripes while those with high adhesion site formation rates results in motility only parallel to the substrate stripes. We explore the effects of varying the substrate pattern geometry and the strength of actin polymerization on the directionality of the crawling cell. These results reveal that high strength of actin polymerization results in motion perpendicular to substrate stripes only when the substrate is relatively nonadhesive; in particular, this suggests potential applications in motile cell sorting and guiding on engineered substrates.