proceedings of the american mathematical society Volume 81, umber 4, April 1981 A OTE O PERIODIC SOLUTIOS I THE VICIITY OF A CETER BAU-SE DU Abstract. We show that a small polynomial perturbation of a center will result in periodic solutions provided the coefficients of the polynomial satisfy inequality (2) below. The proof is based on an application of degree theory. 1. Introduction. In her book [1] (see also [2]), Cronin stated the following. Let hx(x,y, p) and h2(x, y, p) be polynomials in x, v and p and let /^(sin 9, cos 9, p), &2(sin 9, cos 9, p) be polynomials in sin 9, cos 9 and p. Assume that n( p) is a C ' function and i)0 = t/(0) ^ 0. Let w = [ 1 + pn( p)]_1 and t = wt. Then the following two-dimensional nonlinear differential system x' = y + (i[hx(x,y, p) + kx(sin t, cos t, p)], y' = -x + n[h2(x,y, p) + k2(sin r, cos t, p)], I' = J, has, for all sufficiently small p, at least one solution of period 2w(l + pij(p)). For the proof of this result, Dr. Cronin used the results of degree theory, in particular, the homotopy theorem and the odd mapping theorem as applied to a function ^C) which depends on the terms hx, h2, kx and k2. The quantity C represents the initial data for the solution which is periodic precisely when ^C) = 0. But it seems that, in some cases, one cannot apply the odd mapping theorem. For example, if we let rj( p) = 1, to = (1 + p)-', t = on and consider x' = y + p(x + 3v + cos t), y' = -x + p(x v sin t), (*) then 0(C) = (2rJ). (From now on, we shall use the same notation as Cronin [1, pp. 91-93].) So the degree of 0(C) in any domain is 0. In this case, we cannot conclude from degree theory that the differential system (*) has, for all sufficiently small p, at least one solution of period 2tt(1 + p). The purpose of this paper is to compute directly 0(C). We will show that if the coefficients of hx and h2 satisfy a simple inequality, then the differential system (1) has a periodic solution for p sufficiently small. The precise statement of our result is given in the next section, but we note that the inequality required is a generic condition on hx and h2, and consequently most systems satisfy the conclusion of Cronin's Theorem. The proof of our result appears in 3 and 4. Received by the editors March 6, 1980. 1980 Mathematics Subject Classification. Primary 34C25. 579 1981 American Mathematical Society 0002-9939/81/0000-0167/S02.50
580 BAU-SE DU 2. Statement of the theorem. For nonnegative integers n and k, we define i(-o*-^k'%+-;1 "«**<- 2n+\-k 2~Vl)*+7~"( *.)(2" + 1_ k) iîn + l<k<2n + l. Then, it is not hard to show that 4" r2* akn = ^- ( cos2n+i-k9-(-l)k sin* 0- e m Jo Let - 2 1 3 5 (2n + 1) n m j.. 2 A 6 (2n + 2) ' A" i* A" r iik = 0, 1 3 5 (2n + 1 - *) 1 3 5 (* - 1) 2m 2 4 6 (2n + 2) if k is even and 0 < k < 2n + 1, 1 3 5 (2/1- k) 1 3 5 k 2m 2 4 6 (2/i + 2) if k is odd and 0 < k < 2«+ 1, (_1)" + 1.4" i.3. 5... (2n n + 1)... - 2tt - ; :-h-rr, if Jfc = 2/i + 1 2 4 6 (2«+ 2) ' 2«+l páx'y) = 2 a*,n-x;2',+1 ~^*' *-0 Ôn(^7) = 2nf,6^2n+1^\ fc-0 and let An = S (-l)'(%/2t^ - *2*+l, «2it+l>»). fc-0 fin = 2 (-1)*(û2*+l,n«2*+l,n + ^/2t.J' fc-0 ow we can state the following theorem. Theorem. Assume that hx(x,y, p) and h2(x,y, p) are polynomials in x, y and p. When p = 0, let hx(x,y, 0) = P(x,y) + 2 Pn(x,y), n-0 h2(x,y, 0) = Q(x,y) + 2 Qn(x, y), n-0
