Periodic, quasi-periodic and unbounded solutions of radially symmetric systems with repulsive singularities at resonance
The authors deal with a radially symmetric system of the type \[ u''(t)=-h(|u(t)|,t)\,\frac{u(t)}{|u(t)|}\,, \] where \(h:(0,+\infty)\times{\mathbb R}\to{\mathbb R}\) is continuous and \(2\pi\)-periodic in \(t\). Writing \[ h(x,t)=\frac{n^2}{4} x +f(x,t)\,, \] where \(n\) is a positive integer, they consider the function \(\sigma_f:{\mathbb R}\to{\mathbb R}\) defined as \[ \sigma_f(\theta):=\int_0^{2\pi}\left|\sin\left(\frac{\theta}{2}+\frac{n}{2}t\right)\right|f(+\infty,t)\,dt\,. \] They prove that, if \(\sigma_f(\theta)\) doesn't change sign, then there are infinitely many periodic solutions rotating around the origin. Otherwise, if there is a simple zero of \(\sigma_f(\theta)\), all solutions starting sufficiently far from the origin will be unbounded, either in the past, or in the future.
scientific article; zbMATH DE number 6494619
scientific article