In the Poincare map method, After huge amount of trajectories integrated in quite enough time, point data is picked from chosen surface. This points data in the chosen surface present the Poincare map (or surface of section). And the formations of these points give us about dynamical insight of the non-liner system. Such as, if the points form some circles which lie one inside the other (create like an island), so that motion is the quasi-periodic motion, and middle of the circle gives exact periodic obits or resonance orbits. If the points spread around so chaotically, this means that motion in that area would be chaotic as also. So that, In the Poincare map, the picture is generally as circles inside to another (which is called islands) and chaotically spread points (which is called either sea or ocean). This plot with island and ocean, also presents an initial condition map, which initial condition of desired trajectories can be picked on it.
In -planar-CR3BP, the Poicaré Map is very useful for finding stable periodic and quasi-periodic orbits around M1 and M2. Also they are used in invariant manifold intersection, to determine finest point to jump one manifold to another in low-cost. But in this section, I’ll give an example how stable periodic and quasi-periodic orbits around M1 is found in CR3BP. You may also look this page for another example for Earth-Moon Case.
Let’s take Sun-Jupiter system, and integrated a bunch of random trajectories around Sun (M1). Then pick the point data from chosen surface. In the bottom figure, the chosen surface is (x, Vx) while y=0. So in 4 dimensional phase space (position x,y and velocity Vx,Vy, total 4), we already know 3 value of them (x, dx come from map, because of surface section y=0), and the forth one (Vy) can be found from using conservation of energy (Jacobi Integral: V²=2U-C ). As it is seen, the Poincare map method is more suitable in 2-D space of planar-CR3BP.
The red numbers on the map shows some resonance value (ratio) of the periodic orbits, which’s appeared on Poincaré Map. To see these resonance more clearly, the point data in surface of section can be tranformed and plotted with transfering to Delaunay variables [Koon, W.S., M.W. Lo, J.E. Marsden and S.D. Ross (2000); page 455, 456]. In the right below, the poincare map in reflection with Delaunay variables are presented; the left plot, which is transformed from upper plot, is plotted for inner side of zero-velocity curves. And the right plot is plotted for outer side of zero-velocity curves.
Shapes of these resonances in position space are shown in below. Note that, orbits were intentionally rotated as their orbits tips point the neck region (like a cursor), to demonstrate possible conncetion between inner and outer region on neck realm;
Amongs of these resonance, the 3:2-2:3 resonance is very spacial, because it was practically observed in Oterma’s Trajectory. According to the source [Koon, W.S., M.W. Lo, J.E. Marsden and S.D. Ross (2000)], Oterma Comet resonated in effect of Sun-Jupiter gravitation in 1910, but because of some disturbing effects of solar system it was exited from this route in 1980. The upper figure actually is not exact Oterma route in its part of in neck region, but it is totally monolithic orbit, which makes Homocilinic (the flow which joins only one equilibrium points; either L1 or L2 in this example) inner and outer motion and Heteroclinic (the flow which joins two equilibrium points; both L1 and L2 in this example) neck connection to connect these 2 Homocilinic motion. This is also called Homocilinic-heteroclinic chain.
Oterma’s interesting route inspired the scientists; Using these resonance, spacecrafts may sit or travel in the periodic orbits in very low-cost. Of course, as in Oterma’s example, disturbing effect of solar system disturb or break the resonance, but still making small control maneuvers, the spacecraft can be kept in its route, relativly cheaply. So inspiring from Oterma, poincare maps, which show these resonace, is plotted for all binary system of solar system by researchers, to find these adventageous orbits.
Lastly note that; In 3-D space, dimension of phase space increase to 6 dimension (x,y,z,Vx,Vy,Vz), and so the surface of section must be 4-dimension. 4-D surface of section is not so comfortable for analysing the motion, visually and also the variable number which increase the complexity of solution, is too much. Therefore, it is needed to develop new mapping method for 3-D space, see motion map, which will be explaned later.