Figure 10

The initial setand the target set${\mathcal{L}}^{\mathbf{\prime }}$represented in the$\mathbf{(}\mathit{\theta }\mathbf{,}\mathit{E}\mathbf{)}$-plane. The present figure is drawn for the parameters $k=1$, ${\mu }_1=2$, ${\mu }_0=1/2$. The internal region ℰ corresponds to the strip $\mathbb{R}\times [{\ell }_1,{\ell }^{\ast }]$. Since we are interested in the evolution of the set ℒ through the Poincaré map, we consider only the half-strip $\mathcal{E}(-\mathrm{\infty },2\pi ]=(-\mathrm{\infty },2\pi ]\times [{\ell }_1,{\ell }^{\ast }]$. For the flow associated to (2.16), all the points of ℒ move from the right to the left on lines parallel to the θ-axis. The point R (as well as the points on the line $E={\ell }_1$) moves faster than the points $P_{\pm }$ which are on the line $E={\ell }^{\ast }$. During all the evolution, the distance between the images of $P_{-}$ and $P_{+}$ remains constantly equal to π.