On the stability of an equilibrium of a system of differential equations in the critical case of two pure imaginary and two zero roots of the characteristic equation (Q2703886)

The real-analytic autonomous system NEWLINE\[NEWLINE\begin{aligned}\dot x_1=-x_2+X_2 (x_1,x_2,y_1,y_2),\quad & \dot y_1=y_2+ Y_2(x_1,x_2,y_1,y_2),\\ \dot x_2=x_1+X_2 (x_1,x_2, y_1,y_2),\quad & \dot y_2=Y_2 (x_1,x_2,y_1,y_2), \end{aligned} \tag{1}NEWLINE\]NEWLINE where the series \(x_j\), \(y_j\), \(j=1,2\), begin with terms of order \(\geq 2\), is considered. System (1) has two invariant surfaces: \(x_1=x_2=0\) and \(y_1=y_2=0\). The problem to distinguish between a center and a focus on these surfaces has to be solved.NEWLINENEWLINENEWLINEThe author performs and polar substitution of the variables and obtains equations for the new variables \(r,\rho, \varphi\) and \(\theta\). Then, he proves that the zero solution to (1) is asymptotically stable if in the equations which are obtained after successively averaging of the right-hand sides in the equations for \(r,\rho,\varphi\) and \(\theta\), has nonzero terms of \(n\)th order \((n\geq 3)\), and if the zero solution to the corresponding truncated system, whose right-hand side contains only the indicated terms is asymptotically stable.