Analysis of resonances by action map comparing tracking and normal forms

The network of resonances for a 4D map is obtained by two different methods and displayed in the plane of invariant actions. Starting from tracking data the tunes are determined for every initial point in a plane and the actions are computed whenever the tunes are non-resonant. In the action plane a resonance appears as an empty channel, whose width is the true resonance width. The normal forms locate the single resonance lines in the action plane and provide their width. The tracking method matches the normal form results close to the origin and allows to explore the neighborhood of the dynamic aperture, where the normal forms are inaccurate. The proposed procedure applies to any real lattice and is suitable to analyze the diffusion due to a slow modulation of the linear frequency since the actions are adiabatic invariants.