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In this work, we apply the Lindstedt-Poincaré method in order to seek the periodic solutions of the Barbanis-Contopoulos nonintegrable Hamiltonian system. We first prove that this system admits six nontrivial periodic families in the neighborhood of the origin. Then we compute the series representing these families up to O(e^20A^21) and their periods up to O(e^20A^20) by means of the computer algebra system 'Mathematica', where A is the zeroth-order amplitude and e is a perturbative parameter. We also test the validity f the LP series using a numerical integration technique. Moreover we give the periods up to O(e^20E^10), where E is the energy, and prove that the period of the two 'oblique' periodic families is exactly equal to a Gauss hypergeometric series. Using the Bulirsch-Stoer algorithm we compute with good accuracy the radius of convergence of the 'circular' period. Finally, we compare our results with those of a 'geometrical' method.
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