Normal form and stability of the zero solution of a second-order periodic invertible ODE with a small parameter

Abstract

The problem of the stability of the zero solution of differential equation x''(t) + \beta^2 x^{2n-1}+\varepsilon b(t) x^{n - 1} x' + X(t, x, x') = 0 under the following assumptions, is considered: n is a natural number, \varepsilon is a non-negative parameter, X is real analitic in x, x', continuous and 2\pi-periodic in t function, expansion of X does not contain terms of order < 2n if the order of x' is ascribed to be equal to n. Order of x is equal to 1. Also, b(t) is continuous 2\pi-periodic odd function and X(-t, x, -x' ) = X(t, x, x'). Thus, the equation is invariant under change t \to -t. Such equations are named reversible. There are two possibilities. Either all terms of the expansion of X must be taken into consideration or only a finite number of them is needed. After Liapunov, the first case is named transcendental and the second one is named algebraic. Reversible equations are transcendental. In 2022, it was established that the trivial solution of the equation is stable if n >= 2 (\beta = 1). This paper is devoted to the case n = 1, \beta is irrational (this condition may be weakened). For proof methods of KAM-theory modified for reversible systems are used. According to this theory in any neighborhood of the origin there exist invariant two-dimension tori dividing three-dimensional phase space. This implies the stability of the trivial solution. There are some differences between application of the KAM-theory if n >= 2 and if n = 1. Small parameter is treated as one of the variables with the same order as the order of x, x' , and the obtained system can be reduced to the normal form with the constant and pure imaginary coefficients. It is shown that the normalization process is more convenient, if it is assumed that X = X(t, x, x' , \varepsilon), and its expansion starts with terms of order two or greater. The obtained formal normal form presents an interesting subject on its own. The stability is proven under condition that the first coefficient of the normal form, that doesn’t contain the small parameter, is not zero. The degenerate case, where this coefficient is a zero, requires additional research, because the main theorems of the KAM-theory cannot be applied without further modifications.

Keywords:

second-order differential equations, periodic perturbations, stability, normal form, reversibility, transcendence