Encyclopedia of Nonlinear Science
The simplest coupled oscillator is a pair of linearly coupled harmonic oscillators, which is used as a model for a wide variety of physical systems—including the interactions of musical instruments and tuning forks, lattice vibrations, electrical resonances, and so on—in which energy tunnels back and forth between two sites at a difference (beat) frequency. If there are many elementary oscillators that are nonlinear, coupled systems exhibit more varied nonlinear phenomena. There are two types of coupled nonlinear oscillators: those described by Hamiltonian (energy-conserving) dynamics, and systems in which energy is not conserved. In addition to coupled pendula, examples of the first kind include the Fermi-Pasta-Ulam model and the Toda lattice.
Coupled nonlinear oscillators that do not conserve energy can be viewed as coupled limit cycle oscillators. A limit cycle oscillator (also called a self-sustained oscillator) is described as an attractor in a dissipative dynamical system. A typical dissipative dynamical system that exhibits a limit cycle oscillation is van der Pol’s equation
in which the character of the oscillation varies from sinusoidal and energy-conserving to a strongly dissipative (blocking or relaxation) oscillation through the variation of a parameter (ε) from zero to large values (van der Pol, 1934).
Among the varieties of limit cycle oscillators, the behavior of a quasilinear oscillator (small ε) can be expressed by a sinusoidal wave,
The wave shape of a relaxation oscillator (large ε), on the other hand, is composed of alternating fast and slow motions, similar to the spikes and slow recovery motions in a firing neuron, and stick-slip oscillations in frictional motions.
Although the limit cycle oscillation has a certain natural amplitude and frequency, the phase variable, for example, for 
a quasilinear oscillator, is a neutral mode, sensitively perturbed by an external force. If the external force is periodic with a frequency close to the natural frequency of the limit cycle oscillator, the phase of the limit cycle oscillator tends to approach the phase of the external periodic force. If the external force is sufficiently strong, the phase difference 
between the limit cycle oscillator and the external force is fixed. This phenomenon—termed phase or frequency locking—occurs more easily when ε is large, the frequency of the limit cycle oscillator is close to that of the external force, and the coupling (K) is large.
Regions in the (ω, ε, K) parameter space where frequency locking is observed are termed “Arnol’d
tongues” owing to their peculiar shape. The frequency ratio between the limit cycle and the external force is 1:1 in the above frequency locking. In general, n:m frequency lockings are possible, where n and m are small integers.
For a collection of coupled limit cycle oscillators with slightly different natural frequencies, frequency locking (called mutual entrainment) also occurs, as was first observed by Christiaan Huygens in the 17th century. He found that the motions of pendulum clocks suspended from the same wooden beam come to coincide with each other perfectly. Nobert Wiener analyzed such systems in the 1950s, showing that the power spectrum of the waves should have a peak close to 10 Hz, and he inferred that a similar shape of the power spectra of electroencephalogram (EEG) is due to mutual entrainment in coupled neural oscillators (Wiener, 1958). Buck and Buck reported that rhythmical flashes of South Asian fireflies were mutually synchronized (Buck & Buck, 1976).
Mutual entrainment of coupled limit cycle oscillators has been studied by Winfree (2000) and also by Kuramoto, who considered a coupled phase oscillator model, noting the neutrality of phase variables (Kuramoto, 1984). The simplest model with global coupling has the form
where 
and ωi represent the phase and the natural frequency of the ith oscillator, N is the total number of oscillators, and K is a coupling constant.
For K<Kc, the motion of each oscillator is independent and the frequency of the ith oscillator is the same as ωi. However, for K>Kc, collective oscillation appears and a number of oscillators are entrained to the collective oscillation. Figure 1 displays a typical frequency distribution for K<Kc and K>Kc. The δ-function peak in the frequency distribution implies mutual entrainment and a depression is seen around the deserved frequency for K>Kc.
The Josephson junction is a quantum device composed of two weakly coupled superconductors. With the current bias current below a critical value, the superconducting current flows without a voltage drop. If the bias is above the critical current, the phase difference 
between the Josephson junction is not constant in time, and the voltage drop (V) between the Josephson junction equals 
This is called the AC Josephson effect. Thus the Josephson junction behaves as a kind of limit cycle oscillator above the critical current. If microwaves with frequency ω0 are applied to the Josephson junction, n:1 frequency locking occurs, and the voltage becomes 
With N Josephson junctions coupled in series, the total voltage across the array is given by 
Such series arrays are currently used to establish the international standard of voltage (See Josephson junction arrays).
HIDETSUGU SAKAGUCHI
See also Chaotic dynamics; Phase dynamics; Synchronization; Van der Pol equation
Further Reading
Buck, J. & Buck, E. 1976. Synchronous fireflies. Scientific American, 234:74–85
Kuramoto, Y. 1984. Chemical Oscillations, Waves, and Turbulence, Berlin: Springer
van der Pol, B. 1934. The nonlinear theory of electric oscillations. Proceedings of the IRE, 22:1051–1086
Wiener, N. 1958. Nonlinear Problems in Random Theory, Cambridge, MA: MIT Press
Winfree, A.T. 2000. When Time Breaks Down, Berlin and New York: Springer
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