This is a preprint of: “Limit cycles of piecewise polynomial differential systems with the discon- tinuity line xy=0”, Tao Li, Jaume Llibre, Commun. Pure Appl. Anal., vol. 20(11), 3871–3893, 2021. DOI: [10.3934/cpaa.2021136] LIMIT CYCLES OF PIECEWISE POLYNOMIAL DIFFERENTIAL SYSTEMS WITH THE DISCONTINUITY LINE xy = 0 TAO LI1 AND JAUME LLIBRE2 Abstract. In this paper we study the maximum number of limit cycles bifurcating from the periodic solutions of the period annulus of the center ẋ = −y((x2 + y 2 )/2)m , ẏ = x((x2 + y 2 )/2)m with m ≥ 0 under discontinuous piecewise polynomial perturbations of degree n with four zones separated by the discontinuity set Σ = {(x, y) ∈ R2 : xy = 0}. For such perturbations of degree n we provide an upper bound for the maximum number of the bifurcated limit cycles using the averaging theory up to an arbitrary order N , and this upper bound is reached at least for orders one and two. In particular, if m = 0, namely the center is linear, we improve the result of [Nonlinear Analysis: Real World Applications 41 (2018), 384–400] providing more limit cycles that bifurcate from the unperturbed periodic orbits of the period annulus of the center. 1. Introduction and statement of main result Hilbert’s 16th problem, an important subject in the qualitative theory of differential sys- tems, asks for the maximum number of limit cycles that planar polynomial differential systems with a fixed degree can have. Since David Hilbert [15] proposed it in 1900, a large number of works were devoted to the study of this problem, see the survey paper [19]. But it is still an open problem up to now, even for quadratic differential systems. As Hilbert’s 16th problem turns out to be extremely difficult, some researchers have particularized it to identify the maximum number of limit cycles bifurcating from a periodic annulus, when we perturb it inside the class of all planar polynomial differential systems with a fixed degree n ≥ 1. In essence, this is the weak Hilbert’s 16th problem, see [2, 16, 19]. If this periodic annulus is formed by the linear center ẋ = −y, ẏ = −x, Iliev [16] proved that [3(n − 1)/2] is a lower bound for the maximum number, where [·] denotes the integer part function. They also gave that [N (n − 1)/2] is an upper bound for the maximum number using the Melnikov method of order N . However the exact maximum number is unclear so far except for some special perturbations, e.g. Liénard family [14, 21]. In 2010, Buică, Giné and Llibre [6] extended the work of Iliev [16] considering the polynomial perturbations of the center  2 m  2 m x + y2 x + y2 (1) ẋ = −y , ẏ = x 2 2 with m ≥ 0. Note that H(x, y) = (x2 + y 2 )/2 is a first integral of system (1). Discontinuous events are widespread in the real world, such as stick-slip motion in oscilla- tors with dry friction [12, 20], switching in electronic circuits [4, 5], and impact in mechanical 2010 Mathematics Subject Classification. 34C29, 34C25, 34C05. Key words and phrases. Averaging method, Hilbert’s 16th problem, limit cycle bifurcation, piecewise poly- nomial system. 1 2 TAO LI AND JAUME LLIBRE devices [4, 27]. To mathematically depict them, many models were established by discon- tinuous piecewise smooth differential systems, which consist of multiple smooth differential systems defined on different regions separated by smooth lines or curves, usually called dis- continuity set [4]. The solution of such systems can be defined by the Filippov convention [13]. For discontinuous piecewise smooth differential systems, an isolated periodic orbit is said to be a crossing limit cycle if it intersects the discontinuity set only at the so-called crossing region, a subset of the discontinuity set where both vector fields are transverse to it and their normal components have the same sign, see [18] for more details. Analogous to polynomial differential systems, we can extend the weak Hilbert’s 16th problem to discontinuous piecewise