Synchronization theory for forced oscillations in second-order systems

Abstract

We consider differential equations of the form

$$\ddot x + \in f(x,\dot x) + x = \in u$$

, where ε >0 is supposed to be small. For piecewise continuous controlsu(t), satisfying |u(t)| ≤ 1, we present sufficient conditions for the existence of 2π-periodic solutions with a given amplitude. We present a method for determining the limiting behavior of controlsū _ε for which the equation has a 2π-periodic solution with a maximum amplitude and for determining the limit of this maximum amplitude as ε tends to zero. The results are applied to the linear system$\ddot x + \in \dot x + x = \in u$, the Duffing equation$\ddot x + \in (x - 1)\dot x + x = \in u$, and the Van der Pol equation$\ddot x + \in (x^2 - 1)\dot x + x = \in u$.