Tan Su

Numerical integration of ODEs by standard numerical methods reduces continuous time problems to discrete time problems. Discrete time problems have intrinsic properties that are absent in continuous time problems. As a result, numerical solution of an ODE may demonstrate dynamical phenomena that are absent in the original ODE. We show that numerical integration of systems with one fast rotating phase leads to a situation of such kind: numerical solution demonstrates phenomenon of scattering on resonances that is absent in the original system.

Let the adiabatic invariant of action variable in a slow-fast Hamiltonian system with two degrees of freedom have limits along the trajectories as time tends to plus and minus infinity. The difference of these two limits is exponentially small in analytic systems. An isoenergetic reduction and canonical transformations are applied to transform the slow-fast system to form of a system depending on a slowly varying parameter in a complexified phase space. On the basis of this method an estimate for the accuracy of conservation of adiabatic invariant is given.