polynomial differential systems, asking for the maximum number of crossing limit cycles bifurcating from a periodic annulus under discontinuous piecewise polynomial perturbations of a fixed degree. The simplest case is the piecewise polynomial perturbations of the linear center ẋ = −y, ẏ = x in two zones separated by a straight line. In this case for any given degree n ≥ 1 Buzzi, Lima and Torregrosa [7] proved that N n − 1 is an upper bound for the maximum number using Melnikov method up to order N . Moreover, they showed that this upper bound is reached for orders one and two, which means that 2n − 1 is a lower bound. According to the works [8] (resp. [24]), this lower bound can be improved to be 3 (resp. 8 and 13) for n = 1 (resp. n = 2 and n = 3) up to a study of order seven (resp. five). Since models in applications with the discontinuity that is located on multiple straight lines or curves are ubiquitous, see e.g. [1, 12, 26], this encourages us to research the crossing limit cycles of discontinuous piecewise smooth differential systems having multiple straight lines or curves as the discontinuity set, e.g. [10, 11, 25, 29, 31, 32] for some contributions. In this paper, motivated by real applications and the works [7, 16], we bring the weak Hilbert’s 16th problem to discontinuous piecewise polynomial perturbations of the center (1) in four zones Rk (k = 1, 2, 3, 4) separated by the discontinuity set Σ = {(x, y) ∈ R2 : xy = 0}, where R1 = {(x, y) ∈ R2 : x > 0, y > 0}, R2 = {(x, y) ∈ R2 : x < 0, y > 0}, R3 = {(x, y) ∈ R2 : x < 0, y < 0}, R4 = {(x, y) ∈ R2 : x > 0, y < 0}. More precisely, we consider the discontinuous piecewise polynomial system  2 m X ∞  2 m X ∞ x + y2 i k x + y2 (2) ẋ = −y + ε fi (x, y), ẏ = x + εi gik (x, y) 2 i=1 2 i=1 if (x, y) ∈ Rk , where ε ∈ R is a small perturbation parameter, fik and gik are real polynomials of degree n ≥ 1. Our main goal is to study the maximum number of crossing limit cycles of system (2) bifurcating from the unperturbed periodic annulus. The following theorem is our main result. Theorem 1. Let MN (m, n) be the maximum number of crossing limit cycles of system (2) with m ≥ 0, n ≥ 1 which can be obtained using the averaging theory of order N for |ε| > 0 sufficiently small. We have the following statements. (i) If n ≥ 2m + 1, then MN (m, n) ≤ N n, and this upper bound is reached for N = 1, 2. (ii) If n ≤ 2m, then MN (m, n) ≤ (2m + 1)(N − 1) + n. Moreover, this upper bound is reached for N = 1, and N = 2 and n ≥ m, while for N = 2 and n ≤ m − 1, it can be reduced to 3n + 1 and the new upper bound is reached. LIMIT CYCLES OF PIECEWISE POLYNOMIAL DIFFERENTIAL SYSTEMS 3 We remark that Theorem 1 improves the results of [9, 29, 31]. In [29], using the first order Melnikov method for system (2) with m = 0 and any fixed n, Wang, Han and Constantinescu proved that n crossing limit cycles can bifurcate from the unperturbed periodic annulus. Thus n is a lower bound for the maximum number of crossing limit cycles bifurcating from the periodic annulus of the linear center ẋ = −y, ẏ = x, when we perturb it inside discontinuous piecewise polynomial systems with four zones separated by Σ. After this, the lower bound is updated as 2n − 1 in [31] by introducing multiple small parameters into the considered systems and deriving a generalized first order Melnikov function. According to statement (i) of Theorem 1, we can further update the lower bound as 2n and thus improve the result of [31] providing one more crossing limit cycle. On the other hand in [9] the authors considered discontinuous piecewise polynomial per- turbations of the center (1) with m ≥ 1 in k zones separated by k rays originating at the origin. According to [9, Theorem 1], for given m, k and degree n the first order averaging theory at most provides n crossing limit cycles and this number is reached. Therefore if we restrict the perturbations considered in [9] to system (2), i.e. k = 4 and the four rays are exactly the coordinate axes, all the results of Theorem 1 with N = 1 is just [9, Theorem 1]. Although system (2) is special one, we obtain new results using the higher order averaging theory as stated in Theorem 1 and thus we improve the work [9]. The paper is organized as follows. In section 2 we recall the averaging theory for non- autonomous discontinuous piecewise smooth differential systems with many zones. Then we prove Theorem 1 in section 3. 