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  • Proceedings of the Royal Society of Edinburgh: Section A Mathematics, Volume 104, Issue 3-4
  • January 1986, pp. 235-259

Some applications of Hausdorff dimension inequalities for ordinary differential equations

  • Russell A. Smith (a1)
  • DOI: http://dx.doi.org/10.1017/S030821050001920X
  • Published online: 14 November 2011
Abstract
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.

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6Ky Fan . Maximum properties and inequalities for the eigenvalues of completely continuous operators. Proc. Nat. Acad. Sci. U.S.A. 37 (1951), 760766.

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11E. N. Lorenz . Deterministic non-periodic flow. J. Atmospheric Sci. 20 (1963), 130141.

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21H. Weyl . Inequalities between two kinds of eigenvalues of a linear transformation. Proc. Nat. Acad. Sci. U.S.A. 35 (1949), 408411.

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Proceedings of the Royal Society of Edinburgh Section A: Mathematics
  • ISSN: 0308-2105
  • EISSN: 1473-7124
  • URL: /core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics
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