Visualization and comparison of classical structures and quantum states of four-dimensional maps

For generic 4D symplectic maps we propose the use of 3D phase-space slices, which allow for the global visualization of the geometrical organization and coexistence of regular and chaotic motion. As an example, we consider two coupled standard maps. The advantages of the 3D phase-space slices are presented in comparison to standard methods, such as 3D projections of orbits, the frequency analysis, and a chaos indicator. Quantum mechanically, the 3D phase-space slices allow for the comparison of Husimi functions of eigenstates of 4D maps with classical phase-space structures. This confirms the semiclassical eigenfunction hypothesis for 4D maps.