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.