Invariant manifolds with asymptotic phase for nonautonomous difference equations
The concept of asymptotic phase originally occurred in connection with the approach of a solution of an autonomous ordinary differential equation to an arbitrary asymptotically stable periodic solution [cf. \textit{B. Aulbach}, Nonlinear Anal. Theory Methods Appl. 6, 817--827 (1982; Zbl 0509.58034)] for autonomous difference equations with an invariant manifold. Conditions are known which guarantee that a solution approaching this manifold eventually behave like a solution on this manifold [cf. \textit{B. Aulbach}, Continuous and discrete dynamics near manifolds of equilibria. (1984; Zbl 0535.34002); \textit{J. López-Fenner} and \textit{M. Pinto}, Comput. Math. Appl. 33, No.~10, 105-124 (1997; Zbl 0880.39012)]. In this paper the authors generalize the main result of \textit{B. Aulbach} [1984; loc. cit.] for autonomous difference equations to the case of a nonautonomous difference equation \( x(k+1) = f(k,x(k)) \) whose right-hand side is allowed to be invertible and whose invariant manifold doesn't necessarily consist of stationary solutions. The paper consists of 5 sections and an appendix. The main result is given in section 5, that is if \( x(k+1) = f(k,x(k)) \) is a difference equation with invariant compact manifold \(M\), then under 5 conditions \(M\) possesses an asymptotic phase.
scientific article; zbMATH DE number 2082241