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| In this talk, we will present a methodology to investigate the processes that lead to chaos in area preserving maps (APM). Concretely, we will study some dynamical properties of orientation-preserving and orientation-reversing quadratic H\'enon maps concerning the stability region, the size of the chaotic zones, its evolution with respect to parameters and the splitting of the separatrices of fixed and periodic points plus its role in the preceding aspects. Several techniques involving the computation of rotation numbers, estimates of the fraction of regular and chaotic points by means of Lyapunov exponent indicators and the measure of the splitting of the separatrices (using extended precision arithmetics when needed) will be used. These techniques can be also applied to study 2 degrees of freedom Hamiltonian systems in suitable Poincar\'e sections, in some level of energy. The role of H\'enon maps in the diffusion which takes place in the Chirikov standard map for large values of the parameter will be stressed. |
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