Synchronization theory for forced oscillations in second-order systems

We consider differential equations of the form x¨+∈f(x,x˙)+x=∈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 controlsūε 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 x¨+∈x˙+x=∈u, the Duffing equation x¨+∈(x−1)x˙+x=∈u, and the Van der Pol equation x¨+∈(x2−1)x˙+x=∈u.