On the stability of the zero solution of a diﬀerential equation of the second order under periodic perturbation of a center
Small periodic perturbations of the oscillator x¨ + x2n-1 = 0 are considered, where 1 < n is a natural number, and the right-hand side is an analytic function in the origin neighborhood with variables x˙, x. The equilibrium position of the given equation on stability is investigated. As a result, suﬃcient conditions for asymptotic stability and instability are formulated. In this work new periodic functions of the Lyapunov type are introduced. With their help, a transition to the system of equations is performed, similar to the transition to a system in polar coordinates. A system of two diﬀerential equations is obtained, the unknown functions of which are the amplitude and the “angular” variable. Then, polynomial change of variables in powers of the amplitude is made. The coeﬃcients are periodic in time and “angular” variable functions. This replacement leads to a system of diﬀerential equations with a Lyapunov constant which in general is non-zero. Its sign determines the stability of the zero solution of the original equation. It is important that cases of even and odd n diﬀer from each other. For an even n a non-zero Lyapunov constant can be found from one equation, and for an odd one it can be found from a system of three equations. The system is solved recurrently.