One dimensional invariant manifolds of Gevrey type in real-analytic maps
The authors consider the map \(F:U\subset \mathbb{R}\times \mathbb{R}^d\rightarrow \mathbb{R}\times \mathbb{R}^d, z=(x,y)\rightarrow F(x,y)=f(x,y),g(x,y))\) defined in an open neighborhood \(\mathcal{U}\) of \(0=(0,0)\) giving a discrete dynamical system \[ \begin{matrix} \overline{x}=x-a x^N+ f_N(x,y)+f_{\geq N+1}(x,y)\\ \overline{y}= Ay+g_{\geq 2} (x,y), \end{matrix} \eqno(1) \] where \(a\) is a nonzero constant; \(1 \not \in\sigma (A); N\geq 2, \widehat{f}_{N}(x,y)\) is an \(N\)-degree homogeneous polynomial \(\widehat{f} _N (x,0)=0; f_{\geq N+1}\) has the order \(N+1;g\geq 2\) has the order \(2\) and the map \(F\) is real-analytic, i.e., it can be extended to a holomorphic function defined in a complex neighborhood \(\mathcal{U}_c\) of \(\mathcal{U}\). The questions of existence, uniqueness, differentiability, analyticity and computability of parabolic invariant manifold (center manifold) tangent to the simple eigenvalue are discussed. The parametrization method reduces the dynamics on the parabolic manifold to a polynomial. Under additional assumptions it is proved that the asymptotic expansion of this manifold is of Gevrey type.
scientific article; zbMATH DE number 5359475