LIMIT CYCLES OF DIFFERENTIAL SYSTEMS VIA THE AVERAGING METHODS

Transcription

1 CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 17, Number 1, Spring 29 LIMIT CYCLES OF DIFFERENTIAL SYSTEMS VIA THE AVERAGING METHODS DINGHENG PI AND XIANG ZHANG ABSTRACT. In this article we first recall the averaging methods which are useful in studying limit cycle bifurcations from simple differential systems under small perturbations. We then review some recent results on limit cycle bifurcations obtained by using the averaging methods. Finally we present a new result by applying the averaging method to study the Hopf bifurcation of some smooth differential systems in R n, we show that the function in the number of limit cycles bifurcated from one singularity of the systems can be an exponential function in the dimension of the system with exponent n 2, and that the function can also be a power function in the degree of the system with its nonlinear part a polynomial. As we know this last phenomena is first found using the averaging method, which is an extension of Theorem 1 of [29] 1 Introduction and the averaging methods Studying the existence of periodic orbits of differential systems via the averaging methods has a long history see for instance Marsden and McCracken [33], Chow and Hale [6], Sanders and Verhulst [35], Luo et al. [31], Verhulst [37], Buică and Llibre [2], Buică, Françoise and Llibre [1] and the references therein). The problem on the number, distribution and stability of limit cycles for a differential system is difficult see for instance Ye et al. [38], Zhang et al. [39], Li [2], Christopher and Li [7] and the references therein). We will show that the averaging methods are useful tools for investigating the number of limit cycles for some differential systems. Add also this method can be used to obtain the shape, stability and the approximate expression of limit cycles. In the rest of this section we will recall the basic theory on the averaging methods. Of course, the averaging methods have different presentations depending on the problems which will be studied. Here we AMS subject classification: 34A34, 34C2, 34C4. Keywords: Differential systems, limit cycles, averaging method. Copyright c Applied Mathematics Institute, University of Alberta. 243

2 244 DINGHENG PI AND XIANG ZHANG restrict to those for which the number, shape and stability of limit cycles can be obtained. The first one is to present the first-order averaging method as it was obtained by Buică and Llibre in [2], which does not need differentiability of the vector field. The conditions imposing the existence of a simple isolated zero of the averaged function are given in terms of the Brouwer degree. Theorem 1. First order averaging method) Consider the differential system 1) ẋt) = εft, x) + ε 2 gt, x, ε), where f : R D R n, g : R D ε f, ε f ) R n are continuous functions, T -periodic in the first variable, and D is an open subset of R n. Define f : D R n by 2) f z) = 1 T T fs, z)ds, and assume that i) f and g are locally Lipschitz with respect to x; ii) there exists b D such that f b) = and a neighborhood V D of b in which f z) for all z V \ {b} and d B f, V, b) where d B f, V, b) denotes the Brouwer degree of f in the neighborhood V of b). Then, for ε > sufficiently small, there exists an isolated T -periodic solution φ, ε) of system 1) such that φb, ) = b. Buică et al. [3] provided the asymptotic stability of the limit cycles obtained from averaging theory with f having C 1 differentiability. Theorem 2. Assume that the functions f and g of 1) are C 1 and Lipschitz respectively in a neighborhood of the limit cycle φ, ε) given in Theorem 1 by the simple zero b of f, then for ε > sufficiently small if the eigenvalues of the Jacobian matrix of f at b all have negative respectively positive) real part, then the limit cycle φ, ε) is asymptotically stable respectively unstable). Furthermore, if the functions f and g of 1) have better regularity on their smoothness then we can get more information on the limit cycle φ, ε) which are obtained in Theorem 1.

3 LIMIT CYCLES OF DIFFERENTIAL SYSTEMS 245 Theorem 3. Assume that the functions f and g of 1) are C 2 and C 1, respectively. If there exists an b D such that f b) = and detd z f b)), where D z f denotes the Jacobian matrix of f z) with respect to z, then for ε > sufficiently small the following hold. i) there exists a T -periodic limit cycle φ, ε) of 1) such that φ, ε) b as ε ; ii) the stability of the limit cycle φ, ε) given in i) is the same as that of the singularity b of the averaged system ż = εf z) In fact the singularity b has the stability behavior of the Poincaré map associated to the limit cycle φ, ε). A proof of this result can be found in [37] or in [13]. This shows that using the averaging methods we obtain not only the existence of limit cycles but also their stability and approximate expressions. We now recall the second and third order averaging methods see, e.g., [2]). Theorem 4 Second order averaging method). Consider the differential system 3) ẋt) = εf 1 t, x) + ε 2 f 2 t, x) + ε 3 gt, x, ε), where f 1, f 2 : R D R n, g : R D ε f, ε f ) R n are continuous functions, T -periodic in the first variable, and D is an open subset of R n. Assume that i) f 1 t, ) C 1 D) for all t R, f 2, g and D x f 1 are locally Lipschitz with respect to x, and g is differentiable with respect to ε. Define f 1, f 2 : D R n by 4) 5) f 1 z) = 1 T f 2 z) = 1 T T T f 1 s, z)ds, D x f 1 s, z) s ) f 1 t, z)dt + f 2 s, z) ds; ii) for V D an open and bounded set and for ε ε f, ε f )\{}, there exists a ε V such that f1 a ε)+εf2 a ε) = and d B f1 +εf 2, V, ). Then, for ε > sufficiently small, there exists a T -periodic solution φ, ε) of system 3).

4 246 DINGHENG PI AND XIANG ZHANG Theorem 5 Third order averaging method). Consider the differential system 6) ẋt) = εf 1 t, x) + ε 2 f 2 t, x) + ε 3 f 3 t, x) + ε 4 gt, x, ε), where f 1, f 2, f 3 : R D R n, g : R D ε f, ε f ) R n are continuous functions, T -periodic in the first variable, and D is an open subset of R n. Assume that i) f 1 t, ) C 2 D), f 2 t, ) C 1 D) for all t R, f 3, g, D xx f 1, D x f 2 are locally Lipschitz with respect to x, and g is twice differentiable with respect to ε, where D xx f 1 is the Hessian matrix of f 1 with respect to x. Define f1, f 2, f 3 : D Rn with f1 and f 2 as in 4) and 5), respectively, and f3 as f 3 z) = 1 T T 1 2 D zzf 1 s, z)y 1 s, z)) 2 + D z f 1 s, z)y 2 s, z) + D z f 2 s, z)y 1 s, z) + f 3 s, z)) ds, where y 1 s, z) = y 2 s, z) = s s f 1 t, z) dt, D z f 1 t, z) t ) f 1 r, z)dr + f 2 t, z) dt; ii) for V D an open and bounded set and for ε ε f, ε f ) \ {}, there exists a ε V such that f1 a ε ) + εf2 a ε ) + ε 2 f3 a ε ) = and d B f1 + εf 2 + ε2 f3, V, ). Then, for ε > sufficiently small, there exists a T -periodic solution φ, ε) of system 6). For the averaging methods of order higher than 3 we can get similar results than the previous ones, but the expression is more complicated, and will not be presented here. The above averaging methods ask the terms in the right hand side of equations are all o1) in terms of the small parameter ε. If it is not the case we will have the following developed averaging methods to deal with them. We consider the following differential system 7) ẋt) = f t, x) + εf 1 t, x) + ε 2 gt, x, ε),

