Commentaries to Arnol′d Problems

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.

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...

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