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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. ==Standard circle map == 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 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Arnold tongue」の詳細全文を読む スポンサード リンク
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