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On smoothness of carrying simplices

Author: Janusz Mierczynski
Journal: Proc. Amer. Math. Soc. 127 (1999), 543-551
MSC (1991): Primary 34C30, 34C35; Secondary 58F12, 92D40
MathSciNet review: 1606000
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Abstract: We consider dissipative strongly competitive systems $\dot{x}_{i}=x_{i}f_{i}(x)$ of ordinary differential equations. It is known that for a wide class of such systems there exists an invariant attracting hypersurface $\Sigma$, called the carrying simplex. In this note we give an amenable condition for $\Sigma$ to be a $C^{1}$ submanifold-with-corners. We also provide conditions, based on a recent work of M. Benaïm (On invariant hypersurfaces of strongly monotone maps, J. Differential Equations 136 (1997), 302-319), guaranteeing that $\Sigma$ is of class $C^{k+1}$.

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Additional Information

Janusz Mierczynski
Affiliation: Institute of Mathematics, Wrocław University of Technology, Wybrzeże Wyspiań- skiego 27, PL-50-370 Wrocław, Poland

Received by editor(s): June 2, 1997
Additional Notes: The author’s research was supported by KBN grant 2 P03A 076 08 (1995–97).
Communicated by: Hal L. Smith
Article copyright: © Copyright 1999 American Mathematical Society