FM 12-2009 (mp_arc 09-185; arXiv:0910.0752 [math.DS])

Authors: Michele V. Bartuccelli, Jonathan H.B. Deane, Guido Gentile, and Frank Schilder

Title: Arnold tongues for a resonant injection-locked frequency divider: analytical and numerical results
 
Abstract: In this paper we consider a resonant injection-locked frequency divider which is of interest in electronics, and we investigate the frequency locking phenomenon when varying the amplitude and frequency of the injected signal. We study both analytically and numerically the structure of the Arnold tongues in the frequency-amplitude plane. In particular, we provide exact analytical formulae for the widths of the tongues, which correspond to the plateaux of the devil's staircase picture. The results account for numerical and experimental findings presented in the literature for special driving terms and, additionally, extend the analysis to a more general setting.

Keywords: Nonlinear dynamics; Bifurcation theory; Subharmonic bifurcation; Periodic solutions; Arnold tongues; Frequency locking; Devil's staircase; Injection-locked frequency divider.

Michele Bartuccelli
Department of Mathematics and Statistics
University of Surrey
Guildford, GU2 7HX, UK
e-mail: m.bartuccelli@surrey.ac.uk

Jonathan Deane
Department of Mathematics and Statistics
University of Surrey
Guildford, GU2 7HX, UK
e-mail: j.deane@surrey.ac.uk

Guido Gentile
Dipartimento di Matematica
UniversitÓ di Roma Tre
Largo San Leonardo Murialdo 1, 00146 Roma, Italy
e-mail: gentile@mat.uniroma3.it

Frank Schilder
Department of Mathematics and Statistics
University of Surrey
Guildford, GU2 7HX, UK
e-mail: f.schilder@surrey.ac.uk