Synchronization Theory for Forced Oscillations in Second-Order Systems ~

We consider differential equations of the form ~ + ~f(x, ~ ) + x = Eu, where E > 0 is supposed to be small. For piecewise continuous controls u(t), satisfying lu( t ) l -1 , we present sulIicient conditions for the existence of 2¢r-periodic solutions with a given amplitude. We present a method for determining the limiting behavior of controls fi, for which the equation has a 27r-periodic solution with a maximum amplitude and for determining the limit of this maximum amplitude as • tends to zero. The results are applied to the linear system 5/+ e2+ x = eu, the Duffing equation 5i+ e (2+ x 3) + x = eu, and the Van der Pol equation 5/+ ~(x 2 1 )2+x = eu.