The transition to chaos for a special solution of the area-preserving quadratic map

Following previous work on chaotic boundaries of half-plane Hamiltonian maps a special solution of the area-preserving quadratic map is introduced and investigated. The breakdown of regular bounded motion on invariant curves is found from the radius of convergence of a power series whose successive terms oscillate wildly due to the presence of small divisors. Previous techniques for taming such series are found to be insufficient and new ones are introduced.It is found that half-plane Hamiltonian maps appear to have certain universal features and that the chaotic boundary has similarities to the boundaries of Siegel domains of complex conformal maps.The chaotic boundary function α_c(ν) has some interesting new features which are not fully understood.