ИСТИНА

We study an autonomous Hamiltonian system with two degrees of freedom and the following properties: 1. It is a small perturbation of an integrable system; 2. There exists a formal integral (a formal series in powers of epsilon); 3. Numerical experiments suggest a possibility of diffusion. The system is a symplectic reduction of a Hamiltonian system, which can be viewed as a Hamiltonian union of the 1D unharmonic oscillator and its system in variations, in 5-dimensional phase space with a degenerate Poisson bracket. The latter system also appears as a truncation of an infinite-dimensional system, which describes evolution of quantum fluctuations and is formally equivalent to the Schroedinger equation with unharmonic potential. References: [1] Sadov S.Yu.// Math.Notes 56 (1994), 960--970. [2] Kondratieva M.F., Sadov S.Yu.// Proc. 4th ISAAC Congress (Toronto, 2003), Kluwer Acad.Publ., 2004. [3] Belov V.V., Kondratieva M.F. // Math.Notes 56 (1994), 1228--1236.