By Arkady Pikovsky, Michael Rosenblum, Jürgen Kurths
Structures as various as clocks, making a song crickets, cardiac pacemakers, firing neurons and applauding audiences show a bent to function in synchrony. those phenomena are common and will be understood inside a standard framework in response to smooth nonlinear dynamics. the 1st 1/2 this ebook describes synchronization with out formulae, and relies on qualitative intuitive principles. the most results are illustrated with experimental examples and figures, and the historic improvement is additionally defined. the second one half the ebook offers the most results of synchronization in a rigorous and systematic demeanour, describing either classical effects on synchronization of periodic oscillators, and up to date advancements in chaotic platforms, huge ensembles, and oscillatory media.
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Additional resources for Synchronization
Example text
Plotting this dependence we get a curve as shown in Fig. ). , synchronization takes place. We emphasize that the frequencies f 1,2 and F1,2 have to be measured for the same objects, but in different experimental conditions: f 1,2 characterize free (uncoupled, or autonomous) oscillators, whereas the frequencies F1,2 are obtained in the presence of coupling. Generally, we expect the width of the synchronization region to increase with coupling strength. A close examination of synchronous states reveals that the synchronization of two clocks can appear in different forms.
To understand the origin of the term “limit cycle” we have to determine how it differs from all other trajectories in the phase plane. For this purpose we consider the behavior of the trajectories in the vicinity of the cycle. In other words, we look at what happens if a phase point is pushed off the limit cycle. ) system this would mean that we somehow perturb its periodic motion. , the phase 3 The number of required variables depends on the particular dynamical system and is called its dimension.
A firefly communicates via light pulses with all other insects in the population, and a person in a theater hears every other member of the audience. In this case one speaks of global (all-to-all) coupling. There are other situations when oscillators are ordered into chains or lattices, where each element interacts only with its several neighbors. Such structures are common for man-made systems, examples are laser arrays and series of Josephson junctions, but may also be encountered in nature. So, mammalian intestinal smooth muscle may be electrically regarded as a series of loosely coupled pacemakers having intrinsic frequencies.