Existence of dichotomies and invariant splittings for linear differential systems I

Robert J. Sacker, George R. Sell

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Abstract

This paper is concerned with linear time-varying ordinary differential equations. Sufficient conditions are given for the existence of an exponential dichotomy for a class of equations which includes those with Bohr almost-periodic coefficients. The problem is treated in the context of linear skew-product flows, where it becomes clear how to generalize to the case of fiber-preserving flows on vector bundles. Both continuous and discrete flows are treated and the results apply to the linearized variational equation for a time-varying vector field on a manifold as well as the linearization of a diffeomorphism acting on a manifold. Sufficient conditions are given for a diffeomorphism on a manifold to be an Anosov diffeomorphism. For linear skew-product flows arising from ordinary differential equations our theory is a partial generalization of Floquet theory to the almost-periodic case.

Original languageEnglish (US)
Pages (from-to)429-458
Number of pages30
JournalJournal of Differential Equations
Volume15
Issue number3
DOIs
StatePublished - May 1974

Bibliographical note

Funding Information:
* This research was begun while visiting at the Istituto di IMatematica versita di Firenze under auspices of the Italian Research Council (C.N.R.). partially supported by U.S. Army Grant DA-ARO-D-31-124-71-Cl76 by NSF Grant No. GP-38955.

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