5 LIMIT CYCLES OF DIFFERENTIAL SYSTEMS 247 where ε is a small parameter, f, f 1 : R D R n and g : R D ε f, ε f ) R n are C 2 functions, T -periodic in the first variable, and D is an open subset of R n. Assume that the unperturbed system of 7) 8) ẋt) = f t, x), has a solution, denoted by x t, z), satisfying x, z) = z. Write the linearization of 8) along the solution x t, z) in 9) ẏ = P t, z)y with P t, z) = D x f t, x t, z)). Let Y, z) be some fundamental matrix solution of 9). Theorem 6. Let β : V R n k be a C 2 function, where V R k is open and bounded. Assume that i) Z = {z α = α, β α)), α V } D and that for each z α Z, the unique solution x α of 8) with x) = z α is T -periodic; ii) for each z α Z, there exists a fundamental matrix solution of 9), denoted by Y α t) = Y t, z α ), such that the matrix Yα 1 1 ) Yα T ) has in the upper right corner the null k n k) matrix, while in the lower right corner has the n k) n k) matrix α with det α ). We consider the function F 1 : V R k given by 1) F 1 α) = P T Yα 1 t)f 1t, x α t)) dt, where P is the projection from R k R n k to its first k coordinates R k. If there exists a b V satisfying F 1 b) = and detd α F 1 b)), then there exists a T -periodic solution φ, ε) of system 7) such that φ, ε) z α as ε. This result was presented by Malkin 1956) and Roseau 1966) see, e.g., Françoise [12]). For a new and short proof, see [1]. Here we use the presentation of [1]. This last theorem has some very nice corollaries see also [1]). The first one is on perturbation of isochronous systems.

6 248 DINGHENG PI AND XIANG ZHANG Corollary 7 Perturbations of isochronous systems). Assume that there exists an open subset V with V D such that for each z V, the solutions x, z) of 8) is T -periodic. Let F 1 : V R n be given by F 1 z) = T Y 1 t, z)f 1 t, xt, z)) dt. If there exists b V such that F 1 b) = and detd z F 1 b)), then there exists a T -periodic solution φ, ε) of system 7) satisfying φ, ε) b as ε. The second one is on the perturbation of linear systems. Corollary 8 Perturbation of linear systems). Consider system 7) with f t, x) = P t)x + qt). Assume that system ẏ = P t)y, has k T -periodic linearly independent solutions. Denote by Y t) some fundamental matrix solution of this last linear system, and assume that i) P T Y 1 s)qs) ds =, where P is the projection from R k R n k to R k ; ii) det ), where is the n k) n k) matrix from the lower right corner of the matrix Y 1 ) Y 1 T ). Then the following statements hold. a) there exists β : R k R n k such that, for all α R k, z α = α, β α)) satisfies Y 1 T ) Y 1 ) ) T z = Y 1 s)qs) ds and the unique solution x α t) of ẋ = P t)x + qt) with x) = z α is T -periodic. b) Let F 1 : R k R k be the function defined by F 1 α) = P T Y 1 t)f 1 t, x α t)) dt. If there exists b V such that F 1 b) = and detd α F 1 b)), then there exists a T -periodic solution φ, ε) of system 7) such that φ, ε) z b as ε.

7 LIMIT CYCLES OF DIFFERENTIAL SYSTEMS 249 Based on perturbation of linear periodic differential systems via averaging method, there had been also Han s work [14], who established conditions for the existence of a unique periodic solution and gave estimates for its supremum and for the values of the small parameter for which this solution exists. We will not present it here. Finally we consider a special case of systems 7) with f t, x) = f 1t, u), f 2t, u, v)), f 1t, x) = f1 1t, u, v), f 1 2 t, u, v)) and gt, x, ε) = g 1 t, u, v, ε), g 2 t, u, v, ε)), where f 1, f1 1, g 1 R k and f 2, f1 2, g 2 R n k. Corollary 9 Perturbation of a special case). Assume that there exists an open subset V with V PD such that for each α V, the unique solution u α t) of the system u = f 1 t, u) satisfying u) = α is T -periodic, and the system v = f 2 t, u α t), v) has a unique T -periodic solution, denoted v α t). Let U α t) and V α t) be respectively fundamental matrix solutions of the systems u = D u f 1t, u αt))u and v = D v f 2t, u αt), v α t)). Define F 1 α) = T Uα 1 t)f 1 t, u α t), v α t)) dt. Assume that α = Vα 1 ) Vα 1 T ) has det α ), and that there exists b V such that F 1 b) = and detd α F 1 b)). Then there exists a T -periodic solution φ, ε) of system 7) such that φ, ε) b, v α )) as ε. There are also some other forms of averaging method for studying the existence of invariant tori, subharmonic bifurcations and so on, which are beyond our main concern of this paper. We refer the interesting readers to [16, 4] and the reference therein. At the end of this section we present the Bezout s theorem, which gives the maximum number of zeros of a system of polynomial functions see, e.g., [36]). We will use it in Section 3. Theorem 1 Bezout s theorem). Let P i be polynomials in the variables x 1,..., x n ) R n of degree d i for i = 1,..., n. Consider the following polynomial system P i x 1,..., x n ) =, i = 1,..., n. If the number of solutions of this system is finite, then it is bounded by d 1 d n.

