we come back to definition (\[vp8\]). The captured motion is integrable and $\Psi (h,\sg)$ is the Hamiltonian.

While the streamline moves, the flux through the separatrix loop, that is a function of slow variables, changes and, if it returns to the initial value, the streamline is released from the resonance. If the flux keeps increasing, the once captured streamline stays captured forever (in real settings, until it reaches the limits of the system).

As it was discussed in Sect. 3, capture can be considered as a probabilistic phenomenon: initial conditions for streamlines that are or are not captured are mixed. Consider a point $M$ far from the resonance such that streamline passing through $M$ intersects the resonance. Let $V^{\dt}$ be a sphere of radius $\dt$ centered at $M$ and $V^{\dt, \eps}_c$ be the part of $V^{\dt}$ formed by initial conditions of trajectories with a capture into the resonance (see [@Neishtadt:1997; @Neishtadt:1999; @Itin:2000]). We define the probability of capture for the streamlines starting at $M$ as $$P(M) = \sqrt{\eps} \; \lim_{\dt \to 0} \: \lim_{\eps \to 0} \frac{V^{\dt, \eps}_c/\sqrt{\eps}}{\vol
V^{\dt}}.$$ Following [@Neishtadt:1999], we have: P(M) = \~, (S\_R/)\_\* &gt;0, \[vp9a\] where the subscript ‘$*$’ indicates that the corresponding quantity must be 