Stability and Almost Periodicity in Dynamical Systems1

let x=(xi, • • ■ , xn) and /=(fi, • • • ,/„), and suppose that / is of class C1. A set, 12, of points x will be called unrestricted if, whenever a point x0 is in 12, the solution path x = x(t), where x0 = x(0), exists for — co 0 a 5 = 5€>0 such that |x(2)— y{t)\ 0 a 5 = 5(>0 such that \x(t)—y(t)\ <e whenever y(t) is in 12 and |x(/o) — y(to)\ <8 holds for some value t = t0, — co </0< oo. Of course, in both types of stability, the number 6 depends not only on e but also on the particular solution x(/) considered. The notion of A -stability is that associated with Minding, Dirichlet (in the case of equilibrium solutions), and Liapounoff [5, pp. 98-99; 3, pp. 210-211]. The .B-stability was considered by Hartman and Wintner [l]. It is easy to see that a solution which is 5-stable is surely A -stable. The converse is false however. In fact, one need only consider the single differential equation (w = l)