8 25 DINGHENG PI AND XIANG ZHANG 2 Review: recent results on limit cycle bifurcations via averaging method In this section we review some recent results on limit cycle bifurcations using the averaging methods. 2.1 Application of the averaging methods We first recall some results on the bifurcation of limit cycles of differential systems via averaging methods. Consider a smooth differential system of the form ẋ = Ax + F x, ε), where F x, ) = O x 2 ), x R n, n 3 and b A = diag b ) ),, b. For ε small the system may have a limit cycle near the origin. Furthermore, if F x, ) =, then the system when ε may have a limit cycle near a closed curve of the form L h : x x2 2 = h > ), x j = for j = 3,..., n. For the limit cycle bifurcation near either the origin or L h a general theory together with a vector bifurcation function via Liapunov-Schmidt method was established in Theorems 1 4 and Remark 2 by Han in [15], see also Theorems and Remark 4 of [18] by Han and Zhu. In the case of n = 3 by introducing a suitable rescaling of variables transform the original system to the normal form system and then using the averaging method the condition for the existence of one or two limit cycles was given in [18] and [17]. For n 3 following the same idea one can also introduce the averaging method to study the bifurcation of limit cycles. Here we only generally mentioned the results on the existence of limit cycles via averaging methods, but did not present them carefully for details we refer the readers to the original papers). Because in this paper we are mainly interested in the number of limit cycles bifurcated using the averaging methods. We now study the bifurcation of limit cycles from a class of C m+1 smooth differential systems in R n with n 3 and m 2. Assume that these systems have a singularity at the origin with the linear part having eigenvalues εa ± bi and εc k for k = 3,..., n, where ε > is a

9 LIMIT CYCLES OF DIFFERENTIAL SYSTEMS 251 small parameter. Such systems can be written in ẋ = εax by + a i1,...,i n x i1 y i2 z i zin n + A, 11) ẏ = bx + εay + z k = εc k z k + i i n=m i i n=m i i n=m b i1,...,i n x i1 y i2 z i zin n + B, c k) i 1,...,i n x i1 y i2 z i zin n + L k, k = 3,..., n, where a i1,...,i n, b i1,...,i n, c k) i 1,...,i n, a, b and c k are real parameters and b. The functions A, B and L k are the Lagrange expressions of the error function of m + 1)th order in the expansion of the functions of the system in Taylor series. The first result was given in Theorem 1 of [29] by Llibre and Zhang for system 11) with m = 2, where the limit cycles came from the Hopf bifurcation via the first order averaging method. Theorem 11. For m = 2 there exist C 3 systems 11) for which l {, 1,..., 2 n 3 } limit cycles bifurcate from the origin at ε =, i.e. for ε sufficiently small the system has exactly l limit cycles in a neighborhood of the origin and these limit cycles tend to the origin when ε. In the next section we will generalize this result to all natural number m 3. The following result is an immediate consequence of the proof of Theorem 11. Corollary 12. There exist quadratic polynomial differential systems 11) i.e. with m = 2 and A = B = L k = ) for which l {, 1,..., 2 n 3 } limit cycles bifurcate from the origin at ε =, i.e. for ε sufficiently small the system has exactly l limit cycles in a neighborhood of the origin and these limit cycles tend to the origin when ε. For lower dimensional systems 11) we can get more detail information on the limit cycles bifurcated from the Hopf zero. For three dimensional system of form 11) restricted to quadratic polynomial differential systems, Buzzi, Llibre and de Silva [4] obtained some sufficient conditions for the existence or not of one limit cycle and its kind of stability. We now recall only the results in the four dimensional case from Theorem 3 of Llibre and Zhang [29] because if we set the fourth equation vanishes we have a three dimensional differential system, and then the results of

10 252 DINGHENG PI AND XIANG ZHANG [4] follows from those of [29]. Write system 11) as a four dimensional one in the form ẋ = εax by + a ijkl x i y j z k w l + A, 12) ẏ = bx + εay + ż = εcz + ẇ = εdw + i+j+k+l=2 i+j+k+l=2 i+j+k+l=2 i+j+k+l=2 b ijkl x i y j z k w l + B, c ijkl x i y j z k w l + C, d ijkl x i y j z k w l + D, where a ijkl, b ijkl, c ijkl, d ijkl, a, b, c and d are real parameters, ab, and A, B, C and D are the higher order terms. We assume without loss that b >. Set A = KG 1 LG 2, 13) B = 2aN 2 F 1 2N 3 F 2 ) + cf1 2 G 2 + df 1 F 2 G 1, C = 2a2aN 3 df 1 G 1 ), D = 2a M 1 F M 2 F 1 F 2 + M 3 F2 2 ) + ckf1 + dlf 2, E = a2am 2 F 1 + 2aM 3 F 2 + dl), Λ = df 2 G 1 2aN 2, Ϝ = cf 2 G 2 + 2aN 1, = Λ 2 8aN 3 Ϝ, Γ = 8a 2 c 2 N acF 2 N3 2 2ac 11 N 3 Λ c 2 Λ 2 + )/2, Φ = 2c 11 CF 1 2aB + CF 2 ) 2cBCF 2 1 2c 2 C 2 F 2 1 2c 22aB + CF 2 ) 2, Ψ = 4a cf 2 2ac 2 )Λ 2 + 2ac 11 F 2 ΛϜ 2ac 2 F 2 2 Ϝ 2), where F 1 = a 11 + b 11, F 2 = a 11 + b 11, G 1 = c 2 + c 2, G 2 = d 2 + d 2, K = d 2 F1 2 d 11F 1 F 2 + d 2 F2 2, L = c 2 F1 2 c 11 F 1 F 2 + c 2 F2 2, M 1 = c 2 d 11 c 11 d 2, M 2 = c 2 d 2 c 2 d 2,

