Arnold tongues

In mathematics, particularly in dynamical systems theory, an Arnold tongue is a phase-locked or mode-locked region in a driven (kicked) weakly-coupled harmonic oscillator. Arnold tongues are observed in a large variety of complex vibrating systems, including the inharmonicity of musical instruments, orbital resonance and tidal locking of orbiting moons, mode-locking in fiber optics and phase-locked loops and other electronic oscillators, as well as in cardiac rhythms and heart arrhythmias.

One of the simplest physical models that exhibits mode-locking consists of two rotating disks connected by a weak spring. One disk is allowed to spin freely, and the other is driven by a motor. Mode locking occurs when the freely-spinning disk turns at a frequency that is a rational number of the driven rotator.

The simplest mathematical model that exhibits mode-locking is the circle map, which attempts capture the motion of the spinning disks at discrete time intervals.

Arnold tongues are named after Vladimir Arnold.

Arnold tongues were first investigated for a family of dynamical systems on the circle first defined by Andrey Kolmogorov. Kolmogorov proposed this family as a simplified model for driven mechanical rotors, and specifically, for a free-spinning wheel weakly coupled by a spring to a motor. The circle map also provides a simple model of the phase-locked loop in electronics, of mechanically or acoustically coupled musical instruments, and of cardiac tissue. The map exhibits certain regions of its parameters where it is locked to the driving frequency, commonly referred to as phase-locking or mode-locking.

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