We analyze a model for the synchronization of nonlinear oscillators due to reactive coupling and nonlinear frequency pulling motivated by the physics of arrays of nanoscale oscillators. We study the model for the mean field case of all-to-all coupling, deriving results for the onset of synchronization as the coupling or nonlinearity increase, and the fully locked state when all the oscillators evolve with the same frequency.PACS numbers: 85.85.+j, 05.45.Xt, 62.25.+g In the last decade we have witnessed exciting technological advances in the fabrication of nanoelectromechanical systems (NEMS). Such systems are being developed for a host of nanotechnological applications, as well as for basic research in the mesoscopic physics of phonons and the general study of the behavior of mechanical degrees of freedom at the interface between the quantum and the classical worlds [1,2]. Among the outstanding features of nanomechanical resonating elements is the fact that at these dimensions their normal frequencies are extremely high-recently exceeding the 1GHz mark [3]-facilitating the design of ultra-fast mechanical devices. Since with diminishing size output signals diminish as well, there is a need to use the coherent response in large arrays of coupled nanomechanical resonators (like the ones that were recently fabricated [4,5]) for signal enhancement and noise reduction. One potential obstacle for achieving such coherent response is the fundamental problem of the irreproducibility of NEMS devices. Clearly, as the size of a resonating beam or cantilever decreases to the point that its width is only that of a few dozen atoms, any misplaced atomic cluster dramatically can change the normal frequency or any other property of the resonator. Thus, it is almost inevitable that an array of nanomechanical resonators will contain a distribution of normal frequencies. Here we propose to overcome this potential difficulty by making use of another typical feature of nanomechanical resonators-their tendency to behave nonlinearly at even modest amplitudes. We shall demonstrate here that systems of coupled nonlinear nanomechanical resonators (like the one we studied recently [6]) can self-synchronize to one common frequency through the dependence of their frequencies on the amplitude of oscillation.The synchronization of systems of coupled oscillators that have a distribution of individual frequencies is important in many disciplines of science [7,8]. The coherent oscillations can be used to enhance the sensitivity of detectors or the power output from sources, as proposed here. Synchronization is also important in biological phenomena, for example the collective behavior in populations of animals, such as the synchronized flashing of fire flies, and the coherent oscillations observed in the brain.Although synchronization is often put forward as an example of the importance of understanding a nonlinear phenomenon, the intuition for the phenomenon, and indeed the subsequent mathematical discussion, can often be developed in terms of ...

We present a physical analysis of the dynamics and mechanics of contractile actin rings. In particular, we analyze the dynamics of ring contraction during cytokinesis in the Caenorhabditis elegans embryo. We present a general analysis of force balances and material exchange and estimate the relevant parameter values. We show that on a microscopic level contractile stresses can result from both the action of motor proteins, which cross-link filaments, and from the polymerization and depolymerization of filaments in the presence of end-tracking cross-linkers.

We show that long chaotic transients dominate the dynamics of randomly diluted networks of pulse-coupled oscillators. This contrasts with the rapid convergence towards limit cycle attractors found in networks of globally coupled units. The lengths of the transients strongly depend on the network connectivity and varies by several orders of magnitude, with maximum transient lengths at intermediate connectivities. The dynamics of the transient exhibits a novel form of robust synchronization. An approximation to the largest Lyapunov exponent characterizing the chaotic nature of the transient dynamics is calculated analytically. 87.10.+e The dynamics of complex networks [1] is a challenging research topic in physics, technology and the life sciences. Paradigmatic models of units interacting on networks are pulse-and phase-coupled oscillators [2]. Often attractors of the network dynamics in such systems are states of collective synchrony [3,4,5,6,7,8,9,10,11,12]. Motivated by synchronization phenomena observed in biological systems, such as the heart [3] or the brain [13], many studies have investigated how simple pulse-coupled model units can synchronize their activity. Here a key question is whether and how rapid synchronization can be achieved in large networks. It has been shown that fully connected networks as well as arbitrary networks of non-leaky integrators can synchronize very rapidly [5,8]. Biological networks however, are typically composed of dissipative elements and exhibit a complicated connectivity.In this Letter, we investigate the influence of diluted network connectivity and dissipation on the collective dynamics of pulse-coupled oscillators. Intriguingly, we find that the dynamics is completely different from that of globally coupled networks or networks of non-leaky units, even for moderate dissipation and dilution: long chaotic transients dominate the network dynamics for a wide range of connectivities, rendering the attractors (simple limit cycles) irrelevant. Whereas the transient length is shortest for very high and very low connectivity, it becomes very large for networks of intermediate connectivity. The transient dynamics exhibits a robust form of synchrony that differs strongly from the synchronous dynamics on the limit cycle attractors. We quantify the chaotic nature of the transient dynamics by analytically calculating an approximation to the largest Lyapunov exponent on the transient.We consider a system of N oscillators [5,6] that interact on a directed graph by sending and receiving pulses. For concreteness we consider asymmetric random networks in which every oscillator i is connected to an other oscillator * present address: Max-Planck-Institut für Physik komplexer Systeme, 01187 Dresden, Germany j = i by a directed link with probability p. A phase variable φ j (t) ∈ [0, 1] specifies the state of each oscillator j at time t. In the absence of interactions the dynamics of an oscillator j is given by dφ j (t)/dt = 1 .(1)When an oscillator j reaches the threshold, φ j (t) = 1, its...

We present a detailed analysis of a model for the synchronization of nonlinear oscillators due to reactive coupling and nonlinear frequency pulling. We study the model for the mean field case of all-to-all coupling, deriving results for the initial onset of synchronization as the coupling or nonlinearity increase, and conditions for the existence of the completely synchronized state when all the oscillators evolve with the same frequency. Explicit results are derived for the Lorentzian, triangular, and top-hat distributions of oscillator frequencies. Numerical simulations are used to construct complete phase diagrams for these distributions.