We provide a complete analysis of the Kuramoto model of coupled nonlinear oscillators with uncertain natural frequencies and arbitrary interconnection topology. Our work extends and supersedes existing, partial results for the case of an all-to-all connected network. Using tools from spectral graph theory and control theory, we prove that for couplings above a critical value all the oscillators synchronize, resulting in convergence of all phase di#erences to a constant value, both in the case of identical natural frequencies as well as uncertain ones. We further explain the behavior of the system as the number of oscillators grows to infinity.

Summary. This paper studies connections between phase models of coupled oscillators and kinematic models of groups of self-propelled particles. These connections are exploited in the analysis and design of feedback control laws for the individuals that stabilize collective motions for the group. 1

A system of ordinary differential equations (ODEs) is derived from a discrete system of Vicsek, Czirók et al. [35], describing the motion of a school of fish. Classes of linear and stationary solutions of the ODEs are found and their stability explored using equivariant bifurcation theory. The existence of periodic and toroidal solutions is also proven under deterministic perturbations and structurally stable heteroclinic connections are found. Applications of the model to the migration of the capelin, a pelagic fish that undertakes an extensive migration in the North Atlantic, are discussed and simulation of the ODEs presented. 1

In this dissertation I develop both general results on the dynamics of neural oscil-lators and integrators and specific applications of these results to brain areas involved in simple cognitive tasks. The scientific motivation is broad: neural networks inside our brains are able to adapt to changing information processing demands by exercising cognitive control, for example focussing on salient components of noisy sensory inputs when making specific decisions based on these inputs, but relaxing this focus when other needs become prominent. But what free variables or parameters can account for the observed adaptability? And does this adaptation occur optimally, with respect to simple economic metrics and physiological limitations? Here I address these questions via reduced models of neurons and populations near bifurcations, which characterize the dynamics of a brainstem nucleus involved in adaptive cognitive control, and via variational problems arising from neural signal processing, which clarify the role of this nucleus, and other dynamical mechanisms in decision tasks. First, I study and apply nonlinear oscillator dynamics. I develop and extend phase

. We present an experimental as well as theoretical study of kink motion in one-dimensional arrays of Josephson junctions connected in parallel by superconducting wires. The boundaries are closed to form a ring, and the waveform and stability of an isolated circulating kink is discussed. Two one-dimensional rings can be coupled which provides an interesting and clean platform to study interactions between kinks. These studies form foundations for investigating the more difficult two-dimensional arrays in which vortices move along rows but with some inter-row coupling. We introduce recent progress in the analysis of vortex dynamics in 2D arrays. Key words. Josephson junction, kink, vortex, patterns, experiments. 1. Introduction. Over several decades, many applied mathematicians have studied coupled systems of Josephson junctions. They have certainly been intrigued by the apparent simplicity of the governing equations in contrast to the variety of the dynamics they exhibit. Josephson ar...

to a homoclinic orbit to a saddle. The system we study is the following x = y \Gamma z y = 2:657466x +2:328733y + x 2 + xy + x 2 +0:83893461z (8.14) z = e 2:657466x \Gamma 0:83893461y 3:361277 \Gamma 0:83893461z This system was derived by augmenting a truncated unfolding of the normal form of the Takens-- Bogdanov bifurcation x = y y = 2cx + (1 + c)y + x 2 + xy (8.15) which is numerically observed to have a homoclinic connection to the origin at c 1:328733,

We consider ensembles of sine-coupled phaseoscillators consisting of subpopulations of identical units, with a general heterogeneous coupling between subpopulations. Using the Watanabe-Strogatz ansatz we reduce the dynamics of the ensemble to a relatively smallnumberofdynamicalvariables plusmicroscopicconstants ofmotion. This reduction is independent of the sizes of subpopulations and remains valid in the thermodynamic limits, where these sizes or/and the number of subpopulations are infinite. We demonstrate that the approach to the dynamics of such systems, recently proposed by Ott and Antonsen, correspondsto a particular choice of microscopic constants of motion. The theory is applied to the standard Kuramoto model and to the description of two interacting subpopulations, exhibiting a chimera state. Furthermore, we analyze the dynamics of the extension of the Kuramoto model for the case of nonlinear coupling and demonstrate the multistability of synchronous states.

journal homepage: www.elsevier.com/locate/physd Self-organized partially synchronous dynamics in populations of nonlinearly

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Stability of periodic solutions in series arrays of Josephson junctions with internal capacitance