General solution of systems of nonlinear difference equations with continuous argument (Q2722242)

This paper deals with the functional equation of the form \(x(t+1)=\Lambda x(t)+f(t,x(t)),\) where \(t\in \mathbb R\), \(\Lambda \) is a diagonal matrix with real eigenvalues \(\lambda_1,\ldots,\lambda_n\), the mapping \(f:\mathbb R\times \mathbb C^n \to \mathbb C^n\) is continuous, the mapping \(f(\cdot,x):\mathbb R \to \mathbb C^n\) is bounded, and, for any fixed \(x\in \mathbb C^n\), \(f(t,\cdot):\mathbb C^n \to \mathbb C^n\) is a \(C^k\)-mapping which vanishes at \(x=0\) together with its first derivative. The case is studied where \(0<|\lambda_k|<1\), \(k=1,\ldots,n\). NEWLINENEWLINENEWLINEGeneralizing the results of \textit{S. Sternberg} [Am. J. Math. 79, 809-824 (1957; Zbl 0080.29902), Am. J. Math. 80, 623-631 (1958; Zbl 0083.31406)] the author constructs changes of variables which linearize the equation in nonresonant case (i.e. when \(\lambda_i\neq\lambda_1^{i_1}\cdots \lambda_n^{i_n}\), \(i=1,\ldots,n\), \(\sum_{j=1}^{n}i_j\leq k\)), and which reduce this equation to a finite normal form in general case (when resonant terms of order \(\leq k\) are present in Taylor expansion of \(f(t,x)\)). The existence and properties of invariant manifolds for the equation are studied under the assumption that the linear mapping \(x \mapsto \Lambda x\) is hyperbolic.