Nonlinear Analysis. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Theor y , Mr r hods & Applicanons, Vol 5. No 2. pp. 151-165 0 Per gamon Press Ltd 1981 Prmted in Great Britain 0362.546X~ 81.0201.0157 SO2.0010 zyxwvutsrqp PERIODIC SOLUTIONS OF POLYNOMIAL FIRST ORDER DIFFERENTIAL EQUATIONS S. SHAHSHAHANI Department of Mathematics and Computer Science, Sharif University of Technology, P.O. Box 3406, Tehran, Iran Department of Mathematics, University of California, Berkeley, California 94720, U.S.A. (Received 15 November 1979; revised 20 February 1980) Key words andphrases: Polynomial differential equation, periodic solution, equation of variation, bifurcation, hyperbolic periodic solution, multiplicity of a periodic solution, averaging method. 0. INTRODUCTION IT HAS been suggested by C. Pugh that as an easier version of Hilbert’s 16th problem [2] on the determination of the number of limit cycles of a polynomial differential equation in the plane, one might investigate the number of periodic solutions of the first-order differential equation 1 = xn + U,_i(t)Xn-l + . . . + a,(t) (0) in which the coefficients a,(t) are real polynomials in t, or more generally real-analytic functions defined on [0, 11. By a periodic solution of (0), one means a solution x(t) defined on [0, l] for which x(0) = x(1). It is a simple consequence of the equation of variation for (0) that if n < 2, then n itself is an upper bound for the number of periodic solutions. For n = 3, Smale has ob- tained the same result in an unpublished work. His method consists of extending the time-one map of the differential equation to a map CP’ -+ CP’, studying the behavior of the extension at co, and applying the Lefschetz Index Formula. Note that the fued points of time-one map correspond to periodic solutions. For n = 4, independent examples by Nirenberg, Yorke, and Lins show that n is no longer an upper bound for the number of periodic solutions. If the degree in t of all q(t) is bounded by p, it remains an open problem to obtain an upper bound in terms of p and n, even for n = 4. The present paper contains some new results in this direction, and provides a self-contained introduction to the subject. In Section 1 we state some preliminary ideas. For convenience, a slightly broader class of equations than (0) is considered which, however enjoys the same features. Section 2 is devoted to the local study of periodic solutions. In particular, we investigate the number of periodic solutions that can bifurcate off a given one. In (2.1) and (2.5) we observe that the proliferation of periodic solutions for n z 4 is already present at the local level. In Section 3 we discuss some global results. Theorem (3.1) is analogous to results on the perturbations of Hamiltonian systems and is based on the averaging method. In (3.3) we give a new proof of Smale’s 157