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- Vadim Kaloshin, Mark Levi
- 2008

In this paper, using the ideas of Bessi and Mather, we present a simple mechanical system exhibiting Arnold diffusion. This system of a particle in a small periodic potential can be also interpreted as ray propagation in a periodic optical medium with a near-constant index of refraction. Arnold diffusion in this context manifests itself as an arbitrary… (More)

The model of a bicycle is a unit segment AB that can move in the plane so that it remains tangent to the trajectory of point A (the rear wheel is fixed on the bicycle frame); the same model describes the hatchet planimeter. The trajectory of the front wheel and the initial position of the bicycle uniquely determine its motion and its terminal position; the… (More)

- Vadim Kaloshin, Mark Levi, Maria Saprykina
- 2014

The main model studied in this paper is a lattice of pendula with a nearest-neighbor coupling. If the coupling is weak, then the system is near-integrable and KAM tori fill most of the phase space. For all KAM trajectories the energy of each pendulum stays within a narrow band for all time. Still, we show that for an arbitrarily weak coupling of a certain… (More)

1. INTRODUCTION: THE PONCELET CLOSURE THEOREM. The Poncelet closure theorem (or Poncelet porism) is a classical result of projective geometry. Given nested ellipses γ and , with γ inside , one plays the following game: starting at a point x on , draw a tangent line to γ until it intersects at point y, repeat the construction , starting with y, and so on.… (More)

- Mark Levi
- 2008

The goal of this paper is to present to nonspecialists what is perhaps the simplest possible geometrical picture explaining the mechanism of Arnold diffusion. We choose to speak of a specific model—that of geometric rays in a periodic optical medium. This model is equivalent to that of a particle in a periodic potential in R n with energy prescribed and to… (More)