4 Stability criterion

In the present dynamical system the largest of the quasi-periodic orbits corresponds to the maximum amplitude of oscillation around the periodic one. Following the approach adopted by Winter (2000), the stability of these kinds of periodic orbits can be measured in terms of the maximum amplitude of oscillation about the periodic orbit. This maximum amplitude of oscillation can be quantified from the Poincaré surface of section considering, for each Jacobi constant, the width of the largest island (quasi-periodic orbit) in the line of conjunction (values of x when $\dot x=0$).

We have measured such amplitudes from our surfaces of section and the results are presented in Fig. 9. The thick lines indicate the location of the Moon and the values of the Jacobi constant associated with the Lagrangian points L1 and L2. There is a thin line that divides the figure into pericentric and apocentric regions. The thin lines labeled h1 and h2 indicate the location of the periodic orbits, while the lines around them indicate the maximum amplitude of oscillation in each case. From this figure we can see the evolution of the stability for both families of periodic orbits as a function of the Jacobi constant. In the case of the periodic orbit h1 there is no stability region (in the sense defined by Winter 2000) for $C_j\leq 3.1969$. In the case of h2, the stability region is confined to the interval $3.1820\leq C_j\leq 3.1945$.