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Generalized theorem of Hartman-Grobman on measure chains

Part of: Qualitative theory Difference equations Difference and functional equations

Published online by Cambridge University Press:  09 April 2009

Stefan Hilger
Stefan Hilger
Affiliation:
Mathematisch-Geographische FakultättKatholische Universität EichstättD-85071 EichstätGermany e-mail: mga052@eo-nwfs-1.ku-eichstaett.de
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Abstract

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We will prove the Theorem of Hartman-Grobman in a very general form. It states the topological equivalence of the flow of a nonlinear non-autonomous differential or difference equation with critical component to the flow of a partially linearized equation. The critical spectrum has not necessearily to be contained in the imaginary axis or the unit circle respectively. Further on we will employ the socalled calculus on measure chains within dynamical systems theory. Within this calculus the usual one dimensional time scales can be replaced by measure chains which are essentially closed subsets of R. The paper can be understood without knowledge of this calculus.

So our main theorem will be valid even for equations defined on very strange time scales such as sequences of closed intervals. This is especially interesting for applications within the theory of differential-difference equations or within numerical analysis of qualitative phenomena of dynamical systems.

MSC classification

Secondary: 39A12: Discrete version of topics in analysis 34C20: Transformation and reduction of equations and systems, normal forms
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 60 , Issue 2 , April 1996 , pp. 157 - 191
Copyright
Copyright © Australian Mathematical Society 1996

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