Stability and Normal Forms

In this chapter our concern is with a system of ordinary differential equations $\dot {\rm x} = {\bf f}({\bf x})$ in ${\rm R}^n$ or ${\rm C}^n$ in a neighborhood of a point ${\bf x}_0$ at which ${\bf f}({\bf x}_0)= 0$. Early investigations into the nature of solutions of the system of differential equations in a neighborhood of such a point were made in the late 19th century by A. M. Lyapunov ([114, 115]) and H. Poincarée ([143]). Lyapunov developed two methods for investigating the stability of ${\bf x}_0$. The so-called First Method involves transformation of the system to normal form; the Second or Direct Method involves the use of what are now termed Lyapunov functions. In the first section of this chapter we present several of the principal theorems of Lyapunov’s Direct Method. Since smoothness of f is not necessary for these results, we do not assume it in this section. The second and third sections are devoted to the basics of the theory of normal forms.