Advanced Studies in Pure Mathematics

Dissipative delay endomorphisms and asymptotic equivalence

Christian Pötzsche

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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.

Article information

Advances in Discrete Dynamical Systems, S. Elaydi, K. Nishimura, M. Shishikura and N. Tose, eds. (Tokyo: Mathematical Society of Japan, 2009), 237-259

Received: 30 October 2006
First available in Project Euclid: 28 November 2018

Permanent link to this document euclid.aspm/1543447658

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 39A11 37D10: Invariant manifold theory 65L06: Multistep, Runge-Kutta and extrapolation methods

Delay difference equation pullback attractor attractive invariant manifold


Pötzsche, Christian. Dissipative delay endomorphisms and asymptotic equivalence. Advances in Discrete Dynamical Systems, 237--259, Mathematical Society of Japan, Tokyo, Japan, 2009. doi:10.2969/aspm/05310237.

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