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In this paper, we seek the periodic solutions of the Hénon-Heiles nonintegrable Hamiltonian system. We apply the Lindstedt-Poincare method, in order, first to enumerate the main periodic families in the neighbourhood of the origin, then to determine the series corresponding to these families and to their periods. All the series will be computed to O(A^21) by means of the computer algebra system 'Mathematica', where A is the zeroth-order amplitude. We also prove that the period of the rectilinear periodic family is exactly equal to a Gauss hypergeometric series. Moreover, we show that the celestial technique of the 'elimation of secular terms' is rigorously equivalent to the 'Fredholm alternative'. We further test the validity of the period families using numerical integration. Finally, we compare our results with those of the Churchill-Pecelli-Rod 'geometrical' method.
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