Stability conditions for second order ordinary differential equations with periodic coefficients

Any stable second order ordinary differential equation with periodic coefficients belongs to exactly one of a countable collection {??n}, n = 0, ??1, ??2,..., of open simply connected sets. In this paper we give conditions on the coefficients of such an equation which places it in a given ??n. That is, conditions which guarantee that all solutions of the differential equation are bounded. The earliest and best known result of this type is due to Liapunov. It states that all solutions of Hill´s equation ?? + p(t)y = 0 are bounded if p(t+T) = p(t) ?? 0, p(t) ?? 0, and if ??0 TP(t)dt < 4/T. Alternatively stated, Liapunov´s result shows that Hill´s equation lies in ??o when these conditions on p(t) are satisfied, Since Liapunov´s time, several authors have given sufficient conditions on p(t) for Hill´s equation to belong to any one of the sets ??n. Our results extend these results to a general class of second order ordinary differential equations.