PERIODIC SOLUTIOS I THE VICIITY OF A CETER 581 where P(x, y), Q(x, y) are polynomials in x, y in which each term is of even degree. Define An, Bn as above, and let kx($rn 9, cos 9, p) and A:2(sin 9, cos 9, p) be polynomials in sin 9, cos 9 and p. Assume that r (n) is a Cl function and t/0 = tj(0). Let 03 = [1 + ptj(p)]"1 and T = at. If Ko + Ko? + (2i)o- «w + M' + 2 K2 + B2) * 0, then the following two-dimensional x' = y + fi{hx(x,y, n-l nonlinear differential system p) + kx(sin r, cos t, p)}, y' = -x + n{h2(x, v, p) + /c2(sin t, cos t, p)}, has, for all sufficiently small p, ai /easi o/ie? periodic solution of period 2nr(\ + p7,(p)). Corollary. //, in the above theorem, we let hx(x,y, 0) = a0 + axx + a2y + H.O.T., h2(x,y, 0) = b0 + bxx + b2y + H.O.T., and assume (ax + b2)2 + (2tj0 a2 + bx)2 = 0, then the conclusion of the above theorem holds. Remark. We call attention to the fact that this theory applies in cases where 3. Computation of 0(C). Let and let z = c, i'c2. When p = 0, we let -CD (2) (x\(x(9,0,c)\ m ( cxcos9 + c2sin0 \ _ {?> \y(9, 0, C)) \-cx sin 9 + c2 cos 9) ~ Í1 (e'9z + e-iez) ^(eiez - e~*z) In particular, when z = 1, one has x = cos 9, y = -sin 9. In Cronin's notation, one has where (C) = f 2wV C2 + //,(C" Cl) + Kl \ \ -2mj]0cx + H2(cx, c2) + K2 ) / Hi(cu ci)\ m cm hx(x,y, 0)cos 9 - h2(x,y, 0)sin 9\ \ H2(cx, c2) ) h \ hx(x, y, 0)sin 9 + h2(x, y, 0)cos 9 j -Í Re{[A1(x,^,0) + ia2(jc,>-,0)]e*} lm{[n,(x,v,0) + in2(x,v,0)]e*}
582 BAU-SE DU and / Kx \ r2mi A:,(sin 9, cos 9, 0)cos 9 - k2(sin 9, cos 9, 0)sin 9 \ \K2J ~ J0 \ kx(sin 9, cos 9, 0)sin 0 + fc2(sin 9, cos 0, 0)cos 9 J is a constant vector. Since hx, h2 are polynomials in x and y, it suffices to compute l" h(x,y)e'e d.9, where h(x,y) is a polynomial in x andy. Given two nonnegative integers m and n, and let h(x,y) = x"y". If m + n is even, then it is trivial that J"2,* h(x, y)e'9 d9 = 0. So we can assume that m + n is odd. If h(x,y) = x2n + l~kyk, where 0 < k < n, then { h(x,y)emd9 = f x2"+1-*yv*<0 i i*7r m, _. _ = ^>r(zz \zz)"z )* «*, -akn. Similarly, if A(x,y) = x2" + 1"fcy*, where n + 1 < it < 2n + 1, then f2nh(x,y)eie d9 = f2v"+1-*y V* d9 ikm, _. _ 2n + *-k. Uk+J-J k \(2n+l-k\ im, _,. _ ow assume *,<*,.y, o) = 2 ^x2"*1-^* = jv(*a (V-=0 /t2(x, y, 0) = ^,Bx2" + 1-*y* = ß (x, v), («> 0). *=o
