Synchronization in parametrically driven Hamiltonian systems

Analytical and numerical techniques of non-linear dynamics are used to study synchronization in certain periodically modulated 2-dimensional Hamiltonian systems. Our model equation describes the behaviour of a «flat» particle beam passing through a periodic array of magnetic dipole focusing elements with sextupole non-linearities. Explicit expressions for the frequency of oscillations about the ideal path are obtained to second order in a small parameter, representing the strength of the instantaneous interactions. These expressions agree very well with the results of numerical computations of the corresponding periodic orbits. Finally, the stability properties of these so-called synchronized periodic solutions are studied numerically and their relevance to the problem of beam stability is discussed.