Non-resonance and double resonance for a planar system via rotation numbers
The authors consider a general planar periodic system and propose two existence results. In the first one they compare the nonlinearity with two positively homogeneous functions with ``rotation numbers'' larger than some \(n\) and smaller than \(n+1\). They thus prove that the system has a periodic solution, by the use of the Poincaré-Bohl fixed point theorem. This is a generalization of some classical ``nonresonance'' results. In the second one the above two functions have rotation numbers exactly equal to \(n\) and \(n+1\). Then, in order to avoid possible resonance phenomena, they add two Landesman-Lazer conditions, and they prove again the existence of a periodic solution. The proofs involve delicate analysis in the phase-plane, in order to precisely estimate the rotational properties of the solutions.
scientific article; zbMATH DE number 7369324
scientific article