PERIODIC SOLUTIOS I THE VICIITY OF A CETER 583 Then -2ir f "[hx(x,y, 0) + i«2(x,y, 0)]e's d9 Jo = 2nî\ak<n + ibkjf2\2"^-ye">d9 /t-o *-0 2n+l i m = 2 («*. + *Jt») ' IS" " (ZZ)"Z "a*,n *-0 Jo = 4»(«) " / -\n- 2 2 (-l)*02*,«a2m - 2 (-0**2*+l,««2* + l.«/t = 0 t-0 + '4>t(zz) z 2 (-1)*a2*+l,n«2*+l, + 2 (-l)%k,na2k,n k-0 k-0 IT IT n = (zz)"zan + i (zz)"zb (where v4n, /? are as defined in 2) = ^(zz)"z(an + ibn) = y(c? + c2)n(c, + ic2)(^ +./? ) w =?(c? + c22)fl[(c^n - c2bn) + i(cxbn + C2A )]. Therefore, the real and imaginary parts of S\?[P (x,y) + iq (x,y)]e (m/a")(c2 + c2)"(cxan - c2bn) and (m/an)(c2 + c2)"(cxbn + C2A ) respectively. Finally, we let hx(x,y, 0) = P(x,y) + 2?_0 P (x,y), and h2(x,y, 0) = Q(x,y) + 2^_o Q (x,y) where P(x,y) and Q(x,y) are polynomials in x, y in which each term is of even degree, and Pn(x, y), Q (x, y) are defined as above. Then C"[hx(x,y, 0) + ih2(x,y, 0)]eie d9 = 2 C'[PÁx,y) + iq (x,y)]e* db. J0 n~0j0 (ote that (2"p(x,y)em d9 = 0= (2 Q(x,y)e* d9 >o Jo = 2 i(*? + c22)"[(c1v4n - c2/? ) + /(c,^ + cvl,)]. n=0 ^ So the real and imaginary parts of /o*[«,(x,y, 0) + i7j2(x, y, 0)]e'9 d9 are 2ir_o(^/4")(c2 + c2)"(cxan - c2bn) and 2^0(^/4")(c2 + c2)\cxbn + ClAn) respectively. Therefore, ">(c c2) = 2 3(c? + c2)"(c^ - c2/? ), n-0 ^ are #2(^1. c2) = 2 (c? + c2)"(cxb + C2An), n-0 H where An, Bn are as defined in 2. So we have got the explicit formulas Hx(cx, c and H2(cx, c^. for
584 BAU-SE DU 4. Proof of the theorem. ow we have obtained the formulas for Hx(cx, c^ and H2(cx, c2>- So (C) = I 2mi C2 + H^Cl' C2) + Kl\ \ -2Tr/0c, + H2(cx, c2) + K2 ) = m m 2irn0c2 + m(cxa0 - c2b0) + 2 7«"(ci + c2)"(cian ~ C2BJ + *i n-l H -2,rr,0c, + m(cxb0 + c^0) + 2 m «(*? + 4)n(cxBn + C2A ) + K2 n-\ * ci(ao,o + *i,o) + c2(2i7o + bo,o - ai,o) + 2 S (*ï + 4) 2\" n-l 4" (c^ - c2bn) ic2 + c2\" Ci(-2i)o - V + ai,o) + c2(a0,o + >i,o) + 2 n-l ' An 4" 2 (c\bn + <VU (*;) If for some n > 1, A2 + B2 = 0, then JV0(C) is dominated at least by (c2 + c2)". For all sufficiently large C, one has 0(C) ^ 0, and we can apply the odd mapping theorem. On the other hand, if An = Bn = 0 for all 1 < n <./V and if (ao,o + 6i,o)2 + (2^,0 + *o,o - ai,o)2 ^ 0. ti"511 tc\ m J ci(ao,o + 6i,o) + c2(2tjo + *o,o- «i.o) \ + / *1 \ = 0 I C!(-2t,0 - b00 + axo) + c2(a00 + bxo) ) \K2) for exactly one C. So, again, for all sufficiently large C, the function Q(C) can never be zero and the odd mapping theorem applies too. But if a00 + bxo = 0 = 2î/o + *o,o ai,o an(* -An = Bn = 0 for 1 < n <, then the function 0(C) is constantly equal to (* ). So its degree is either undefined or equal to zero. Hence no conclusion can be drawn in this case. This completes the proof. Acknowledgments. I want to thank Professor G. R. Sell for his many valuable suggestions. References 1. J. Cronin, Fixed points and topological degree in nonlinear analysis, Math. Surveys, no. 11, Amer. Math. Soc., Providence, R. I., 1964, pp. 91-93. 2.. G. Lloyd, Degree theory, Cambridge Univ. Press, Cambridge and ew York, 1978, pp. 132-133. School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455