Exponential potential, (x,u) Poincaré surfaces of section at E = −0.05 from the numerical and analytical I: (top-left) Numerical section, (top-right) analytical section obtained from a fit at order 12 and orbits at energies from −1 to −0.05, (bottom-left) from a fit at order 18 and orbits with energy E = −0.05, (bottom-right) from three local fits (order 18) of orbits in the neighbourhood of three resonant orbits, the empty space between the islands of stability is not filled correctly by the analytical integrals (red dots are the sections, y = 0 and v > 0, from computed orbits).