We have constructed by a rapidly convergent variational method the periodic solutions, analytic in the time t and with “arbitrary” (long) period, of the well known Henon-Heiles System, described by the Hamiltonian: \(H = \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/\kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} (\dot x^2 + \dot y^2 + x^2 + y^2 ) + x^2 y - y^3 /3\). We obtain, what appears to be, a dense set of 1-parameter families of periodic solutions, wherever there exist bounded solutions, even in the so called ‘stochastic regions’ of this nonintegrable system. The ‘stochastic regions’ manifest themselves in our results as “regions of confluence” of the 1-parameter families, i.e. these families all come together there. Thus the period [i.e. the Poincare Recurrence Time] and the orbits, exhibit a very “sensitive dependence” on the initial conditions there, going from one family to the “next”. These regions show up already in our analytic, “zeroth order” (a-priori-), approximations of the converged results. The variational method is of Newton form and the -asymptotically- quadratic rate of convergence does not deteriorate inside the ‘stochastic regions’ or at very high nonlinearity.