General Stability Analysis of Synchronized Dynamics in Coupled Systems

We consider the stability of synchronized states (including equilibrium point, periodic orbit or chaotic attractor) in arbitrarily coupled dynamical systems (maps or ordinary differential equations). We develop a general approach, based on the master stability function and Gershgorin disc theory, to yield constraints on the coupling strengths to ensure the stability of synchronized dynamics. Systems with specific coupling schemes are used as examples to illustrate our general method.Comment: 8 pages, 1 figur

Plankton lattices and the role of chaos in plankton patchiness

Spatiotemporal and interspecies irregularities in planktonic populations have been widely observed. Much research into the drivers of such plankton patches has been initiated over the past few decades but only recently have the dynamics of the interacting patches themselves been considered. We take a coupled lattice approach to model continuous-in-time plankton patch dynamics, as opposed to the more common continuum type reaction-diffusion-advection model, because it potentially offers a broader scope of application and numerical study with relative ease. We show that nonsynchronous plankton patch dynamics (the discrete analog of spatiotemporal irregularity) arise quite naturally for patches whose underlying dynamics are chaotic. However, we also observe that for parameters in a neighborhood of the chaotic regime, smooth generalized synchronization of nonidentical patches is more readily supported which reduces the incidence of distinct patchiness. We demonstrate that simply associating the coupling strength with measurements of (effective) turbulent diffusivity results in a realistic critical length of the order of 100 km, above which one would expect to observe unsynchronized behavior. It is likely that this estimate of critical length may be reduced by a more exact interpretation of coupling in turbulent flows

Synchronization of coupled limit cycles

A unified approach for analyzing synchronization in coupled systems of autonomous differential equations is presented in this work. Through a careful analysis of the variational equation of the coupled system we establish a sufficient condition for synchronization in terms of the geometric properties of the local limit cycles and the coupling operator. This result applies to a large class of differential equation models in physics and biology. The stability analysis is complemented with a discussion of numerical simulations of a compartmental model of a neuron.Comment: Journal of Nonlinear Science, accepte

Consequential noise-induced synchronization of indirectly coupled self-sustained oscillators

We consider the dynamics of identical self-sustained oscillators coupled via a common linear system (beam), which is perturbed by noise. We demonstrate that increasing the noise intensity induces complete synchronization between the oscillators and, surprisingly, their in-phase synchronization with the beam. This new phenomenon of in-phase synchronization of both the oscillators and the oscillating beam arises when the noise intensity exceeds a threshold value, and can not appear in the deterministic case where the beam stably oscillates in anti-phase with the synchronized oscillators (as it is in the case of the Huygens clocks synchronization). Similar behavior persists for slightly non-identical oscillators

Two van der Pol-Duffing oscillators with Huygens coupling

We consider a system of two Van der Pol-Duffing oscillators with Huygens (speeding up) coupling. This system serves as appropriate model for Huygens synchronization of two mechanical clocks hanging from a common support. We examine the main regimes of complete and phase synchronization, and study the dependence of their onset on the initial conditions. In particular, we reveal co-existence of two chaotic phase synchronized modes and study the structure of their complicated riddled basins

Two van der Pol-Duffing oscillators with Huygens coupling

We consider a system of two Van der Pol-Duffing oscillators with Huygens (speeding up) coupling. This system serves as appropriate model for Huygens synchronization of two mechanical clocks hanging from a common support. We examine the main regimes of complete and phase synchronization, and study the dependence of their onset on the initial conditions. In particular, we reveal co-existence of two chaotic phase synchronized modes and study the structure of their complicated riddled basins

A study of controlled synchronization of Huijgens' pendula

In this paper we design a controller for synchronization problem for two pendula suspended on an elastically supported rigid beam. A relation to Huijgens’ experiments as well as the practical motivation are emphasized