We analyze a large system of globally coupled phase oscillators whose natural frequencies are bimodally distributed. The dynamics of this system has been the subject of long-standing interest. In 1984 Kuramoto proposed several conjectures about its behavior; ten years later, Crawford obtained the first analytical results by means of a local center manifold calculation. Nevertheless, many questions have remained open, especially about the possibility of global bifurcations. Here we derive the system's stability diagram for the special case where the bimodal distribution consists of two equally weighted Lorentzians. Using an ansatz recently discovered by Ott and Antonsen, we show that in this case the infinite-dimensional problem reduces exactly to a flow in four dimensions. Depending on the parameters and initial conditions, the long-term dynamics evolves to one of three states: incoherence, where all the oscillators are desynchronized; partial synchrony, where a macroscopic group of phase-locked oscillators coexists with a sea of desynchronized ones; and a standing wave state, where two counter-rotating groups of phase-locked oscillators emerge. Analytical results are presented for the bifurcation boundaries between these states. Similar results are also obtained for the case in which the bimodal distribution is given by the sum of two Gaussians.

A method to characterize dynamical interdependence among nonlinear systems is derived based on mutual nonlinear prediction. Systems with nonlinear correlation will show mutual nonlinear prediction when standard analysis with linear cross correlation might fail. Mutual nonlinear prediction also provides information on the directionality of the coupling between systems. Furthermore, the existence of bidirectional mutual nonlinear prediction in unidirectionally coupled systems implies generalized synchrony. Numerical examples studied include three classes of unidirectionally coupled systems: systems with identical parameters, nonidentical parameters, and stochastic driving of a nonlinear system. This technique is then applied to the activity of motoneurons within a spinal cord motoneuron pool. The interrelationships examined include single neuron unit firing, the total number of neurons discharging at one time as measured by the integrated monosynaptic reflex, and intracellular measurements of integrated excitatory postsynaptic potentials ͑EPSP's͒. Dynamical interdependence, perhaps generalized synchrony, was identified in this neuronal network between simultaneous single unit firings, between units and the population, and between units and intracellular EPSP's. ͓S1063-651X͑96͒04012-3͔

We design and analyze the dynamics of a large network of theta neurons, which are idealized type I neurons. The network is heterogeneous in that it includes both inherently spiking and excitable neurons. The coupling is global, via pulselike synapses of adjustable sharpness. Using recently developed analytical methods, we identify all possible asymptotic states that can be exhibited by a mean field variable that captures the network's macroscopic state. These consist of two equilibrium states that reflect partial synchronization in the network and a limit cycle state in which the degree of network synchronization oscillates in time. Our approach also permits a complete bifurcation analysis, which we carry out with respect to parameters that capture the degree of excitability of the neurons, the heterogeneity in the population, and the coupling strength (which can be excitatory or inhibitory). We find that the network typically tends toward the two macroscopic equilibrium states when the neuron's intrinsic dynamics and the network interactions reinforce one another. In contrast, the limit cycle state, bifurcations, and multistability tend to occur when there is competition among these network features. Finally, we show that our results are exhibited by finite network realizations of reasonable size.

