The object of Theorem 1 below is to establish the existence of periodic solutions of an autonomous differential equation y = ƒ (y) by an extension of the Poincaré method of sections (see [2; 4]). The following situation is envisaged: the equation is defined on a subset D of euclidean space and has unique solutions y(x, t) jointly continuous in / and the initial point x\ D contains a compact subset K with the property that the positive trajectories starting from points of K remain in K. The

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... nment to x in K and / in [0, ) of the point Tt(x) =y(x, t) in K defines a continuous one-parameter semigroup T t acting on K % i.e., T% is jointly continuous in x and t, TQ is the identity on K and r« + «= T 8 o Tu THEOREM 1. Let Kbea connected finite complex, let T t be a continuous one-parameter semigroup acting on K and let co be a closed 1-form on K (defined over a portion of euclidean space containing K) with integervalued periods. Make the following two assumptions on K, T t and o): A. For each x in K there is a t for which the integral of o) over the trajectory from x to Tt(x) is positive. B. The classes of closed paths in K over which the integral of oe vanishes form a subgroup of the fundamental group of K. Assume that the corresponding covering space K* has nonvanishing Euler characteristic. Conclusion: T t has a periodic trajectory, i.e., there is an x in K and a period p>0 such that T t+P (x) = T t (x) for all t*zO. REMARK a. If we denote the integral of oe over the trajectory from x to T t (x) by A(x, t) , assumption A implies that there exists a positive constant a such that at<A(x, t) for sufficiently large t. Thus A(x, t) converges uniformly to + oo. If T t is engendered by the differential equation y^fiy), A(x, t) can be written as the integral with respect to t of the scalar product co-/, evaluated along the trajectory from x to T t (x). REMARK b. Although the covering space K' is not a finite complex, assumption A implies that K' has finite Betti numbers, so that its Euler characteristic is defined. REMARK C. The period of the periodic trajectory disclosed by the theorem is bounded by a number depending on the uniform rate of 409