11 LIMIT CYCLES OF DIFFERENTIAL SYSTEMS 253 M 3 = c 11 d 2 c 2 d 11, N 1 = d 2 G 1 c 2 G 2, N 2 = d 11 G 1 c 11 G 2, N 3 = d 2 G 1 c 2 G 2. Assume that F F 2 2, G2 1 + G2 2. We have the following results which come from the Hopf bifurcation of system 11) at the origin. Theorem 13. For a C 3 system 12) with G 1 the following statements hold. a) a.1) For F 1 and A, if B 2 4AC > and 4AE DB + D B 2 4AC)F 1 > resp. 4AE DB D B 2 4AC)F 1 > ), system 12) has a limit cycle Γ 1ε resp. Γ 2ε ) tending to a singular point as ε. Moreover for suitable choice of the parameters system 12) can have the two limit cycles Γ 1ε and Γ 2ε. In these last case both limit cycles tend to different singular points when ε. a.2) For F 1 and A =, if B and ΦG 1 >, system 12) has a limit cycle Γ 1ε tending to the origin as ε. a.3) For F 1 =, F 2 and N 2, if > and Γ 2ac 11 N 2 + c 2 Λ) )G 1 > resp. Γ + 2ac 11 N 2 + c 2 Λ) )G 1 > ), system 12) has a limit cycle Γ 3ε resp. Γ 4ε ) tending to the origin as ε. Moreover for suitable choice of the parameters system 12) can have the two limit cycles Γ 3ε and Γ 4ε. In these last case both limit cycles tend to different singular points when ε. a.4) For F 1 =, F 2 and N 2 =, if Λ and ΨG 1 >, system 12) has a limit cycle Γ 3ε tending to the origin as ε. b) For ε > sufficiently small the limit cycle Γ 1ε, Γ 2ε, Γ 1ε, Γ 3ε, Γ 4ε or Γ 3ε of statement a) is given respectively by the graph rθ) = ε 4AE DB + D ) B 2 4AC /F 1 A 2 ) + Oε 2 ), zθ) = ε B ) B 2 4AC /2A) + Oε 2 ), wθ) = ε 4aA + F 2 B ) B 2 4AC) /2AF 1 ) + Oε 2 );

12 254 DINGHENG PI AND XIANG ZHANG rθ) = ε 4AE DB D ) B 2 4AC /F 1 A 2 ) + Oε 2 ), zθ) = ε B + ) B 2 4AC /2A) + Oε 2 ), wθ) = ε 4aA + F 2 B + ) B 2 4AC) /2AF 1 ) + Oε 2 ); rθ) = ε Φ/G 1 B 2 F1 2) + Oε2 ), zθ) = εc/b + Oε 2 ), wθ) = ε 2aB + CF 2 )) /BF 1 ) + Oε 2 ); rθ) = ε Γ 2ac 11 N 2 + c 2 Λ) ) /G 1 F2 2N 2 2) + Oε2 ), zθ) = ε2a/f 2 + Oε 2 ), wθ) = ε Λ + ) /2N 2 F 2 ) + Oε 2 ); rθ) = ε Γ + 2ac 11 N 2 + c 2 Λ) ) /G 1 F2 2N 2 2) + Oε2 ), zθ) = ε2a/f 2 + Oε 2 ), wθ) = ε Λ ) /2N 2 F 2 ) + Oε 2 ); or rθ) = ε Ψ/G 1 F2 2Λ2 ) + Oε 2 ), zθ) = ε2a/f 2 + Oε 2 ), wθ) = ε2aϝ/λ + Oε 2 ), where θ S 1 and the coordinates r, z, w) and θ come from x, y, z, w) via the cylindrical coordinate transformation. c) For the results of this section we need that system 12) be at least C 4. c.1) For F 1 > and A, the limit cycle Γ 1ε resp. Γ 2ε ) has at least one-dimensional stable resp. unstable) manifold,

13 LIMIT CYCLES OF DIFFERENTIAL SYSTEMS 255 and consequently the limit cycle is not a global repeller resp. attractor). For F 1 < and A the one-dimensional invariant manifold of the limit cycle has converse stability than for F 1 >. c.2) For F 1 >, A = and B, the limit cycle Γ 1ε has at least one-dimensional stable resp. unstable) manifold provided that B > resp. B < ). For F 1 < the one-dimensional invariant manifold of the limit cycle has a different stability than for F 1 >. c.3) For F 1 =, F 2 > and N 2 the limit cycle Γ 3ε resp. Γ 4ε ) has at least one-dimensional stable resp. unstable) manifold. For F 1 =, F 2 < and N 2, the one-dimensional invariant manifold of the limit cycles has a different stability than for F 2 >. c.4) For F 1 =, F 2, N 2 = and Λ, the limit cycle Γ 3ε has at least one-dimensional stable resp. unstable) manifold provided that > resp. < ). We have seen from Theorem 13 that the averaging methods provide not only the existence of limit cycles but also their stability and asymptotic expressions. Theorem 13 has some applications in forth order differential equations [29] d 4 x dt 4 + x pd3 dt 3 + q d2 x dt 2 + k dx dt + lx = f x, dx ) dt, d2 x dt 2, d3 x dt 3, and also in the simplified system of immune response without influence of damaged organ and time delay given in [32] by Marchuk 14) dx dy = β γz)x, dt dz dt = ρy µ 2 + ηγx)z, dt = αxz µ 1Y δ), dw = σx µ 3 W. dt Theorem 14 [29]). There is an open set in the parameter spaces for which system 14) has at least one limit cycle. Until now we consider only the application of averaging methods to higher dimensional systems. We now turn to two dimensional systems, especially the Liénard differential systems, which were presented by Liénard in 1928, and are of the forms ẍ + fx)ẋ + gx) =.

14 256 DINGHENG PI AND XIANG ZHANG This equation looks simpler in form, but in fact is very complicated in dynamics. We consider the cases that f and g are polynomials of degrees n and m, respectively. Denote by Hm; n) the Hilbert number, i.e., the maximum number of limit cycles that the Liénard differential equation with m and n prescribed can have. On the Hilbert number of Liénard systems, it is well known that H1; 1) = and H1; 2) = 1 by Lins, de Melo and Pugh [21] in 1977; H2; 1) = 1 by Coppel [8] in 1988; H3; 1) = 1 by Dumortier and Rousseau [11] in 199 and Dumortier and Li [9] in 1996; and H2; 2) = 1 first by Li [19] in 1986 and then reproved by Dumortier and Li [1] in 1997 and by Luo et al. [31, Theorem 3.3.7] in For other cases the Hilbert number is unknown. Llibre et al. [24] applied the first three order averaging methods to the Liénard systems, and obtained some nice results. Theorem 15 [24]). For the polynomial Liénard differential systems 15) ẋ = y, ẏ = x εfx)y + gx)), if fx) and gx) are polynomials of degree n and m respectively, then for ε > sufficiently small, the maximum number of limit cycles of the polynomial Liénard differential systems 15) bifurcating from the periodic orbits of the linear center ẋ = y, ẏ = x, using the first order averaging theory, is [n/2],, where [ ] denotes the integer part function. Theorem 16 [24]). For the polynomial Liénard differential systems 16) ẋ = y, ẏ = x εf 1 x)y + g 1 x)) ε 2 f 2 x)y + g 2 x)), with f i and g i polynomials of degree n and m respectively, if ε > is sufficiently small, then the maximum number of limit cycles of the polynomial Liénard differential systems 16) bifurcating from the periodic orbits of the linear center ẋ = y, ẏ = x, using the second order averaging theory, is max{[n 1)/2] + [m/2], [n/2]}. Theorem 17 [24]). For the polynomial Liénard differential systems ẋ = y, 17) ẏ = x εf 1 x)y + g 1 x)) ε 2 f 2 x)y + g 2 x)) ε 3 f 3 x)y + g 3 x)), with f i and g i polynomials of degree n and m respectively, if ε > is sufficiently small, then the maximum number of limit cycles of the

