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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

A remark on linear differential systems with the same invariant subbundles


Author: Robert E. Vinograd
Journal: Proc. Amer. Math. Soc. 86 (1982), 305-306
MSC: Primary 58F25
MathSciNet review: 667294
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Abstract: Given a flow $ \sigma (\omega ,t) = \omega \cdot t$ on a compact metric space $ \Omega $ and a continuous $ n \times n$-matrix $ A(\omega )$, the family of ODE systems $ \dot x = A(\omega \cdot t)x$ defines a linear skew-product flow on $ W = \Omega \times X$, $ X = {R^n}$ or $ {C^n}$. Let $ W = U \oplus V$ be a Whitney sum and $ P:W \to W$ be the correspondent projector. Result: the subbundles $ U$ and $ V$ are invariant for the flows induced by $ A(\omega )$ and $ B(\omega )$ iff $ A(\omega )$- $ B(\omega )$ commutes with $ P(\omega )$ for all $ \omega \in \Omega $.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1982-0667294-8
PII: S 0002-9939(1982)0667294-8
Keywords: Almost periodic, ordinary differential equations, projector, skew-product flow, subbundle, Whitney sum
Article copyright: © Copyright 1982 American Mathematical Society