Rapid Deterioration of Convergence in Taylor Expansions of Linearizing Maps of Hénon Maps at Hyperbolic Fixed Points

In this paper, we prove that the Taylor expansions of analytic functions appearing in the linearization of quadratic maps at hyperbolic fixed points do not successfully approximate invariant manifolds, such as stable and unstable manifolds, when higher-order terms are truncated. This fact was pointed out by Newhouse et al. in their numerical experiments, and implies that the Taylor expansions are inadequate for quantitatively studying dynamical systems such as quadratic maps. In fact, it is shown that the computational complexity for the approximations by the Taylor expansions grows exponentially.