New approach to resonance crossing.

Time-varying nonlinear oscillatory systems produce phenomena of resonance crossing and trapping of particles in resonance islands. Traditionally, such processes have been analyzed in terms of adiabatic conditions. Considering, as an example, a simplified one-dimensional model describing the "electron-cloud pinch" during a bunch passage in a particle accelerator, here we present an approach to resonance trapping which does not require any adiabatic condition. Instead we introduce the concept of the attraction point and investigate invariance and scaling properties of motion close to the attraction point, considering a single resonance crossing.