# Synchronization of Oscillations

## Synchronization of Oscillations

the establishment and maintenance of a situation wherein the oscillation frequencies of two or more systems are equal to, or are multiples of, one another. Let us consider, for example, a coupled system consisting of two self-oscillating systems with the frequencies ω_{1} and ω_{2}. When ω_{2} is close to ω, the oscillations are synchronized— that is, the systems begin oscillating at the same frequency ω. The tighter the coupling between the systems, the larger the frequency difference Δω = ǀω_{2} - ω_{1}ǀ at which the oscillations can be brought into synchronism; Δω is known as the locking bond.

A distinction is made between mutual and mutual synchronization. In the mutual synchronization of oscillations in coupled systems, each system acts on the other, and the frequency of the synchronized oscillations differs from both original frequencies. In the forced synchronization of oscillations, or frequency capture, the coupling between the systems is such that one system (the synchronizing system) influences the other (the synchronized system), and the reverse influence does not occur. In this case, the synchronized system oscillates at the frequency of the synchronizing system.

The reason for the mutual synchronization of two systems is that when the systems are coupled together, in each of them forced oscillations occur under the action of the other system. These forced oscillations occur in addition to the natural oscillations of the system. The forced oscillations in a self-oscillating system (for example, in an oscillator) have a double effect on the natural oscillations of the system. On the other hand, the frequency of the natural oscillations is pulled toward the frequency of the external force; on the other hand, the forced oscillations decrease the amplitude of the natural oscillations and may completely cancel out the natural oscillations.

The mutual synchronization of oscillations occurs at frequencies that are nearly multiples of ω_{1}/ω_{2} = *n/m*, where *n* and *m* are integers. The larger the *n* and *m*, the narrower the region of synchronization for the oscillations. Consequently, when *n* and *m* are large, synchronization is observed only when at least one of the interacting generators of oscillations is a relaxation oscillator, for example, a sawtooth-wave oscillator. When mutual synchronization occurs with two generators that differ greatly in power, the more powerful generator plays the role of a synchronizing generator, and the less powerful generator is the synchronized generator. This situation is an intermediate case between the mutual and the forced synchronization of oscillations.

In technology, synchronization of oscillations is of great importance since it permits self-excited oscillators, AC generators, synchronous motors, and other nonlinear systems to lock in synchronism and to operate stably within a finite range of frequencies. Synchronization makes possible the stable operation of several generators in the same power network and allows several radio transmitters to use the same antenna. Frequency multipliers and dividers are based on synchronization. In complex nonlinear systems that generate several frequencies, synchronization is possible at various combination frequencies of the system. For example, the synchronization of oscillations at a difference frequency is used in synchronizing the modes of a laser. Synchronization also finds application in medicine, where cardiac pacemakers are implanted in patients with an abnormal heart rhythm.

### REFERENCES

Teodorchik, K. F.*Avtokolebatel’nye sistemy*. Moscow-Leningrad, 1952.

Blekhman, I.I.

*Sinkhronizatsiia dinamicheskikh sistem*. Moscow, 1971.

Hayashi, C.

*Nelineinye kolebaniia v fizicheskikh sistemakh*. Moscow, 1968. (Translated from English.)

V. N. PARYGIN