15 LIMIT CYCLES OF DIFFERENTIAL SYSTEMS 257 polynomial Liénard differential systems 17) bifurcating from the periodic orbits of the linear center ẋ = y, ẏ = x, using the third order averaging theory, is [n + m 1)/2]. The works on the number of limit cycles for Liénard differential systems using the averaging methods of order higher than 3 are in the progress. But the calculations are extremely tedious and complicated. 2.2 Application of the developed averaging methods We now consider the application of the developed averaging methods, i.e., Theorem 6 and its corollaries, which deal with the cases in which some terms on the right hand side of differential systems have no small parameter as their coefficients. As the first application we consider the polynomial differential system 18) ẋ = y + εp x, y, z), ẏ = x + εqx, y, z) + ε cos t, ż = az + εrx, y, z), with P, Q, R polynomials in the variables x, y, z of degree n. When a and ε are zero the unperturbed system has all R 3 except the z-axis filled of periodic orbits. This case was studied by Cima, Llibre and Teixeira [5] in 28 without the perturbation due to ε cos t. Theorem 18 [5]). Assume that a =. For ε sufficiently small, system 18) without ε cos t has at most nn 1)/2 limit cycles bifurcating from the periodic orbits of the linear system ẋ = y, ẏ = x, ż =. Moreover, there are such perturbed systems having nn 1)/2 limit cycles. When a and ε = the unperturbed system 18) only has the plane z = except the origin filled of periodic orbits, that is, system 18) with ε = restricted to the plane z = has a global center at the origin. This is a nondegenerate center. Theorem 19 [27]). Assume that a. For convenient polynomials P, Q and R, system 18) with ε sufficiently small has at least m {1, 2,..., 2[n 1)/2] + 1} limit cycles bifurcating from the periodic orbits of the linear center contained in z = when ε =. Now we consider the case that the unperturbed system has a degenerate global center.

16 258 DINGHENG PI AND XIANG ZHANG Theorem 2 [27]). For the polynomial differential system in R 3 ẋ = y3x 2 + y 2 ) + εp x, y, z), 19) ẏ = xx 2 y 2 ) + εqx, y, z), ż = zx 2 + y 2 ) + εrx, y, z), with max{deg P, deg Q, deg R} = n, which when ε = restricted to the plane z = has a degenerate global center at the origin. For convenient polynomials P, Q and R, system 19) with ε sufficiently small has at least m {1, 2,..., [n 1)/2]} limit cycles bifurcating from the periodic orbits of the center contained in z = when ε =. The usefulness of Theorem 6 is also shown by the following result, in which Llibre and Zhang [3] provided the first analytic proof on the existence of a Hopf-zero bifurcation for the Michelson system [34] at c =. ẋ = y, ẏ = z, ż = c 2 y x2 2, Theorem 21 [3]). For c sufficiently small the Michelson system has a Hopf-zero bifurcation at the origin for c =. Moreover the bifurcated periodic orbit satisfies xt) = 2c cos t + oc), yt) = 2c sin t + oc) and zt) = 2c cot t + oc) for c > sufficiently small. We should mention that the averaging method not only provides the existence of periodic orbits of the Michelson system but also estimates the shape of the periodic orbit in function of c > sufficiently small. On the existence of periodic orbit of Michelson system there have appeared lots of works, such as Michelson 1986) using a series in sinus for all small c, Kent and Elgin 1991) using the continuation code AUTO, Lamb, Teixeira and Webster 25) using the Liapunov-Schmidt reduction method and so on. The developed averaging method Theorem 6) can also be applied to higher order differential equations for obtaining the existence of their periodic solutions. Theorem 22 [28]). a) The third-order differential equation... x µẍ +ẋ µx = ε sin x x + 12 ) cos t,

17 LIMIT CYCLES OF DIFFERENTIAL SYSTEMS 259 for ε sufficiently small, has at least two limit cycles. b) The third-order differential equation 2)... n x ẍ +ẋ x = ε i+j+k=,i,j,k ) a ijk x i ẋ j ẍ k + cos t, when ε sufficiently small, has at least m {1, 2,..., 2[n 1)/2]+1} limit cycles choosing conveniently the coefficients a ijk. c) For the third-order differential equation... x ẍ +ẋ x = ε cosx + t), and for all m N, there exists ε m > such that if ε [ ε m, ε m ] \ {}, the equation has at least m limit cycles. Except the above mentioned results, there are also some other recent results which will not be listed here. We refer the readers to [22, 23, 25, 26]. 3 A new result: limit cycle bifurcations of higher dimensional systems In this section we extend Theorem 11 for m = 2 to general m 2. The main results are the following. Theorem 23. For systems 11) with ac k, k = 3,..., n, the following statements hold. a) If m is even, there exists system 11) for which l {, 1,..., m n 2 /2} limit cycles bifurcate from the origin at ε =, that is, for ε sufficiently small the system has exactly l limit cycles in a neighborhood of the origin and these limit cycles tend to the origin when ε. b) If m is odd, there exists system 11) for which l {, 1,..., m n 2 } limit cycles bifurcate from the origin at ε =. We remark that Statement a) is for arbitrary even number m, and coincides with the result of [29] when m = 2. For m odd statement b) shows that the number of limit cycles can exponentially increase in terms of the dimension n of the system with exponent n 2. From the proof of Theorem 23 we can easily get the following. Corollary 24. For n 3 and m 2, there exist real polynomial differential systems of degree m in R n, consisting of the linear part of 11) and homogeneous polynomials of degree m, which can have l limit cycles with either l = m n 2 /2 for m even or l = m n 2 for m odd.

