Resonant Phenomena Of Radially Symmetric Systems

The radially symmetric system is an important model of diferential systems. Inthis paper, we prove the existence of infinite quasi-periodic solutions and periodicsolutions for the system near resonance point. Moreover, we establish the coexistenceof periodic solution and unbounded solution for the system.According to the characteristics of radially symmetric system, we transform it intoa second order singular equation with a parameter and a first order equation. In orderto investigate the radially symmetric system, we need to study the behavior of thesolution of singular equation. In the case of the perturbed term being unbounded, wedon’t know whether there is a twist property of Poincar′e map or not. So we considerthe successor map and prove the twist property of the successor map. In the case of theperturbed term being bounded, we introduce some new action-angle variables to provethe twist property. Then we prove the existence of periodic solutions for the system.When the perturbed term is small enough, we give the condition for the coexistenceof periodic solution and unbounded solution for the radially symmetric system.