It has been predicted that in the semiclassical regime the level statistics of a classically chaotic system correspond to that of the Gaussian unitary ensemble (GUE) of random matrices when time reversal symmetry is broken. This Letter presents the first experimental test of this prediction. The system employed is a microwave cavity containing a thin ferrite strip adjacent to one of the walls. When a sufficiently large magnetic field is applied to the ferrite (thus breaking the time reversal symmetry) good agreement with GUE statistics is obtained. The transition from Gaussian orthogonal ensemble (GOE) (which applies in the absence of the applied field) to GUE is also investigated.PACS numbers: 05.45.+b It has been conjectured that, for chaotic systems in the semiclassical limit, the spectral statistics of the Schrodinger equation correspond to that of random matrices with the same symmetry [1]. In particular, when the system is time reversible, the statistical fIuctuations of the energy levels are conjectured to be the same as those for the "Gaussian orthogonal ensemble" (GOE) of random matrices. As a simple example of this class of systems, consider a charged particle in a scalar potential. By reversing the direction of the momentum of the particle, the classical particle will retrace its own path. The wave equation for this particle is real and the corresponding GOE consists of real random symmetric matrices. On the other hand, when a magnetic field 8 is applied, the time reversal symmetry is broken. A classical charged particle will no longer retrace its own path when the direction of its momentum is reversed. In this case, the Schrodinger equation is complex, p -ihV -qA(r), and (in the absence of special symmetries) the statistical fIuctuations of the energy levels are conjectured to be the same as those for the "Gaussian unitary ensemble" (GUE) of random Hermitian matrices. Although the predictions of GOE statistics in actual physical systems have been observed by others [2 -5], there has been no experimental verification of the GUE predictions. The purpose of the present work is to verify the GUE predictions in an experimental setting using a 2D microwave cavity with a thin magnetized ferrite strip adjacent to one of the walls. To see how a magnetized ferrite breaks the time reversal symmetry in the electromagnetic wave equation, consider the situation when a plane wave with the electric field E = E, exp(ik, x + ikYy)z perpendicular to the plane of incidence is incident from the left (x~0) on a slab of magnetized ferrite (0 ( x ( d) which is placed adjacent to a perfect conductor on the right (x = d). In the presence of a static magnetic field B = Bi perpendicular to the plane of incidence, the magnetic permeability p, of the ferrite, in the absence of losses, is @II lK 0 P l K P'lI 0 l (1) 0 0 p, , where p,~~, K, and p, , are real. At the interface between the ferrite and the empty cavity, the boundary conditions require the continuity of both E, and the tangential component of H, which, in the ferrite, i...

The onset of synchronization in networks of networks is investigated. Specifically, we consider networks of interacting phase oscillators in which the set of oscillators is composed of several distinct populations. The oscillators in a given population are heterogeneous in that their natural frequencies are drawn from a given distribution, and each population has its own such distribution. The coupling among the oscillators is global, however, we permit the coupling strengths between the members of different populations to be separately specified. We determine the critical condition for the onset of coherent collective behavior, and develop the illustrative case in which the oscillator frequencies are drawn from a set of (possibly different) Cauchy-Lorentz distributions. One motivation is drawn from neurobiology, in which the collective dynamics of several interacting populations of oscillators (such as excitatory and inhibitory neurons and glia) are of interest.In recent years, there has been considerable interest in networks of interacting systems. Researchers have found that an appropriate description of such systems involves an understanding of both the dynamics of the individual oscillators and the connection topology of the network. Investigators studying the latter have found that many complex networks have a modular structure involving motifs [1], communities [2,3], layers [4], or clusters [5]. For example, recent work has shown that as many kinds of networks (including isotropic homogeneous networks and a class of scale-free networks) transition to full synchronization, they pass through epochs in which well-defined synchronized communities appear and interact with one another [3]. Knowledge of this structure, and the dynamical behavior it supports, informs our understanding of biological [6], social [7], and technological networks [8].Here we consider the onset of coherent collective behavior in similarly structured systems for which the dynamics of the individual oscillators is simple. In seminal work, Kuramoto analyzed a mathematical model that illuminates the mechanisms by which synchronization arises in a large set of globally-coupled phase oscillators [9]. An important feature of Kuramoto's model is that the oscillators are heterogeneous in their frequencies. And, although these mathematical results assume global coupling, they have been fruitfully applied to further our understanding of systems of fireflies, arrays of Josephson junctions, electrochemical oscillators, and many other cases [10]. In this work, we study systems of several interacting Kuramoto systems, i.e., networks of interacting populations of phase oscillators. Our motivation is drawn not only from the applications listed above (e.g., an amusing application might be interacting populations of fireflies, where each population inhabits a separate tree), but also from other biological systems. Rhythms arising from coupled cell populations are seen in many of the body's organs (including the heart, the pancreas, and the kidney...

A general nonlinear method to extract unstable periodic orbits from chaotic time series is proposed. By utilizing the estimated local dynamics along a trajectory, we devise a transformation of the time series data such that the transformed data are concentrated on the periodic orbits. Thus, one can extract unstable periodic orbits from a chaotic time series by simply looking for peaks in a finite grid approximation of the distribution function of the transformed data. Our method is demonstrated using data from both numerical and experimental examples, including neuronal ensemble data from mammalian brain slices. The statistical significance of the results in the presence of noise is assessed using surrogate data.

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