A geometric approach to periodically forced dynamical systems in presence of a separatrix (Q1763668)

The main result of the article gives some sufficient conditions under which the periodic boundary problem for the nonlinear nonautonomous second-order ifferential equation \[ x''+F(x,x')=e(t) \] has a \(T\)-periodic solution. Here, \(e(t)\) is a continuous \(T\)-periodic forcing term and the function \(F(x,y)\) satisfies a standard sign condition and has properties guaranteeing the uniqueness of the solution. The authors propose three alternative conditions for the existence, that are related to properties of the trajectories of the associated autonomous equation. The main tool is a continuation theorem. A particular interest is addressed to the case when the nonautonomous equation and the autonomous one posses an unbounded separatrix, whose behavior provides a partial bound for the \(T\)-periodic solutions of the homotopic equation \[ x''+F(x,x')=\lambda e(t),\quad \lambda \in [0,1]. \] The last section of the paper contains some applications.