Arnold tongue

In mathematics, particularly in dynamical systems theory, an Arnold tongue of a finite-parameter family of circle maps, named after Vladimir Arnold, is a region in the space of parameters where the map has locally-constant rational rotation number. In other words, it is a level set of a rotation number with nonempty interior.

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 (specifically, a free-spinning wheel weakly coupled by a spring to a motor). These circle map equations also describe a simplified model of the phase-locked loop in electronics. The map exhibits certain regions of its parameters where it is locked to the driving frequency (phase-locking or mode-locking in the language of electronic circuits). Among other applications, the circle map has been used to study the dynamical behaviour of a beating heart.

The circle map is given by iterating the map

where ${\displaystyle \theta }$ is to be interpreted as polar angle such that its value lies between 0 and 1.

It has two parameters, the coupling strength K and the driving phase Ω. As a model for phase-locked loops, Ω may be interpreted as a driving frequency. For K = 0 and Ω irrational, the map reduces to an irrational rotation.

For small to intermediate values of K (that is, in the range of K = 0 to about K = 1), and certain values of Ω, the map exhibits a phenomenon called mode locking or phase locking. In a phase-locked region, the values ${\displaystyle \theta _{n}}$ advance essentially as a rational multiple of n, although they may do so chaotically on the small scale.

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