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The parameterization method for invariant curves associated to parabolic points

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Treballs finals del Màster en Matemàtica Avançada, Facultat de matemàtiques, Universitat de Barcelona, Any: 2018, Director: Ernest Fontich Julià

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[en] In the first part of this work we present the parameterization method for invariant manifolds and we apply it to prove the existence of stable invariant curves of planar maps associated to a fixed point with an eigenvalue $\lambda$ such that $0 < |\lambda| < 1$. We study both the case in which the map is analytic and the case in which it is differentiable. In the second part we apply the parameterization method to obtain the existence of a stable analytic curve associated to a nilpotent parabolic fixed point of an analytic map. The main result of this master thesis is the existence of such a stable curve. Finally, we perform a numerical simulation in order to estimate the growth of the coefficients of a parameterization of this curve.

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The parameterization method for invariant curves associated to parabolic points