Twist periodic solutions of repulsive singular equations
The paper deals with the periodic problem \[ x^{\prime\prime}+g(t,x)=0, \quad x(0)=x(2\pi),\quad x^{\prime}(0)=x^\prime(2\pi), \] where \(g(t,x)\) is \(2\pi\)-periodic in \(t\) and \(C^{0,4}\) in \((t,x)\). A motivation to study this problem is due to particular equations of this type: the Lazer-Solimini equation describing the motion of a charged particle in an electric field and the Brillouin equation governing a focusing system for an electron beam immersed in a periodic magnetic field. A solution is said to be of twist type if the first coefficient of Birkhoff, also called the twist coefficient, is nonzero. If strong resonance is avoided, a solution of twist type is Lyapunov-stable. In the paper, by using a compact expression of the twist coefficient, a new stability criterion is derived for a (positive) periodic solution to be of twist type. On this base an estimation on the region of parameters is obtained, where the equations have twist periodic solutions. An application to the motivating equations is given.
scientific article; zbMATH DE number 2101854