1980 Preface * 1999 Preface * 1999 Acknowledgements * Introduction * 1 Circular Logic * 2 Phase Singularities (Screwy Results of Circular Logic) * 3 The Rules of the Ring * 4 Ring Populations * 5 Getting Off the Ring * 6 Attracting Cycles and Isochrons * 7 Measuring the Trajectories of a Circadian Clock * 8 Populations of Attractor Cycle Oscillators * 9 Excitable Kinetics and Excitable Media * 10 The Varieties of Phaseless Experience: In Which the Geometrical Orderliness of Rhythmic Organization Breaks Down in Diverse Ways * 11 The Firefly Machine 12 Energy Metabolism in Cells * 13 The Malonic Acid Reagent ('Sodium Geometrate') * 14 Electrical Rhythmicity and Excitability in Cell Membranes * 15 The Aggregation of Slime Mold Amoebae * 16 Numerical Organizing Centers * 17 Electrical Singular Filaments in the Heart Wall * 18 Pattern Formation in the Fungi * 19 Circadian Rhythms in General * 20 The Circadian Clocks of Insect Eclosion * 21 The Flower of Kalanchoe * 22 The Cell Mitotic Cycle * 23 The Female Cycle * References * Index of Names * Index of Subjects

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/596B270197FE27DE5FABB598B6210622/S0025557200150892a.pdf/div-class-title-span-class-italic-the-geometry-of-biological-time-span-by-a-t-winfree-pp-544-dm68-corrected-second-printing-1990-isbn-3-540-52528-9-springer-div.pdf

The geometry of biological time , by A. T. Winfree. Pp 544. DM68. Corrected Second Printing 1990. ISBN 3-540-52528-9 (Springer)

The Kuramoto model in complex networks

Chimera states describing the stable coexistence of synchronous and incoherent dynamics have so far only been realized numerically. An experimental demonstration of these states in a network of discrete chemical oscillators reveals behaviour that differs from that predicted by existing phase-oscillator models.

/pdf/chimera-and-phase-cluster-states-in-populations-of-coupled-3brugtps2s.pdf

Chimera and phase-cluster states in populations of coupled chemical oscillators

Role of local network oscillations in resting-state functional connectivity

Exploring the network dynamics underlying brain activity during rest.

In order to discover generic effects of heterogeneous communication delays on the dynamics of large systems of coupled oscillators, this Letter studies a modification of the Kuramoto model incorporating a distribution of interaction delays. Corresponding to the case of a large number N of oscillators, we consider the continuum limit (i.e., N --> infinity). By focusing attention on the reduced dynamics on an invariant manifold of the original system, we derive governing equations for the system which we use to study the stability of the incoherent states and the dynamical transitional behavior from stable incoherent states to stable coherent states. We find that spread in the distribution function of delays can greatly alter the system dynamics.

Large coupled oscillator systems with heterogeneous interaction delays.

We consider systems of many spatially distributed phase oscillators that interact with their neighbors. Each oscillator is allowed to have a different natural frequency, as well as a different response time to the signals it receives from other oscillators in its neighborhood. Using the ansatz of Ott and Antonsen [Chaos 18, 037113 (2008)] and adopting a strategy similar to that employed in the recent work of Laing [Physica D 238, 1569 (2009)], we reduce the microscopic dynamics of these systems to a macroscopic partial-differential-equation description. Using this macroscopic formulation, we numerically find that finite oscillator response time leads to interesting spatiotemporal dynamical behaviors including propagating fronts, spots, target patterns, chimerae, spiral waves, etc., and we study interactions and evolutionary behaviors of these spatiotemporal patterns.

Dynamics and pattern formation in large systems of spatially-coupled oscillators with finite response times

We consider systems of many spatially distributed phase oscillators that interact with their neighbors. Each oscillator is allowed to have a different natural frequency, as well as a different response time to the signals it receives from other oscillators in its neighborhood. Using the ansatz of Ott and Antonsen (Ref. \cite{OA1}) and adopting a strategy similar to that employed in the recent work of Laing (Ref. \cite{Laing2}), we reduce the microscopic dynamics of these systems to a macroscopic partial-differential-equation description. Using this macroscopic formulation, we numerically find that finite oscillator response time leads to interesting spatio-temporal dynamical behaviors including propagating fronts, spots, target patterns, chimerae, spiral waves, etc., and we study interactions and evolutionary behaviors of these spatio-temporal patterns.

Dynamics and Pattern Formation in Large Systems of Spatially-Coupled Oscillators with Finite Response Times

This paper addresses the behavior of large systems of heterogeneous, globally coupled oscillators each of which is described by the generic Landau-Stuart equation, which incorporates both phase and amplitude dynamics of individual oscillators. One goal of our paper is to investigate the effect of a spread in the amplitude growth parameter of the oscillators and of the effect of a homogeneous nonlinear frequency shift. Both of these effects are of potential relevance to recently reported experiments. Our second goal is to gain further understanding of the macroscopic system dynamics at large coupling strength, and its dependence on the nonlinear frequency shift parameter. It is proven that at large coupling strength, if the nonlinear frequency shift parameter is below a certain value, then there is a unique attractor for which the oscillators all clump at a single amplitude and uniformly rotating phase (we call this a single-cluster “locked state”). Using a combination of analytical and numerical methods, we show that at higher values of the nonlinear frequency shift parameter, the single-cluster locked state attractor continues to exist, but other types of coexisting attractors emerge. These include two-cluster locked states, periodic orbits, chaotic orbits, and quasiperiodic orbits.

Phase and amplitude dynamics in large systems of coupled oscillators: growth heterogeneity, nonlinear frequency shifts, and cluster states.

In order to discover generic effects of heterogeneous communication delays on the dynamics of large systems of coupled oscillators, this paper studies a modification of the Kuramoto model incorporating a distribution of interaction delays. By focusing attention on the reduced dynamics on an invariant manifold of the original system, we derive governing equations for the system which we use to study stability of the incoherent states and the dynamical transitional behavior from stable incoherent states to stable coherent states. We find that spread in the distribution function of delays can greatly alter the system dynamics.

Wai Shing Lee

Papers

Large coupled oscillator systems with heterogeneous interaction delays.

Dynamics and pattern formation in large systems of spatially-coupled oscillators with finite response times

Dynamics and Pattern Formation in Large Systems of Spatially-Coupled Oscillators with Finite Response Times

Phase and amplitude dynamics in large systems of coupled oscillators: growth heterogeneity, nonlinear frequency shifts, and cluster states.

Large coupled oscillator systems with heterogeneous interaction delays