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Zbl 0622.34040
Smith, Russell A.
Some applications of Hausdorff dimension inequalities for ordinary differential equations. (English)
[J] Proc. R. Soc. Edinb., Sect. A 104, 235-259 (1986). ISSN 0308-2105; ISSN 1473-7124/e

Author's summary: "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 $\omega$-periodic differential equations to have only a finite number of $\omega$-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."
[H.Hochstadt]

MSC 2000:: *34C25 Periodic solutions of ODE
34C15 Nonlinear oscillations of solutions of ODE
28D99 Measure-theoretic ergodic theory

Keywords: Upper bounds; Hausdorff dimension; autonomous equations; Lorenz equation; forced Duffing equation

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Zentralblatt MATH Berlin [Germany]