As it is mentioned in the section Invariant Manifold Tubes, these tubes can be very useful in low-energy tranfer, and planetary transfer like a highway. So the transfer between tubes and jumping one tube to another precisely would be important problem for space mission. Spacecraft must capture the tail of these tubes accuratly to get the desired target (planet or any lanrange points), without lost in eternity of space. Fortunately, the space mission desiners are already know a very proactical way thanks to again H. Poincaré. Using Poincaré mapping method (as similar way in finding Heteroclinic cycles), the “jumping point” between two monifolds can be determined very easily. All we do to take a plane Poincareé cut (so map) between the manifolds, and look at this map wheter there is any suitable “jumping point”, or not ! Let’s see in example;
The above plot is for Earth-Moon binary system in C=3.16. The red tubes are unstable (moving away from equilibrium points) and green one are stable (approaching to equilibrium points) IMT. Also U1, U2, U3 and U4 are the section, which can be ideal for Poincare Map. So let’s See U1 and U4 section, whether there is any suitable jump;
As it is seen from plot, there is only one suitable jumping place in U1 section for (ΔV=0) condition and in C=3.16 energy level (other place need velocity change), and there is no place for “ΔV=0” jump in U4 (there is no intersections). So it means, while travaling Earth-Moon tubes for C=3.16, we can conveniantly jump to another tube in U1-inner region of Hill’s region, but we need to make velocity change to jump other manifold for outer region of Hill’s region. To determine magnitude of velocity change and direction, we can look U1 and U4 sections in upper figures, (but because this a little bir advance level, I hide the details of these for my celestial mechanics book, which I’ll write and show later ^_^). So In shorti we need to find intersection areas between the manifold in surface of section to determine zero-cost (ΔV=0) jumps.
Now, let’s investigate manifold capture around the neck region (around M2) in case of asymmetric neck (L1 and L2 gate open in different size, such as C=3.18);
The colors are asigned as; Pink is unstable manifols of L1, Blue is stable manifols of L1, Green is stable manifols of L2, and Red is unstable manifols of L2. The Poincare cut is taken in vertical; x=1-μ (y, Vy). And as it is seen there is no place for (ΔV=0) jump. So let’s try this;
To avoid too much crowded, only Pink and Green manifold is showed (they are symmetric with blue and red). As it is seen, if the manifold is extended more and take again same cut in vertical; x=1-μ (y, Vy), we can observe a intersection (jump) place. Furthermore, so more alternativly, we can also take a cut between same manifold of equilibrium points, see below, which there are plenty of space for jump;
Now let’s intevtigate another binary system, such as Sun-Jupiter system, in C=3.03;
The notation system is borrowed from [W.S. Koon, M.W. Lo, J.E. Marsden and S.D. Ross]. Again the red tubes are unstable and green one are stable IMT. Also U1, U2, U3 and U4, as showed above, can be again ideal for Poincare Map. Also, black trajectory is Oterma coment, which travels along these manifold tubes in its pure natural dynamics flow. So let’s see neck region cut U2, U3;
To try to avoid confusion, the colur is asigned differantly in this cut. But you can easily figure out which cut is for which manifold from the notation of Γ and W in plots. As it is seen there are many place for jump in this energy level. So then, Let’s see U1 and U4 cut;
The “jump” place can be seen from plots, and black dots are Oterma Trajectory in these cut. Again as it is seen, the manifold captures (so jumps) are very suitable in this energy level, C=3.03. Moreover, we can invastigate more alternative cut, such as in below;
So in short, if you know how to take the Poincaré cut, then almost everything about trajectory design in multi-body of celestial mechanics can become very easy. But note that; this is just easily in planer-CR3BP. But in 3-D space, dimension of phase space increase to 6 dimension (x,y,z,dx,dy,dz), and so the surface of section must be 4-dimension, instead of 2D as in shown in this post. And 4-D surface of section is not so comfortable for analysing the motion, 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, which I reserached and design in my PhD thesis, I will show later ^_^ !