2. Averaging theory It is well known that the averaging theory can be used to study the number of limit cycles of smooth differential systems, see [28]. With the averaging theory we can obtain some information on limit cycles bifurcating from a periodic annulus through studying the zeros of the so-called averaged functions. In recent years, stimulated by both theoretical development and real applications, the classical averaging theory developed for smooth differential systems has been extended to piecewise smooth differential systems, e.g. [17,22,23,30]. In this section we summarize the averaging theory for studying the crossing limit cycles of the discontinuous piecewise polynomial system (2) following [23]. Using the change to polar coordinates (r, θ) where x = r cos θ and y = r sin θ we write system (2) as ∞ ∞   dr X i k dθ r2m 1X i k (k − 1)π kπ (3) = ε pi (θ, r), = m − ε qi (θ, r) if θ ∈ , , dt i=1 dt 2 r i=1 2 2 where k = 1, 2, 3, 4 and pki (θ, r) = cos θfik (r cos θ, r sin θ) + sin θgik (r cos θ, r sin θ), qik (θ, r) = sin θfik (r cos θ, r sin θ) − cos θgik (r cos θ, r sin θ). Taking θ as the new independent variable, system (3) is transformed into N dr X i (4) = ε Fi (θ, r) + εN +1 R(θ, r, ε), dθ i=1 4 TAO LI AND JAUME LLIBRE with 4 X 4 X (5) Fi (θ, r) = χ[ (k−1)π , kπ ] (θ)Fik (θ, r), R(θ, r) = χ[ (k−1)π , kπ ] (θ)Rk (θ, r, ε), 2 2 2 2 k=1 k=1 and i−1 2m pki (θ, r) X k 22m qi−l k (θ, r) X 23m qjk1 (θ, r)qjk2 (θ, r) Fik (θ, r) = 2m + pl (θ, r) 4m+1 + +··· r r r6m+2 l=0 j1 +j2 =i−l 1≤j1 ,j2 ≤i−l (6) ! X 2(i−l+1)m qjk1 (θ, r)qjk2 (θ, r) · · · qjki−l (θ, r) + , j1 +j2 +···+ji−l =i−l r(2m+1)(i−l)+2m 1≤j1 ,j2 ,...,ji−l ≤i−l where we are assuming that pk0 (θ, r) ≡ 0 for the sake of convenience, χA (θ) is the characteristic function on the interval A, i.e. ( 1 if θ ∈ A, χA (θ) = 0 if θ ∈ / A. Let S1 ≡ R/(2πZ). Clearly, Fik : S1 × (0, +∞) → R and Rk : S1 × (0, +∞) × (−ε0 , ε0 ) → R are C N +1 functions which are 2π-periodic in the variable θ. According to [23], the averaged function of order i associated to system (4) is yi (2π, r) (7) Fi (r) = i! where the functions yi (θ, r) : S1 × (0, ∞) → R for i = 1, 2, ..., N are given recurrently by Z θ y1 (θ, r) = F1 (ϕ, r)dϕ, 0 (8) Z θ i X l ! X 1 ∂ L Fi−l (ϕ, r) Y bj yi (θ, r) = i! Fi (ϕ, r) + yj (ϕ, r) dϕ. 0 K ∂rL j=1 l=1 Sl Here 4 ∂ L Fi (θ, r) X ∂ L Fik (θ, r) (9) = χ[(k−1)π/2,kπ/2] (θ) , ∂rL ∂rL k=1 where Sl is the set of all l-tuples of non-negative integers (b1 , b2 , ..., bl ) satisfying b1 + 2b2 + · · · + lbl = l, L = b1 + b2 + · · · + bl and K = b1 !b2 !2!b2 · · · bl !l!bl . Moreover we are assuming that F0 = 0 in (7) for convenience. The next theorem was proved in [23]. Theorem 2. Consider the non-autonomous discontinuous piecewise smooth differential sys- tem (4). Suppose that i0 is the first positive integer such that Fi = 0 for 1 ≤ i ≤ i0 − 1 and Fi0 6= 0. If Fi0 (r∗ ) = 0 and Fi00 (r∗ ) 6= 0 for some r∗ ∈ (0, ∞), then for |ε| > 0 sufficiently small there exists a 2π-periodic solution r(θ, ε) of system (4) such that r(0, ε) → r∗ as ε → 0. Theorem 2 implies that a simple positive real zero of the first non-vanishing averaged function provides a crossing limit cycle of system (2) bifurcating from some unperturbed periodic orbit of the period annulus of the center (1). LIMIT CYCLES OF PIECEWISE POLYNOMIAL DIFFERENTIAL SYSTEMS 5 We state the Descartes Theorem for studying the number of positive real zeros of polyno- mials in the following, for a proof see [3]. Theorem 3. Consider the real polynomial p(x) = ai1 1 xi1 + ai2 xi2 + ... + air xir with 0 = i1 < i2 < ... < ir and r > 1. If aij aij+1 < 0, we say that aij and aij+1 have a variation of sign. If the number of variations of signs is r0 ∈ {0, 1, 2, ..., r − 1}, then the polynomial p(x) has at most r0 positive real roots. Furthermore, we can choose the coefficients of the polynomial p(x) in such a way that p(x) has exactly r − 1 positive real roots. 