A numerical study of the likelihood of phase locking
Let \(f\) be a continuous strictly increasing mapping of the real line to itself \((f(x+1)=f(x)+1)\). By passing to quotients one can obtain a continuous one-to-one mapping \(\tilde f\) of the circle -- identified with the real numbers modulo the integers -- to itself. An orientation-preserving homeomorphism of the circle can be defined to be a mapping obtained in this way. The mapping \(f\) is not uniquely determined by the induced mapping \(\tilde f\); two \(f\)'s induce the same \(\tilde f\) iff they differ by an integer constant. From the dynamical point of view, orientation-preserving homeomorphisms of the circle can be grouped into two classes, those which have at least one periodic cycle and those which do not. A mapping with at least one cycle is said to be phase locking. The author's main purpose is to study the two-parameter family of mappings \[ f_{K,\Omega}(x)=x+\Omega +(K/2\pi)\sin (2\pi x) \] where \(\Omega\) lie in \([0,1)\) and \(| K| \leq 1\) (the last inequality implies: \(f_{K,\Omega}\) is nondecreasing). So the author makes a numerical computation of the size of the set of \(\Omega\)'s for which the homeomorphism of the circle \(\tilde f_{K,\Omega}(x)=x+\Omega +(K/2\pi)\sin (2\pi x) \bmod 1\) is phase locked. The obtained numerical results confirm Arnold's picture, that is, from the topological standpoint, phase locking is generic (cf. \textit{V. I. Arnol'd} [Am. Math. Soc., Transl., II. Ser. 46, 213--284 (1965; Zbl 0152.41905); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 25, 21--86 (1961; Zbl 0135.42603)]).
orientation-preserving homeomorphisms of the circle
scientific article; zbMATH DE number 3965049
scientific article