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Unstable sets, heteroclinic orbits and generic quasi-convergence for essentially strongly order-preserving semiflows

Part of: Qualitative theory Partial differential equations

Published online by Cambridge University Press:  23 September 2009

Taishan Yi  and
Qingguo Li
Taishan Yi
Affiliation:
College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, People's Republic of China; Email: ([email protected])
Qingguo Li
Affiliation:
College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, People's Republic of China; Email: ([email protected]) State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, People's Republic of China; Email: ([email protected]
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Abstract

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We present conditions guaranteeing the existence of non-trivial unstable sets for compact invariant sets in semiflows with certain compactness conditions, and then establish the existence of such unstable sets for an unstable equilibrium or a minimal compact invariant set, not containing equilibria, in an essentially strongly order-preserving semiflow. By appealing to the limit-set dichotomy for essentially strongly order-preserving semiflows, we prove the existence of an orbit connection from an equilibrium to a minimal compact invariant set, not consisting of equilibria. As an application, we establish a new generic convergence principle for essentially strongly order-preserving semiflows with certain compactness conditions.

Keywords

essentially strongly order-preserving semiflowinvariant setunstable set

MSC classification

Secondary: 34C37: Homoclinic and heteroclinic solutions 35B40: Asymptotic behavior of solutions
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 52 , Issue 3 , October 2009 , pp. 797 - 807
Copyright
Copyright © Edinburgh Mathematical Society 2009