REVERSIBLE DIFFERENCE SCHEMES FOR CLASSICAL NONLINEAR OSCILLATORS

We consider difference schemes in which the transitions from layer to layer in time are carried out using birational transformations (Cremona transformations). Such schemes are called reversible. It is shown that reversible difference schemes can be constructed for a wide class of nonlinear dynamic with quadratic right-hand side, which includes both all classical nonlinear oscillators integrable in elliptic functions and systems that are not integrable in classical transcendental functions, e.g., asymmetric tops. In the computer experiments we were surprised to see that the points of approximate solutions found by reversible schemes for classical oscillators line up into curves. Elliptic oscillators correspond to the special case, when the points of not only exact but also approximate solutions lie on elliptic curves. The discrete and continuous theories of elliptic oscillators are described by the same formulas: the quadrature describes the transition from initial to final data, the motion is periodic, it is described by meromorphic functions, and so on. The whole difference lies in the fact that in the discrete theory the birational transformation describing the transition from the old position of the system to the new one is continued to the Cremona transformation (Joint results with Mark Gambaryan, Marina Konyaeva, Lubov’ Lapshenkova)