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Constructive Generation and Verification of Lyapunov Functions around Fixed Points of Nonlinear Dynamical Systems

Abstract

An iterative method is developed that provides a transformation of coordinates of a dynamical system near a fixed point. In the new coordinates, Lyapunov functions and pseudo-Lyapunov functions can be determined. Using differential algebraic methods, the transformations can be chosen to belong to the same symmetry groups as the underlying motion. For area preserving stable motion, the method yields first integrals or near-first integrals of motion, and the convergence properties of the perturbative technique are tied to the question of integrability of the motion. For other stable motion, the method often yields Lyapunov functions that can be used to locally assert stability of the motion. Examples of the use of the method are given.

M. Berz, K. Makino, International Journal of Computer Research 12,2 (2003) 235-244

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