Planar differential systems at resonance
The authors study perturbed Hamiltonian systems of the form \[ J\dot{u} = \nabla \mathcal{H} (u) + f(u) + p(t), \] where \( J= \left(\begin{smallmatrix} 0 & -1 \\ 1 & 0 \end{smallmatrix}\right)\) is the standard symplectic matrix, \(\mathcal{H}: \mathbb{R}^2 \rightarrow \mathbb{R} \) is of class \(C^1\) with locally Lipschitz continuous gradient, \(f: \mathbb{R}^2 \rightarrow \mathbb{R}^2\) is locally Lipschitz continuous and bounded and \(p: \mathbb{R}\rightarrow \mathbb{R}^2\) is measurable, bounded and periodic. If the function \(f\) is given by \[ f(u) =\sum_{k=1}^{m} f_k \left ( \langle u | e^{i\vartheta_k} \rangle \right ) \] where \(\langle\cdot | \cdot \rangle\) denotes the Euclidean scalar product in \(\mathbb{R}^2,\) \( 0 \leq \vartheta_1 < \vartheta_2 < \ldots < \vartheta_m < 2\pi,\) and the functions \(f_k: \mathbb{R} \rightarrow \mathbb{R}^2, \;1 \leq k \leq m,\) are bounded, then the authors prove some new existence theorems for periodic solutions and for unbounded solutions. The proof is based on a convenient change of variables, the associated Poincaré map and topological degree. Applications are given to some second-order equations with separated nonlinearities with the special cases of Liénard and Rayleigh equations.
scientific article; zbMATH DE number 5294332