Modified relative invariants and Lyapunov functions (Q1061917)

Consider the two dimensional system \[ dx/dt=\lambda x+\sum_{i+j\geq 2}a_{ij}x^ iy^ j,\quad x(0)=C_ 1,\quad dy/dt=\mu y+\sum_{i+j\geq 2}b_{ij}x^ iy^ j,\quad y(0)=C_ 2, \] in which \(\| (C_ 1,C_ 2)\|\) is sufficiently small, \(\lambda <0\), \(\mu =m\lambda\), for some positive integer m greater than l, and \(a_{ij}\), \(b_{ij}\) are constants. For such a system, the author shows that there exist modified relative invariants u, v of the form \(u(x,y)=x+\sum_{i+j\geq 2}C_{ij}x^ iy^ j,\) \(v(x,y)=y+\sum_{i+j\geq 2}d_{ij}x^ iy^ j,\) converging in a neighborhood of the origin and satisfying the differential equations \(du/dt=\lambda u\), \(dv/dt=\mu v+B_ mu^ m.\) The author states that the result is valid for higher dimension, and can be extended to the case \(i\lambda =\mu j\). The paper concludes by showing that the modified relative invariants can be used to construct Lyapunov functions which are different from those obtained by Zubov's method.