Renato E. Mirollo

Synchronization of pulse-coupled biological oscillators

A simple model for synchronous firing of biological oscillators based on Peskins model of the cardiac pacemaker (Mathematical aspects of heart physiology, Courant Institute of Mathematical Sciences, New York University, New York, 1975, pp. 268-278) is studied. The model consists of a population of identical integrate-and-fire oscillators. The coupling between oscillators is pulsatile: when a given oscillator fires, it pulls the others up by a fixed amount, or brings them to the firing threshold, whichever is less. The main result is that for almost all initial conditions, the population evolves to a state in which all the oscillators are firing synchronously. The relationship between the model and real communities of biological oscillators is discussed; examples include populations of synchronously flashing fireflies, crickets that chirp in unison, electrically synchronous pacemaker cells, and groups of women whose menstrual cycles become mutually synchronized.

Stability of incoherence in a population of coupled oscillators

We analyze a mean-field model of coupled oscillators with randomly distributed frequencies. This system is known to exhibit a transition to collective oscillations: for small coupling, the system is incoherent, with all the oscillators running at their natural frequencies, but when the coupling exceeds a certain threshold, the system spontaneously synchronizes. We obtain the first rigorous stability results for this model by linearizing the Fokker-Planck equation about the incoherent state. An unexpected result is that the system has pathological stability properties: the incoherent state is unstable above threshold, butneutrally stable below threshold. We also show that the system is singular in the sense that its stability properties are radically altered by infinitesimal noise.

Solvable Model for Chimera States of Coupled Oscillators

Networks of identical, symmetrically coupled oscillators can spontaneously split into synchronized and desynchronized subpopulations. Such chimera states were discovered in 2002, but are not well understood theoretically. Here we obtain the first exact results about the stability, dynamics, and bifurcations of chimera states by analyzing a minimal model consisting of two interacting populations of oscillators. Along with a completely synchronous state, the system displays stable chimeras, breathing chimeras, and saddle-node, Hopf, and homoclinic bifurcations of chimeras.

Dynamics of a large system of coupled nonlinear oscillators

Abstract We consider the interaction of a large number of limit-cycle oscillators with linear, all-to-all coupling and a distribution of natural frequencies. The system exhibits extremely rich dynamics as the coupling strength and the width of the frequency distribution are varied. We find a variety of steady behaviors that can be described by a stationary distribution in phase space: frequency locking, amplitude death, incoherence and partial locking. An unexpected result is that the system can also exhibit unsteady behavior, in which the phase space distribution evolves periodically, quasiperiodically or even chaotically. The simple form of the model allows us to derive several analytical results. The stability boundaries of amplitude death and incoherence are found explicitly. Rigorous results on the existence and stability of frequency locking are also obtained.

Amplitude death in an array of limit-cycle oscillators

We analyze a large system of limit-cycle oscillators with mean-field coupling and randomly distributed natural frequencies. We prove that when the coupling is sufficiently strong and the distribution of frequencies has sufficiently large variance, the system undergoes “amplitude death”-the oscillators pull each other off their limit cycles and into the origin, which in this case is astable equilibrium point for the coupled system. We determine the region in couplingvariance space for which amplitude death is stable, and present the first proof that the infinite system provides an accurate picture of amplitude death in the large but finite system.

Phase-locking and critical phenomena in lattices of coupled nonlinear oscillators with random intrinsic frequencies

Abstract We study phase-locking in a network of coupled nonlinear oscillators with local interactions and random intrinsic frequencies. The oscillators are located at the vertices of a graph and interact along the edges. They are coupled by sinusoidal functions of the phase differences across the edges, and their intrinsic frequencies are independent and identically distributed with finite mean and variance. We derive an exact expression for the probability of phase-locking in a linear chain of such oscillators and prove that this probability tends to zero as the number of oscillators grows without bound. However, if the coupling strength increases as the square root of the number of oscillators, the probability of phase-locking tends to a limiting distribution, the Kolmogorov-Smirnov distribution. This latter result is obtained by showing that the phase-locking problem is equivalent to a discretization of pinned Brownian motion. The results on chains of oscillators are extended to more general graphs. In particular, for a hypercubic lattice of any dimension, the probability of phase-locking tends to zero exponentially fast as the number of oscillators grows without bound. We also consider a less stringent type of synchronization, characterized by large clusters of oscillators mutually entrained at the same average frequency. It is shown that if such clusters exist, they necessarily have a sponge-like geometry.

Dynamics of a globally coupled oscillator array

Abstract We study a set of N globally coupled ordinary differential equations of the form encountered in circuit analysis of superconducting Josephson junction arrays. Particular attention is paid to two kinds of simple time-periodic behavior, known as in-phase and splay phase states. Some results valid for general N , as well as further results for N = 2 and N → ∞, are presented; a recurring theme is the appearance of very weak dynamics near the periodic states. The implications for Josephson junction arrays are discussed.

Identical phase oscillators with global sinusoidal coupling evolve by Möbius group action

Systems of N identical phase oscillators with global sinusoidal coupling are known to display low-dimensional dynamics. Although this phenomenon was first observed about 20 years ago, its underlying cause has remained a puzzle. Here we expose the structure working behind the scenes of these systems by proving that the governing equations are generated by the action of the Mobius group, a three-parameter subgroup of fractional linear transformations that map the unit disk to itself. When there are no auxiliary state variables, the group action partitions the N-dimensional state space into three-dimensional invariant manifolds (the group orbits). The N-3 constants of motion associated with this foliation are the N-3 functionally independent cross ratios of the oscillator phases. No further reduction is possible, in general; numerical experiments on models of Josephson junction arrays suggest that the invariant manifolds often contain three-dimensional regions of neutrally stable chaos.

Collective dynamics of coupled oscillators with random pinning

Abstract We analyze a large system of nonlinear oscillators with random pinning, mean-field coupling and external drive. For small coupling and drive strength, the system evolves to an incoherent pinned state, with all the oscillators stuck at random phases. As the coupling or drive strength is increased beyond a depinning treshold, the steady-state solution switches to a coherent moving state, with all the oscillators moving nearly in phase. This depinning transition is discontinuous and hysteretic. We also show analytically that there is a delayed onset of coherence in response to a sudden superthreshold drive. The time delay increases as the threshold is approached from above. The discontinuous, hysteretic transition and the delayed onset of coherence are directly attributable to the form of the coupling, which is periodic in the phase difference between oscillators. The system studied here provides a simple model of charge-density wave transport in certain quasi-one-dimensional metals and semiconductors in the regime where phase-slip is important; however this paper is intended primarily as a study of a model system with analytically tractable collective dynamics.

Splay-phase orbits for equivariant flows on tori

This paper studies dynamical systems on the n-fold torus equivariant under a cyclic permutation of coordinates. It is proved that under a mild condition, these systems have splay-phase solutions. These are periodic orbits in which the n coordinates are given by the same function of time, but equally separated in phase. Applications to systems of equations used to model Josephson junction arrays are discussed.