tori can actually be identified and characterized according to various analytical and numerical methods (Jorba 2000; Lan, Chandre & Cvitanovi[ć]{} 2006; Laakso & Kaasalainen 2014; Lange et al. 2014). As we already mentioned above, objects in the phase space, like the torus in Fig. \[fig:KAM\_torus\], associated to regular motions are responsible for algebraic decay of very long-lived trajectories.

The Effect of the Initial Velocity
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So far in all our investigations, the test particles started with the same velocity ($v_{x0}$; $v_{y0}$) which is equal to the velocity of the $L_5$ point in the circular RTBP, but it may be interesting to examine what happens to the long- lived island around the $L_5$ point if we change the initial velocity.

In this subsection we investigate how the structure of the ICMs changes for different initial velocities. To do this, several ICMs are generated with given conditions. With this procedure one can scan the $x-y$ plane for different $v_{y0}$ keeping $v_{x0}$ constant. Fig. \[fig:velocity\] shows three ICMs ($T$=1300 days) for different velocities $v_{y0}.$

![image](fig5a){width="55mm"} ![image](fig5b){width="55mm"} ![image](fig5c){width="55mm"}

Panels (a), (b), and (c) show ICMs where initial conditions are chosen from the region $x\in[-1.15,-0.7],\;y\in[-0.4,0.2]$ including the exact position of $L_5$ point in the RTBP 