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On a gradient-like integro-differential equation

Published online by Cambridge University Press:  14 November 2011

Jack K. Hale  and
Krzysztof P. Rybakowski
Jack K. Hale
Affiliation:
Division of Applied Mathematics, Lefschetz Center for Dynamical Systems, Brown University, Providence, R. I. 02912, U.S.A.
Krzysztof P. Rybakowski
Affiliation:
Division of Applied Mathematics, Lefschetz Center for Dynamical Systems, Brown University, Providence, R. I. 02912, U.S.A.
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Synopsis

Under appropriate conditions on b and a function g with 2k +1 simple zeros, the equation

has a maximal compact invariant set Ab,g in C([−1,0]R), consisting of the zeros of g and the one-dimensional unstable manifolds of these zeros. For k =2, it is shown that there may be a saddle connection in the flow on Ab,g for some g. This implies that the zeros of g as elements of the flow on Ab,g cannot be given the natural order of the reals.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 92 , Issue 1-2 , 1982 , pp. 77 - 85
Copyright
Copyright © Royal Society of Edinburgh 1982

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References

1Hale, J. K.. Theory of Functional Differential Equations (Berlin: Springer, 1977).CrossRefGoogle Scholar
2Hale, J. K.. Asymptotic behavior of an integro-diflferential equation. Amer. J. Math., to appear.Google Scholar
3Whitney, H.. Analytic extensions of differentiable functions defined on closed sets. Trans. Amer. Math. Soc. 36 (1934), 369–387.CrossRefGoogle Scholar