Garab Ábel
(Alpen-Adria Universität, Klagenfurt, Ausztria)
Morse decomposition for scalar delay difference equations
Absztrakt: Consider the following class of difference equations
x_{k+1} = f(x_k, x_{k-d}), where f : R^2 › R is a C^1 function, which is strictly increasing in its first variable and fulfils either yf(0, y) > 0 or yf(0, y) < 0 for all y \in R {0}. Under the assumption that the global attractor exists, we give a Morse
decomposition of the attractor. The decomposition is based on an integer valued Lyapunov function introduced by Mallet-Paret and Sell as a discrete-time counterpart of their celebrated discrete Lyapunov function
for delay differential equations. The results can be applied to many known
discrete-time models, e.g. May’s genotype model, the Wazewska-Czyzewska
and Lasota equation and also to discretizations of several continuous-time
equations, e.g. the Krisztin–Walther equation, the Mackey–Glass equation,
Wright’s equation, etc.
