By Arkady Pikovsky
Structures as assorted as clocks, making a song crickets, cardiac pacemakers, firing neurons and applauding audiences express an inclination to function in synchrony. those phenomena are common and will be understood inside of a standard framework in accordance with sleek nonlinear dynamics. the 1st half this e-book describes synchronization with out formulae, and is predicated on qualitative intuitive rules. the most results are illustrated with experimental examples and figures, and the ancient improvement is additionally defined. the second one half the ebook provides 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 structures, huge ensembles, and oscillatory media.
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Extra resources for Synchronization. A universal concept in nonlinear sciences
Example text
If the clocks do not interact, the coupling strength is zero. If the beam is not rigid, but can vibrate longitudinally or bend, then an interaction takes place. 2. Frequency detuning Frequency detuning or mismatch f = f 1 − f 2 quantifies how different the uncoupled oscillators are. In contrast to the coupling strength, in experiments with clocks detuning can be easily measured and varied. 1). 3 Mechanical clocks usually have a mechanism that easily allows one to do this. The process is used to force the clock to go faster if it is behind the exact time, and to force it to slow down if it is ahead.
2 that all trajectories tend to the cycle. 4 The reason why we distinguish this curve from all others is thus that it attracts phase trajectories5 and is therefore called an attractor of the dynamical system. 6 To conclude, self-sustained oscillations can be described by their image in the phase space – by the limit cycle. The form of the cycle, and, hence, the form of oscillation is entirely determined by the internal parameters of the system. If this oscillation is close in form to a sine wave, then the oscillator is called quasilinear (quasiharmonic).
Introduction 14 of the phases4 of two clocks. This helps us to distinguish between two different synchronous regimes. If two pendula move in the same direction and almost simultaneously attain, say, the rightmost position, then their phases φ1 and φ2 are close and this state is called in-phase synchronization (Fig. 10a). If we look at the motions of pendula precisely (we would probably need rather complicated equipment in order to do this), we can detect that the motions are not exactly simultaneous.