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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A generalized Poincaré stability criterion

Authors: Carmen Chicone and R. C. Swanson
Journal: Proc. Amer. Math. Soc. 81 (1981), 495-500
MSC: Primary 58F15; Secondary 58F10
MathSciNet review: 597670
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Abstract: Let $ \Phi _t^\char93 \eta = {\Phi ^{ - t}} \circ \eta \circ {\phi ^t}$ define a semigroup on the Banach space $ \Gamma (M,E)$ of continuous sections of $ E$ over $ M$. It is known that $ ({\Phi ^t},{\phi ^t})$ is hyperbolic iff $ \Phi _\char93 ^t$ has spectrum off the unit circle for $ t \ne 0$. We prove that a third equivalent condition is that the (unbounded!) infinitesimal generator $ L$ of $ \{ \Phi_t^\char93 \}$ have its spectrum disjoint from the imaginary axis. In two dimensions this property coincides with the Poincaré stability criterion for a periodic orbit of a planar dynamical system.

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Keywords: Hyperbolic flow, spectrum, adjoint representation
Article copyright: © Copyright 1981 American Mathematical Society