While the orbits were computed numerically, the transformations, which provide the coordinates used, were computed using Hamiltonian perturbation theory and Lie series. The plot shows the changes in the shape and the orientation of planetary orbits for half million years. The names of the planets refer to the eigenmodes of the secular system. The radial variable describes the size of the orbits, the angular their orientation in space. Resonances between the eigenmodes of the secular system can cause very large variations in the shape and orientation of orbits.
Numerical experiments are useful to obtain insight.
Varying the initial conditions and parameters, the system can traverse the region of resonance. One can make a composite of the results by combining the Fourier spectra of individual cases. The blue line on the surface is the Fourier spectrum for actual Solar System. The imposed changes in the mass of Neptune cause the f7 frequency to increase. Further increase in Neptune's mass takes the system beyond the resonance, associated with another ramp leading down from the ridge.