18 26 DINGHENG PI AND XIANG ZHANG As we know it is the first result showing the phenomena that the function in the number of limit cycles bifurcated from one singularity of higher dimensional polynomial differential systems is a power function in the degree of the systems. Proof of Theorem 23. For using the averaging method we need to transfer system 11) to the form of system 1). Taking the cylindrical transformation x = r cos θ, y = r sin θ, z i = z i, i = 3,..., n, system 11) becomes ṙ = εar + A i1,...,i n θ)r cos θ) i1 r sin θ) i2 z i3 3 zin n i 1+ +i n=m + Om + 1), 21) θ = b + 1 r z k = εc k z k + i i n=m i 1+ +i n=m B i1,...,i n θ)r cos θ) i1 r sin θ) i2 z i3 3 zin n + Om + 1), c k) i 1,...,i n r cos θ) i1 r sin θ) i2 z i3 3 zin n + Om + 1), for k = 3,..., n, where A i1,...,i n θ) = a i1,...,i n cos θ + b i1,...,i n sin θ, B i1,...,i n θ) = b i1,...,i n cos θ a i1,...,i n sin θ, and Om + 1) = O r, z 3,..., z n ) m+1 ) denotes the terms having orders lager than m. Taking a i3...i n = b i3...i n =, it is easy to show that in a suitable neighborhood of r, z 3,..., z n ) =,,..., ), we have θ. Treating θ as the independent variable we get from system 21) the

19 LIMIT CYCLES OF DIFFERENTIAL SYSTEMS 261 following: 22) dr dθ = dz k dθ = r`εar + br + r`εc k z k + br + P i 1 + +i n=m P i 1 + +i n=m P A i1,...,i n θ)r cos θ) i 1 r sin θ) i 2 z i 3 3 zn in + Om + 1) B i1,...,i n θ)r cos θ) i 1 r sin θ) i 2 z i 3 3 zn in + Om + 1) c k) i 1 + +i n=m P i 1 + +i n=m i 1,...,i n r cos θ) i 1 r sin θ) i 2 z i zn in + Om + 1) B i1,...,i n θ)r cos θ) i 1 r sin θ) i 2 z i z in n + Om + 1)., For ε >, set r, z 3,..., z n ) = ρ ε 1 m 1, η3 ε 1 m 1,..., ηn ε 1 m 1 ). We have 23) dρ dθ = εf 1θ, ρ, η 3,..., η n ) + ε 2 g 1 θ, ρ, η 3,..., η n, ε), dη k dθ = εf kθ, ρ, η 3,..., η n ) + ε 2 g k θ, ρ, η 3,..., η n, ε), k = 3,..., n, where aρ + A i1,...,i n θ)ρ cos θ) i1 ρ sin θ) i2 η i ) ηin n i f 1 = 1+ +i n=m, b 24) ck η k + c k) i 1,...,i n ρ cos θ) i1 ρ sin θ) i2 η i ) ηin n i f k = 1+ +i n=m. b The averaged system of 23) is 25) ẏ = εf y), where y = ρ, η 3,..., η n ) and f y) = f1 y), f 3 y),..., f n y)) with f i y) = 1 2π 2π f i θ, ρ, η 3,..., η n )dθ, for i = 1, 3,..., n.

20 262 DINGHENG PI AND XIANG ZHANG In order for calculating the isolated zeros of f y), we need to get the explicit expression of f1 y), f 3 y),..., f n y). For this we will use the obvious fact: 2π cos i θ sin j θdθ = if and only if either i or j is odd. Proof of Statement a). Since m is even, we have f1 y) = aρ b + 1 2bπ 2π fky) = c kη k b 2π m 2)/2 k= + 1 2bπ m/2 for k = 3,..., n, where i 1 +i 2 =2k+1 i 3+ +i n=m 2k 1 k= i 1 +i 2 =2k i 3+ +i n=m 2k A i 1,...,i n θ)ρ i1+i2 η i ηin n dθ, c k) i 1,...,i n ρ cos θ) i1 ρ sin θ) i2 η i ηin n dθ, A i 1,...,i n θ) = a i1,...,i n cos i1+1 θ sin i2 θ + b i1,...,i n sin i2+1 θ cos i1 θ. By the averaging theory we need to calculate the nondegenerate singularities of the averaged system 25) in the region ρ >. It is equivalent to compute the isolated simple roots of the algebraic equations f y) = with ρ >. We claim that there is a system 11) for which the corresponding system of algebraic equations f1 y) =, f 3 y) =,..., f n y) =, has exactly m n 2 /2 isolated simple roots. We now prove the claim. In the expression of f 1 y) we choose a i 1,...,i n = b i1,...,i n = for i 4,..., i n ),..., ) and set 26) A j = 1 2π 2π i 1 +i 2 =j i 3=m j Then f 1 y) = can be simplified to A i 1,i 2,i 3,,..., θ)dθ. 27) aρ + A 1 ρη m A 3 ρ 3 η m A m 1 ρ m 1 η 3 =.

21 LIMIT CYCLES OF DIFFERENTIAL SYSTEMS 263 In the expression of f3 y) we choose c3) i 1,...,i n = for i 4,..., i n ),..., ) and set 28) B l = 1 2π 2π Then f 3 y) = is simply i 1 +i 2 =l i 3=m l c 3) i 1,i 2,i 3,,..., cos θ)i1 sin θ) i2 dθ. 29) c 3 η 3 + B η m 3 + B 2ρ 2 η m B m ρ m =. Since aρ, it follows from 27) and 29) that Consequently, we have B η3 m + + B mρ m A 1 ρη3 m 1 = c 3η A m 1 ρ m 1 η 3 aρ. 3) B c ) ) m 3A 1 η3 a ρ + + B m 2 c 3A m 1 a ) ) 2 η3 + B m =. ρ Since B 2k and A 2k+1 for k =,..., m 2)/2 can be chosen arbitrarily, that is, the coefficients of equation 3) in η 3 /ρ can be arbitrary real numbers, so for the suitable choice of the values of B 2k and A 2k+1 equation 3) can have exactly m/2 simple nonzero real roots η 3 /ρ) 2. Rewriting equation 27) in η3 m 1 a = ) 2 ) m 2. A 1 + A ρ 3 η ρ Am 1 η 3 Since m is even, for each root η 3 /ρ of equation 3) we have a unique real solution a η 3 = ) 2 ) m 2 A 1 + A ρ 3 η ρ Am 1 η 3 1 m 1.

