Date: 19 Dec 2005

Optimal estimates on rotation number of almost periodic systems

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Abstract.

In this paper, we will give some optimal estimates on the rotation number of the linear equation \(\ifmmode\expandafter\ddot\else\expandafter\"\fi{x} + p(t)x = 0,\) and that of the asymmetric equation: \(\ifmmode\expandafter\ddot\else\expandafter\"\fi{x} + p(t)x_{ + } + q(t)x_{ - } = 0,\) where p(t) and q(t) are almost periodic functions and \(x_{ + } = \max \{ x,0\} ,\;x_{ - } = \min \{ x,0\} .\) These estimates are obtained by introducing some kind of new norms in the spaces of almost periodic functions.

Supported by the National Natural Science Foundation of China (no. 10325102), TRAPOYT-M.O.E. of China (2001), and the National 973 Project of China (no. G1999075108).
Received: April 6, 2004; revised: July 7, 2004