# Synchronization of Chaos - Phase Synchronization

Phase Synchronization

Phase synchronization occurs when the coupled chaotic oscillators keep their phase difference bounded while their amplitudes remain uncorrelated This phenomenon occurs even if the oscillators are not identical. Observation of phase synchronization requires a previous definition of the phase of a chaotic oscillator. In many practical cases, it is possible to find a plane in phase space in which the projection of the trajectories of the oscillator follows a rotation around a well-defined center. If this is the case, the phase is defined by the angle, φ(t), described by the segment joining the center of rotation and the projection of the trajectory point onto the plane. In other cases it is still possible to define a phase by means of techniques provided by the theory of signal processing, such as the Hilbert transform. In any case, if φ1(t) and φ2(t) denote the phases of the two coupled oscillators, synchronization of the phase is given by the relation nφ1(t)=mφ2(t) with m and n whole numbers.

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Phase Synchronization

Phase synchronization is the process by which two or more cyclic signals tend to oscillate with a repeating sequence of relative phase angles.

Phase synchronisation is usually applied to two waveforms of the same frequency with identical phase angles with each cycle. However it can be applied if there is an integer relationship of frequency, such that the cyclic signals share a repeating sequence of phase angles over consecutive cycles. These integer relationships are the so called Arnold tongues which follow from bifurcation of the circle map.

One example of phase synchronization of multiple oscillators can be seen in the behavior of Southeast Asian fireflies. At dusk, the flies begin to flash periodically with random phases and a gaussian distribution of native frequencies. As night falls, the flies, sensitive to one another's behavior, begin to synchronize their flashing. After some time all the fireflies within a given tree (or even larger area) will begin to flash simultaneously in a burst.

Thinking of the fireflies as biological oscillators, we can define the phase to be 0° during the flash and +-180° exactly halfway until the next flash. Thus, when they begin to flash in unison, they synchronize in phase.

One way to keep a local oscillator "phase synchronized" with a remote transmitter uses a phase-locked loop.

Odd Sympathy
... to the tendency of two pendulum clocks to synchronize with opposite phases when suspended side by side ... Huygens originally believed the synchronization was due to air currents shared between the two pendulums, but he dismissed the hypothesis himself after several tests ... Huygens only observed anti-phase synchronization of his pendulum clocks ...

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