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Some applications of Hausdorff dimension inequalities for ordinary differential equations

Published online by Cambridge University Press:  14 November 2011

Russell A. Smith
Russell A. Smith
Affiliation:
Department of Mathematical Sciences, University of Durham, Science Laboratories, South Road, Durham DH1 3LE, U.K.
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Synopsis

Upper bounds are obtained for the Hausdorff dimension of compact invariant sets of ordinary differential equations which are periodic in the independent variable. From these are derived sufficient conditions for dissipative analytic n-dimensional ω-periodic differential equations to have only a finite number of ω-periodic solutions. For autonomous equations the same conditions ensure that each bounded semi-orbit converges to a critical point. These results yield some information about the Lorenz equation and the forced Duffing equation.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 104 , Issue 3-4 , 1986 , pp. 235 - 259
Copyright
Copyright © Royal Society of Edinburgh 1986

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References

1Bellman, R.. Introduction to Matrix Analysis (New York: McGraw-Hill, 1960).Google Scholar
2Coddington, E. A. and Levinson, N.. Theory of Ordinary Differential Equations (New York: McGraw-Hill, 1955)Google Scholar
3Cronin, J.. Differential Equations: Introduction and Qualitative Theory (New York: Marcel Dekker, 1980).Google Scholar
4Dieudonné, J.Foundations of Modern Analysis, Vol. 1 (New York: Academic Press, 1960).Google Scholar
5Douady, A..and Oesterlé, J.Dimension de Hausdorff des attractors. C. R. Acad. Sci. Paris Sér. I Math. 290 (1980), 1135–1138.Google Scholar
6Fan, Ky. Maximum properties and inequalities for the eigenvalues of completely continuous operators. Proc. Nat. Acad. Sci. U.S.A. 37 (1951), 760–766.CrossRefGoogle ScholarPubMed
7Fredrickson, P., Kaplan, J. L., Yorke, E. D. and Yorke, J. A.. The Liapunov dimension of strange attractors. J. Differential Equations 49 (1983), 185–207.Google Scholar
8Guckenheimer, J. and Holmes, P.. Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (New York: Springer-Verlag, 1983).CrossRefGoogle Scholar
9Hartman, P.. Ordinary Differential Equations (New York: Wiley, 1964).Google Scholar
10Hartman, P. and Olech, C.. On global asymptotic stability of ordinary differential equations. Trans. Amer. Math. Soc. 104 (1962), 154–178.Google Scholar
11Lorenz, E. N.. Deterministic non-periodic flow. J. Atmospheric Sci. 20 (1963), 130–141.2.0.CO;2>CrossRefGoogle Scholar
12Milnor, J. W.. Topology from the Differentiate Viewpoint (Charlottesville: Virginia Univ. Press, 1965).Google Scholar
13Mori, H.. Fractal dimension of chaotic flows of dissipative systems. Progr. Theoret. Phys. 63 (1980), 1044–1047.Google Scholar
14Nakajima, F. and Seifert, G.. The number of periodic solutions of 2-dimensional periodic systems. J. Differential Equations 49 (1983), 430–440.CrossRefGoogle Scholar
15Olech, C.. On the global stability of an autonomous system on the plane. Contrib. Differential quations 1 (1963), 389–400.Google Scholar
16Pugh, C. C.. An improved closing lemma and general density theorem. Amer. J. Math. 89 (1967), 1010–1021.CrossRefGoogle Scholar
17Sard, A.. The measure of the critical values of differentiable maps. Bull. Amer. Math. Soc. 48 (1942), 883–890.CrossRefGoogle Scholar
18Smith, R. A.. An index theorem and Bendixson's negative criterion for certain differential equations of higher dimension. Proc. Roy. Soc. Edinburgh Sect. A 91 (1981), 63–77.CrossRefGoogle Scholar
19Smith, R. A.. Massera's convergence theorem for periodic nonlinear differential equations. J. Math. Anal. Appl. (to appear).Google Scholar
20Sparrow, C.. The Lorenz Equations: Bifurcations, Chaos and Strange Attractors (New York: Springer-Verlag, 1982).CrossRefGoogle Scholar
21Weyl, H.. Inequalities between two kinds of eigenvalues of a linear transformation. Proc. Nat. Acad. Sci. U.S.A. 35 (1949), 408–411.Google Scholar
22Yoshizawa, T.. Stable sets and periodic solutions in a perturbed system, Contrib. Differential Equations 2 (1963), 407–420.Google Scholar