Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

Some applications of Hausdorff dimension inequalities for ordinary differential equations

Russell A. Smitha1

a1 Department of Mathematical Sciences, University of Durham, Science Laboratories, South Road, Durham DH1 3LE, U.K.


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.

(Received November 13 1985)