The parameterization method for center manifolds. (English) Zbl 1442.37045
Summary: In this paper, we present a generalization of the parameterization method, introduced by X. Cabré et al. [Indiana Univ. Math. J. 52, No. 2, 283–328 (2003; Zbl 1034.37016); Indiana Univ. Math. J. 52, No. 2, 329–360 (2003; Zbl 1034.37017); J. Differ. Equations 218, No. 2, 444–515 (2005; Zbl 1101.37019)], to center manifolds associated to non-hyperbolic fixed points of discrete dynamical systems. As a byproduct, we find a new proof for the existence and regularity of center manifolds. However, in contrast to the classical center manifold theorem, our parameterization method will simultaneously obtain the center manifold and its conjugate center dynamical system. Furthermore, we will provide bounds on the error between approximations of the center manifold and the actual center manifold, as well as bounds for the error in the conjugate dynamical system.
MSC:
| 37D10 | Invariant manifold theory for dynamical systems |
| 37C15 | Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems |
| 37C75 | Stability theory for smooth dynamical systems |
References:
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