The Method of Averaging and Domains of Stability for Integral Manifolds

Liapunov's direct method is a standard and effective approach to computing the domain of stability (or region of attraction) of an autonomous ordinary differential equation. In this paper the author investigates domains of stability of integral manifolds of solutions generated by nonlinear mechanical and electrical oscillatory systems with many degrees of freedom. These manifolds are families of solutions that exhibit stronger stability properties than individual solutions. The problem of estimating the domain of stability of an asymptotically stable integral manifold is reduced to computing the domain of stability of an associated autonomous system of differential equations. This is done by applying the method of averaging to the system generating the integral manifold thus removing angular and time dependences. The stability region of this associated system is then computed and a result is established showing that this region is contained in the stability region of the original system. Several examples, including a coupled van der Pol system of oscillators, are considered.