22 264 DINGHENG PI AND XIANG ZHANG Thus, we get from equations 27) and 29) exactly m/2 simple roots, denoted by ρ i, η 3i ) with ρ i > for i = 1, 2,..., m/2. In the expression of fk y) for k = 4,..., n, we choose ck) i 1,...,i n = for either i 1, i 2 ), ) or i m+1,..., i n ),..., ). Then equations fk y) = can be simplified to 31) c k η k + c k),,i 3,...,i k,,..., ηi ηi k k =. i 3+ +i k =m Equation 31) with k = 4 is 32) c 4 η 4 + c 4) m... ηm 4 + c 4) 1m 1)... ηm 1 4 η c 4) m... ηm 3 =. It can be treated as an algebraic equation in η 4 of degree m. Substituting each solution η 3i, i = 1,..., m/2, of 27) and 29) into 32), then equation 32) can have exactly m different solutions, denoted by η 4ij for j = 1, 2,..., m, provided the suitable choice of the values of c 4),,i 3,i 4,,...,. Substituting the m 2 /2 different solutions η 3i, η 4ij ) for i = 1,..., m/2 and j = 1,..., m into 31) with k = 5. By choosing the suitable values of the coefficients c 5),,i, for each one of η 3,i 4,i 5,,..., 3i, η 4ij ) s equation 31) with k = 5 can have m different solutions, denoted by η 5ijs, s = 1, 2,..., m. So equations 27), 29) and 31) with k = 4 and 5 have m 3 /2 different simple solutions ρ i, η 3i, η 4ij, η 5ijs ) for i = 1,..., m/2 and j, s = 1,..., m. By induction we can prove that equations 27), 29) and 31) can have m n 2 /2 different solutions y q = ρ q, η 3q, η 4q,..., η nq ) with ρ q > for q = 1,..., m n 2 /2. By the Bezout s Theorem [36] see Theorem 1) the m n 2 /2 are the maximum number of the solutions that these systems can have taking into account the equation 3). It follows that each solution y q for q = 1,..., m 2 /2 is simple. This proves the claim. Since all these m n 2 /2 roots are simple, it follows that the determinant of system 25) at these points does not vanish. Hence, by Theorem 3 system 23) can have m n 2 /2 limit cycles. Through the transformation r, z 3,..., z n ) = ρ ε 1 m 1, η3 ε 1 m 1,..., ηn ε 1 m 1 ), system 22) and consequently system 11) has m n 2 /2 limit cycles. Furthermore, these m n 2 /2 limit cycles tend to zero when ε. Working in a similar way to the proof of the above claim, we can prove that equations 27), 29) and 31) can have exactly l {, 1,..., mn 2 2 1} simple roots provided the suitable choice of the values of the coefficients. Then we get that system 11) can have l {, 1,..., m n 2 /2 1}

23 LIMIT CYCLES OF DIFFERENTIAL SYSTEMS 265 limit cycles, which tend to zero when ε. This proves statement a) of Theorem 23. Proof of Statement b). Since m is odd, we have f1 y) = aρ b + 1 2bπ 2π fk y) = c kη k b 2π m 1)/2 k= + 1 2bπ m 1)/2 k= for k = 3,..., n, where i 1 +i 2 =2k+1 i 3+ +i n=m 2k 1 i 1 +i 2 =2k i 3+ +i n=m 2k A i 1,...,i n θ)ρ i1+i2 η i3 3 ηin n dθ, c k) i 1,...,i n ρ cos θ) i1 ρ sin θ) i2 η i3 3 ηin n dθ, A i 1,...,i n θ) = a i1,...,i n cos i1+1 θ sin i2 θ + b i1,...,i n sin i2+1 θ cos i1 θ. We claim that there is a system 11) for which the system of algebraic equations f1 y) =, f 3 y) =,..., f n y) =, has exactly m n 2 simple roots. For proving the claim we choose a i1,...,i n = b i1,...,i n = if i 4,..., i n ),..., ). Then f1 y) = can be written in 33) aρ + A 1 ρη m A 3 ρ 3 η m A m 2 ρ m 2 η A mρ m =, where A j s were defined in 26). Similarly, we choose c 3) i 1,...,i n =,..., ) if i 4,..., i n ),..., ). Then equation f3 y) = can be simplified to 34) c 3 η 3 + B η m 3 + B 2ρ 2 η m B m 1 ρ m 1 η 3 =, where B l s were defined in 28). From 33) and 34) we get B c ) 3A 1 η3 35) a ρ ) m 1 + B 2 c 3A 3 a ) ) m 3 η3 ρ + + B m 1 c 3A m a )) η3 ρ =.

24 266 DINGHENG PI AND XIANG ZHANG Since c 3, A i s and B j s can be chosen arbitrarily, we choose suitable values of these parameters satisfying A j /a < for j = 1, 3,..., m such that equation 35) can have exactly m 1)/2 + 1 roots, one is η 3 = and the others are nonzero given in η3 ρ )2. For η 3 =, equation 33) has a unique positive solution, denoted by ρ. For each nonzero root of 35) we get from equation 33) that η3 m 1 a = A A m 2 ρ η 3 ) m 3 + A m ρ η 3 ) m 1. Since m is odd, we have two solutions η 3 = ± ) 1 m 1 a A A m 2 ρ η 3 ) m 3 + A m ρ η 3 ) m 1. Recall that a/a j < for j = 1, 3,..., m. Hence, each positive real root η 3 /ρ) 2 of 35) corresponds to two solutions ρ, ±η 3 ) of equations 33) and 34). Thus equations 33) and 34) can have the maximum number, saying m, of real roots, denoted by ρ i, η 3i ) with ρ i >, i = 1,..., m. For k = 4,..., n, we set c k) i 1,...,i n = for either i 1, i 2 ), ) or i k+1,..., i n ),..., ). Then equation fk y) = can be written in 36) c k η k + i i k =m c k),,i 3,...,i k,,..., ηi ηi k k =. Working in a similar way to the proof of statement a) we can prove that the system of algebraic equations 33), 34) and 36) can have the maximum number, saying m n 2, of simple real roots ρ, η 3,..., η n ) with ρ >. This proves the claim. Using the last claim and working in a similar way to the proof of statement a) we can finish the proof of statement b). This completes the proof of Theorem 23. Acknowledgements We express our thanks to the referee for his nice comments and suggestions. The second author is partially supported by NNSF of China grants and The both authors are also partially supported by Shanghai Pujiang Program 9PJD13.

