Centre and focus problem

The problem of determining conditions under which all trajectories of an autonomous system of ordinary differential equations

$$\dot x=X(x,y),\quad \dot y=Y(x,y)\label{*}\tag{*}$$

in a certain neighbourhood of an equilibrium position $O$, except the point $O$ itself, are closed curves. The functions $X$ and $Y$ are assumed to be analytic in a certain neighbourhood of $O$. The problem was raised by H. Poincaré in . Fundamental results were obtained by A.M. Lyapunov in [2].

As a rule, one assumes that the characteristic equation of the linearized system at $O$, that is, the system

$$\dot\xi=X_x'(O)\xi+X_y'(O)\eta,$$

$$\dot\eta=Y_x'(O)\xi+Y_y'(O)\eta,$$

has pure imaginary roots. Then the singular point $O$ is for the system \eqref{*} either a centre (surrounded by closed trajectories) or a focus (surrounded by spirals). In this case a necessary and sufficient condition for the existence of a centre is that the system \eqref{*} has in a neighbourhood of $O$ a real-analytic integral $F(x,y)=C$ that is independent of $t$ (see [2]). On the basis of this result, methods have been worked out to obtain conditions for the presence of a centre; these conditions involve that an infinite sequence of polynomials in the coefficients of the series expansions of the right-hand sides of \eqref{*} vanish. In the case of polynomial right-hand sides it follows from Hilbert's theorem on the finiteness of bases of polynomial ideals that in this sequence only finitely many are essential and that the remaining ones are consequences of them. The problem to determine the number of essential conditions for a centre is very complicated and has been completely solved only when $X$ and $Y$ are quadratic polynomials (three conditions). In the case of polynomials of higher degree methods have been worked out to establish conditions for the presence of centres of a certain structure: isochrone, stable or symmetric (see [3], [4]).

References

 [1a] H. Poincaré, "Mémoire sur les courbes définiés par une équation differentielle" J. de Math. , 7 (1881) pp. 375–422 [1b] H. Poincaré, "Mémoire sur les courbes définiés par une équation differentielle" J. de Math. , 8 (1882) pp. 251–296 [1c] H. Poincaré, "Mémoire sur les courbes définiés par une équation differentielle" J. de Math. , 1 (1885) pp. 167–244 [1d] H. Poincaré, "Mémoire sur les courbes définiés par une équation differentielle" J. de Math. , 2 (1886) pp. 151–217 [2] A.M. Lyapunov, "Problème général de la stabilité du mouvement" , Princeton Univ. Press, reprint (1947) (Translated from Russian) [3] V.V. Amel'kin, "On the question of the isochronism of the centre of two-dimensional analytic differential systems" Differential Eq. , 13 (1977) pp. 667–674 , 13 : 6 (1977) pp. 971–980 [4] K.S. Sibirskii, "Algebraic invariants of differential equations and matrices" , Kishinev (1976) (In Russian)