On the stability of the zero solution of differential equation of the second order under critical case

Abstract

Differential equation of the form x¨+x2sgnx = Y (t, x, x˙ ), is considered, where the right-hand side is a small periodic perturbation of t, a sufficiently differentiable function in the origin neighborhood with variables x, x˙ . It is assumed that X perturbation is of order smallness not lower than the fifth, if x is assigned the second order, and x˙ is assigned the third order. Periodic functions are introduced that are the solution of the equation above with zero righthand side. Since differentiability of the quadratic part is bounded, then differentiability of the introduced functions is also bounded. These functions are used to transfer from the initial equation to the system of equations in coordinates similar to polar. This system with the help of the polynomial replacement is reduced to a system with Lyapunov constants. Replacement coefficients are found by the partial fraction decomposition. The conclusion about the nature of stability of the zero solution is made by a sign of the first non zero constant. Due to the bounded differentiability of the introduced functions, the degree of the polynomial replacement must be limited. The system of differential equations for the replacement coefficients is solved recursively. The number of found Lyapunov constants is also bounded. This article considers the case when all found constants are zero. To study this problem there used the method of isolating the main part of the introduced functions and their combinations as a result of the expantion of the latter in the Fourier series. The remainder of the series is assumed to be sufficiently small and it is shown that its presence can be neglected. The transition to the main parts instead of functions allows to compensate for the lack of differentiability of the introduced functions. Considering such systems, polynomial replacement can be again used and the Lyapunov constant for each main part can be found. It is shown that the sign of the constant for any main part is preserved. Sufficient conditions for stability and instability are indicated.