only if the phase portrait in the $(\th, \th^{\pr})$ plane looks like the one shown in Fig. 1a and the phase space $(\th, \th^{\pr})$ has a well-defined separatrix.

Capture into resonance can be described as follows. The flux $\Psi(0,\sg)$ changes as a phase point moves along a streamline. Suppose, $\Psi(0,\sg)$ increases. Then if a streamline comes very close to the hyperbolic fixed point, it may cross the separatrix and, as a result, be caught in the oscillatory domain within the separatrix loop. In this case, the streamline starts moving on the resonant surface.

Captured dynamics consists of fast rotation on the $(\th, \th^{\pr})$ phase plane and slow evolution of the parameters of the orbit, $\sg$ and $h$. The slow equations are \^ &=& , \[vp9\]\
h\^ &=& ( + ), where $ \left< \cdot \right>$ denotes the averaging over the fast, $(\th, \th^{\pr})$, motion. The slow dynamics is governed by a new conservation law: the flux of $f_{2,0}$ through the area inside a closed curve, that we denote by $\Gm_R(h,\sg)$, on the $(\th, \th^{\pr})$ phase plane is constant: $$\Psi (h,\sg) = \int \int_{\Gm_R(h,\sg)} f_{2,0} \; d\th \: d\th^{\pr} = \const.$$ Recall, that the curve $\Gm_R(0,\sg)$ corresponds to the separatrix and 