Duality theorems in the theory of nonlinear oscillations. (Q1128379)

The existence of harmonic solutions in periodic nonautonomous systems and the existence of free oscillations (limit cycles) in autonomous systems of ordinary differential equations can be established by the existence of fixed points of operators in finite-dimensional spaces (e.g. Poincaré-map) or in infinite-dimensional spaces (e.g. Hammerstein operator). The authors study the topological index of those operators and compare them. For example, they prove that the topological index and the functional characteristic of a limit cycle coincide. These results are applied to establish the existence of periodic solutions and their approximation.