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Attractivity for two-dimensional linear systems whose anti-diagonal coefficients are periodic

Authors: Jitsuro Sugie and Ayano Endo
Journal: Proc. Amer. Math. Soc. 137 (2009), 4117-4127
MSC (2000): Primary 34D05, 34D20; Secondary 37B25, 37C75
Published electronically: June 19, 2009
MathSciNet review: 2538573
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Abstract: This paper deals with the linear system $ \mathbf{x}' = A(t)\mathbf{x}$ with $ A(t)$ being a $ 2\times2$ matrix. The anti-diagonal components of $ A(t)$ are assumed to be periodic, but the diagonal components are not necessarily periodic. Our concern is to establish sufficient conditions for the zero solution to be attractive. Floquet theory is of no use in solving our problem, because not all components are periodic. Another approach is adopted. Some simple examples are included to illustrate the main result.

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Additional Information

Jitsuro Sugie
Affiliation: Department of Mathematics and Computer Science, Shimane University, Matsue 690-8504, Japan

Ayano Endo
Affiliation: Department of Mathematics and Computer Science, Shimane University, Matsue 690-8504, Japan

Keywords: Attractivity, linear systems, weakly integrally positive, Floquet theory
Received by editor(s): March 17, 2009
Published electronically: June 19, 2009
Additional Notes: The first author was supported in part by a Grant-in-Aid for Scientific Research, No. 19540182, from the Japan Society for the Promotion of Science
Communicated by: Yingfei Yi
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.