LYAPUNOV FIRST METHOD FOR STRONGLY NONLINEAR-SYSTEMS

Lyapunov's classical First Method is developed fbr strongly non-linear systems. Techniques for ''truncating'' strongly non-linear systems th at possess a well-defined group of symmetries are described. Given suc h a group, it is possible, under fairly general assumptions, to determ ine, by purely algebraic methods, particular solutions of the truncate d systems with prescribed asymptotic expansions. It is shown that thes e solutions can be extended to solutions of the full system by using c ertain series. Sufficient conditions for the existence of parametric f amilies of solutions of the full system that possess certain asymptoti c properties are also derived. The theory is illustrated by a wide ran ge of examples. A new proof is given of one of the inversions of the L agrange-Dirichlet theorem on the stability of equilibrium. It is shown that the method developed here may also be used to construct collisio n trajectories in problems of celestial mechanics in real time. (C) 19 96 Elsevier Science Ltd.