Dissipative delay endomorphisms and asymptotic equivalence

Abstract

Using an invariant manifold theorem we demonstrate that the dynamics of nonautonomous dissipative delayed difference equations (with delay $M$) is asymptotically equivalent to the long-term behavior of an $N$-dimensional first order difference equation (with $N \leq M)$ – assumed the nonlinearity is small Lipschitzian on the absorbing set. As consequence we obtain a result of Kirchgraber that multi-step methods for the numerical solution of ordinary differential equations are essentially one-step methods, and generalize it to varying step-sizes.