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Families of Liapunov Functions for Nonlinear Systems in Critical Cases
Abed, Eyad H.
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Liapunov functions are constructed for nonlinear systems of ordinary differential equations whose linearized system at an equilibrium point possesses either a simple zero eigenvalue or a complex conjugate pair of simple, pure emaginary eigenvalues. The construction is explicit, and yields parametrized families of Liapunov functions for such systems. In the case of a zero eigenvalue, the Liapunov functions contain quadratic and cubic terms in the state. Quadratic terms appear as well for the case of a pair of pure imaginary eigenvalues. Predictions of local asymptotic stability using these Liapunov 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 Liapunov function families thus obtained are amendable to symbolic computer coding.