Computing Arnol′d tongue scenarios

Computing Arnol′d tongue scenarios: some recent advances

Many interesting models in science and engineering involve forced or coupled oscillators. The most striking feature of such systems is the transition between phase locking and quasi-periodicity. Phase locking produces a periodic solution that generically persists under variation of parameters. In contrast, quasi-periodicity is a codimension-one phenomenon, which is thus generically destroyed by...

Arnol'd Tongues Arising from a Grazing-Sliding Bifurcation

The Nĕımark-Sacker bifurcation, or Hopf bifurcation for maps, is a well-known bifurcation for smooth dynamical systems. At a Nĕımark-Sacker bifurcation a periodic orbit loses stability and, except for certain so-called strong resonances, an invariant torus is born; the dynamics on the torus can be either quasi-periodic or phase locked, which is organized by Arnol′d tongues in parameter space. I...

Arnold tongues for a resonant injection-locked frequency divider: analytical and numerical results

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 Arnol′d tongues in the frequency-amplitude plane. In particular, we provide exact analytical formulae for t...

A Weak Liouville-arnol′d Theorem

This paper studies properties of Tonelli Hamiltonian systems that possess n independent but not necessarily involutive constants of motion. We obtain results reminiscent of the Liouville-Arnol′d theorem under a suitable hypothesis on the regular set of these constants of motion. This work continues the work in [30] by the second author.

Commentaries to Arnol′d Problems