BOUNDEDNESS IN A PIECEWISE LINEAR OSCILLATOR AND A VARIANT OF THE SMALL TWIST THEOREM

Abstract

Consider the differential equation

$$\ddot{x} +n^2 x+h_L (x) =p(t),$$

where $n=1,2,\dots$ is an integer, $p$ is a $2\pi$-periodic function and $h_L$ is the piecewise linear function

$$ h_L (x)=\begin{cases} L & \text{if $x\geq 1$},\\

Lx & \text{if $|x|\leq 1$},\\

-L & \text{if $x\leq -1$}.\end{cases}$$

A classical result of Lazer and Leach implies that this equation has a $2\pi$-periodic solution if and only if

\begin{equation}\label{ll} |\hat{p}_n |<{2L\over \pi}, \end{equation}

where

$$\hat{p}_n :={1\over 2\pi}\int_0^{2\pi} p(t)e^{-int}\, dt.$$

In this paper I prove that if $p$ is of class $C^5$ then the condition (\ref{ll}) is also necessary and sufficient for the boundedness of all the solutions of the equation.

The proof of this theorem motivates a new variant of Moser's Small Twist Theorem. This variant guarantees the existence of invariant curves for certain mappings of the cylinder which have a twist that may depend on the angle.

1991 Mathematics Subject Classification: 34C11, 58F35.