25 LIMIT CYCLES OF DIFFERENTIAL SYSTEMS 267 REFERENCES 1. A. Buică, J. P. Françoise and J. Llibre, Periodic solutions of nonlinear periodic differential systems with a small parameter, Commun. Pure Appl. Anal. 6 27), A. Buică and J. Llibre, Averaging methods for finding periodic orbits via brouwer degree, Bull. Sci. Math ), A. Buică, J. Llibre and O. Makarenkov, Asymptotic stability of periodic solutions for nonsmooth differential equations with application to the nonsmooth van der Pol oscillator, SIAM J. Math. Anal. 4 29), C. A. Buzzi, J. Llibre and P. R. de Silva, 3-dimensional hopf bifurcation via averaging theory, Discrete Contin. Dyn. Syst ), A. Cima, J. Llibre and M. A. Teixeira, Limit cycles of some polynomial differential systems in dimension 2, 3 and 4 via averaging theory, Appl. Anal ), S. N. Chow and J. K. Hale, Methods of Bifurcation Thoery, Springer-Verlag, New York, C. Christopher and Chengzhi Li, Limit cycles of differential equations, Series: Advanced Courses in Mathematics, CRM Barcelona, Birkhäuser Verlag, Basel, W. A. Coppel, Some quadratic systems with at most one limit cycles, Dynamics Reported ), F. Dumortier and C. Li, On the uniqueness of limit cycles surrounding one or more sin- gularities for Liénard equations, Nonlinearity ), F. Dumortier and C. Li, Quadratic Liénard equations with quadratic damping, J. Differential Equations ), F. Dumortier and C. Rousseau, Cubic Liénard equations with linear dapimg, Nonlinearity 3 199), J. P. Françoise, Oscillations en biologie: Analyse qualitative et modèles, Collect. Math. Appl. 46, Springer-Verlag, Berlin, J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Appl. Math. Sci. 42, Springer-Verlag, New York, Maoan Han, On parameter size for existence of periodic solutions by the method of averaging, Dynam. Systems Appl. 3 21), Maoan Han, Degenerate Hopf bifurcation for higher dimensional systems with applications, Proceedings of the first conference of youth in China, organized by Science and Technology Society of China, Chinese Science and Technology Society Press, Beijing, Maoan Han, K. Jiang and D. Jr. Green, Bifurcations of periodic orbits, subharmonic solutions and invariant tori of high-dimensional systems, Nonlinear Anal. Theory Methods ), Maoan Han and Yuying Sha, Bifurcation of periodic orbits with codimension two singularities, J. Nanjing Univ. Math. Biquarterly ), Maoan Han and Deming Zhu, Bifurcation Theory of Differential Equations, Coal Industry Press, Beijing, Chongxiao Li, The global bifurcation on the nonlinear oscillator x + a 1 + 2a 2 x + 3a 3 x 2 )x + b 1 x b 2 x 2 =, J. Yunnan Polytechnic Univ. 1986, No. 2, Jibin Li, Hilbert s 16th problem and bifurcations of planar polynomial vector fields, Int. J. Bifur. Chaos 13 23),

26 268 DINGHENG PI AND XIANG ZHANG 21. A. Lins, W. de Melo and C. C. Pugh, On Liénard s Equation, Lect. Notes Math. 597, , Springer, Berlin, J. Llibre, Averaging theory and limit cycles for quadratic systems, Rad. Mat /23), J. Llibre and A. Makholouf, Limit cycles of polynomial differential systems bifurcating from the periodic orbits of a linear differential system in R d, Bull. Sci. Math ), J. Llibre, A. C. Mereu and M. A. Teixerra, Limit cycles of the generalized polynomial Liénard differential equations, Math. Proc. Cambridge Philos. Soc ), J. Llibre, M. A. Teixeira and J. Torregrosa, Limit cycles bifurcating from a k dimensional isochronous center contained in R n with k n, Math. Phys. Anal. Geom. 1 27), J. Llibre and Hao Wu, Hopf Bifurcation for degenerate singular points of multiplicity 2n-1 in dimension 3, Bull. Sci. Math ), J. Llibre, Jiang Yu and Xiang Zhang, Limit cycles coming from the perturbation of 2-dimensional centers of vector fields in R 3, Dynam. Systems Appl ), J. Llibre, Jiang Yu and Xiang Zhang, Limit cycles for a class of third-order differential equations, Rocky Mountain J. Math. 4 21), J. Llibre and Xiang Zhang, Hopf bifurcation in higher dimensional systems via the averaging method, Pacific J. Math ), J. Llibre and Xiang Zhang, On the Hopf-zero bifurcation of the Michelson system, preprint, Dingjun Luo, Xian Wang, Deming Zhu and Maoan Han, Bifurcation theory and methods of dynamical systems, Adv. Ser. Dynam. Systems 15, World Scientific Publishing Co., Inc., River Edge, NJ, G. I. Marchuk, Mathematical Models in Immunology, Nauka, Moscow, J. E. Marsden and M. McCracken, The Hopf bifurcation and its applications, Appl. Math. Sci ). 34. D. Michelson, Steady solutions for the Kuramoto-Sivashinsky equation, Physica D ), J. A. Sanders and F. Verhulst, Averaging methods in nonlinear dynamical systems, Appl. Math. Sci ). 36. I. R. Shafaravich, Basic Algebraic Geometry, Springer-Verlag, New York, F. Verhulst, Nolinear Differential Equations and Dynamical Systems, Springer- Verlag, Berlin, Yanqian Ye, Theory of Limit Cycles, Transl. Math. Monographs 66, Amer. Math. Soc., Providence, RI, Zhifen Zhang, Tongren Ding, Wenzao Huang and Zhenxxi Dong, Qualitative Theory of Differential Equations, Transl. Math. Monographs 11, Amer. Math. Soc., Providence, RI, Deming Zhu and Maoan Han, Invariant tori and subharmonic bifurcations from periodic manifolds, Acta Math. Sinica N.S.) ), Corresponding author: Xiang Zhang Department of Mathematics, Shanghai Jiaotong University, Shanghai 224, The People s Republic of China. address: xzhang@sjtu.edu.cn

27 LIMIT CYCLES OF DIFFERENTIAL SYSTEMS 269 Department of Mathematics, Shanghai Jiaotong University, Shanghai 224, The People s Republic of China. address: pidh@sjtu.edu.cn

28