In a plot like that one those groups of circles that you see are resonances where the winding ratio on the torus (how many degrees you go around per turn) is rational. The area between two resonances (or non-resonant behavior) has a little bit of chaos in those regular regions if you look closely: there is a certain chaotic zone. Those areas of pervasive chaos happen when the chaos area around the rationals covers everything, see
which involves doing a sum over all the rationals.
In the 2 degree of freedom case those tori are a solid wall so even if you have a chaotic zone around a resonance the motion is constrained by the tori. For N>2 though there are more dimensions and the path can go "around" the tori. You could picture our solar system of 8 planets having 24 degrees of freedom (although the problem is terribly non-generic because the three kinds of motion in a 1/r^2 field all have the same period) It sure seems that we are in a regular regime like one of the circles in that plot but we cannot rule out that over the course of billions of years that Neptune won't get ejected. See
which is poorly understood because nobody has found an attack on it. You have the same problem with numerical work that you do in the plot because sensitive dependence on initial conditions magnifies rounding error. Worse than that normal integrators like Runge-Kutta don't preserve all the geometric invariants of
but other than preserving that invariance those perform much worse than normal integrators. This is one of the reasons why the field has been stuck since before I got into it. This
came out of people learning how to find chaotic trajectories to use for transfer orbits and is an exciting development though. Practically though you don't want to take a low energy-long time trajectory to Mars because you'll get your health wrecked by radiation.
In a plot like that one those groups of circles that you see are resonances where the winding ratio on the torus (how many degrees you go around per turn) is rational. The area between two resonances (or non-resonant behavior) has a little bit of chaos in those regular regions if you look closely: there is a certain chaotic zone. Those areas of pervasive chaos happen when the chaos area around the rationals covers everything, see
https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Arnold%E2%8...
which involves doing a sum over all the rationals.
In the 2 degree of freedom case those tori are a solid wall so even if you have a chaotic zone around a resonance the motion is constrained by the tori. For N>2 though there are more dimensions and the path can go "around" the tori. You could picture our solar system of 8 planets having 24 degrees of freedom (although the problem is terribly non-generic because the three kinds of motion in a 1/r^2 field all have the same period) It sure seems that we are in a regular regime like one of the circles in that plot but we cannot rule out that over the course of billions of years that Neptune won't get ejected. See
https://en.wikipedia.org/wiki/Arnold_diffusion
which is poorly understood because nobody has found an attack on it. You have the same problem with numerical work that you do in the plot because sensitive dependence on initial conditions magnifies rounding error. Worse than that normal integrators like Runge-Kutta don't preserve all the geometric invariants of
https://en.wikipedia.org/wiki/Symplectic_geometry
so you know you get the wrong results. There is
https://en.wikipedia.org/wiki/Symplectic_integrator
but other than preserving that invariance those perform much worse than normal integrators. This is one of the reasons why the field has been stuck since before I got into it. This
https://en.wikipedia.org/wiki/Interplanetary_Transport_Netwo...
came out of people learning how to find chaotic trajectories to use for transfer orbits and is an exciting development though. Practically though you don't want to take a low energy-long time trajectory to Mars because you'll get your health wrecked by radiation.