Families of Lyapunov functions for nonlinear systems in critical cases

Lyapunov functions are constructed for nonlinear systems of ordinary differential equations whose linearized system at an equalized point possesses either a simple zero eigenvalue or a complex conjugate pair of simple, pure imaginary eigenvalues. The construction is explicit, and yields parameterized families of Lyapunov functions for such systems. In the case of a zero eigenvalue, the Lyapunov functions contain quadratic and cubic terms in the state. Quartic terms appear as well for the case of a pair of pure imaginary eigenvalues. Predictions of local asymptotic stability using these Lyapunov functions are shown to coincide with those of pertinent bifurcation-theoretic calculations. The development of the paper is carried out using elementary properties of multilinear functions. The Lyapunov function families thus obtained are amenable to symbolic computer coding