3. Proof of Theorem 1 The goal of this section is to prove Theorem 1. To do this, from section 2 we only need to study the maximum number of simple positive real zeros of the averaged functions associated to system (4) satisfying (5) and (6). We start with the following lemma. Lemma 4. Consider the functions Fi (θ, r) and Fik (θ, r), i = 1, 2, ..., N and k = 1, 2, 3, 4, given in (5) and (6) respectively. Let (10) Feik (θ, r) = Fik (θ, r)r(2m+1)i−1 , Fei (θ, r) = Fi (θ, r)r(2m+1)i−1 . If n ≥ 2m + 1 (resp. n ≤ 2m), then both Feik (θ, r) and Fei (θ, r) are polynomials of degree in (resp. (2m + 1)(i − 1) + n) in the variable r. Proof. Since fik (x, y) and gik (x, y) are polynomials of degree n ≥ 1, the functions pki (θ, r) and qik (θ, r) defined in (3) are polynomials of degree n ≥ 1 in the variable r whose coefficients are polynomials in the variables cos θ and sin θ. Thus, if n ≥ 2m + 1 (resp. n ≤ 2m), it follows from (6) that Feik (θ, r) = Fik (θ, r)r(2m+1)i−1 is a polynomial of degree in (resp. (2m + 1)(i − 1) + n) in the variable r whose coefficients are polynomials in the variables cos θ and sin θ. Joining (5) and (10), we have 4 X 4 X Fei (θ, r) = χ[ (k−1)π , kπ ] (θ)Fik (θ, r)r(2m+1)i−1 = χ[ (k−1)π , kπ ] (θ)Feik (θ, r). 2 2 2 2 k=1 k=1 Furthermoe Fei (θ, r) is a polynomial of degree in (resp. (2m + 1)(i − 1) + n) in the variable r whose coefficients are polynomials in the variables cos θ and sin θ if n ≥ 2m + 1 (resp. n ≤ 2m).  Having Lemma 4 we now obtain the following result. Proposition 5. The averaged function of order N associated to system (4) satisfying (5) and (6) has at most N n (resp. (2m + 1)(N − 1) + n) simple positive real zeros if n ≥ 2m + 1 (resp. n ≤ 2m). Proof. Let (11) ỹi (θ, r) = yi (θ, r)r(2m+1)i−1 , where i = 1, 2, ..., N and yi (θ, r) is the function defined in (8). We claim that ỹi (θ, r) is a polynomial of degree in (resp. (2m + 1)(i − 1) + n) in the variable r if n ≥ 2m + 1 (resp. n ≤ 2m). In fact, from (8), (10) and Lemma 4 it follows that Z θ Z θ ỹ1 (θ, r) = y1 (θ, r)r2m = F1 (φ, r)dφr2m = Fe1 (φ, r)dφ 0 0 6 TAO LI AND JAUME LLIBRE is a polynomial of degree n in r, i.e. the claim holds for i = 1. Assuming that this claim also holds for i = 2, 3, ..., i0 − 1 with 2 ≤ i0 ≤ N , by induction we only need to prove it for i = i0 in the following. For 1 ≤ i0 − l ≤ i0 − 1, using (9), (10) and the derivative method for the composition of functions, we have 4 ∂ L Fi0 −l (θ, r) X ∂ L Fik0 −l (θ, r) = χ [ (k−1)π kπ (θ) , 2 ] ∂rL 2 ∂rL k=1 (12) 4 X L X L! ∂ s Feik0 −l (θ, r) dL−s r−2(i0 −l)m−i0 +l+1 = χ[ (k−1)π , kπ ] (θ) 2 2 s=0 s!(L − s)! ∂rs drL−s k=1 Pi0 −l (θ, r) = , r(2m+1)(i0 −l)+L−1 where Pi0 −l (θ, r) is a polynomial in the variable r such that the above equality holds. By Lemma 4 and a direct computation, it is not difficult to get that the degree of Pi0 −l (θ, r) is (i0 − l)n (resp. (2m + 1)(i0 − l − 1) + n) if n ≥ 2m + 1 (resp. n ≤ 2m). We do not provide the specific expression of P (θ, r) because it is not necessary for the rest of the proof. Since b1 + 2b2 + · · · + lbl = l and b1 + b2 + · · · + bl = L, for l ≤ i0 − 1 we have Yl Y l  bj Yl ỹj (θ, r) 1 (13) yj (θ, r)bj = (2m+1)j−1 = (2m+1)l−L ỹj (θ, r)bj , j=1 j=1 r r j=1 Ql and j=1 ỹj (θ, r)bj is a polynomial of degree ln (resp. (2m + 1)(l − L) + Ln) in the variable r if n ≥ 2m + 1 (resp. n ≤ 2m). Here we used the assumption that the claim holds for i = 1, 2, 3, ..., i0 − 1. Thus, combining (8), (12), (13) and F0 = 0 in (8), we get   Z θ 0 −1 X iX L Y l Fi0 (φ, r) + 1 ∂ Fi0 −l (φ, r) yi0 (θ, r) = i0 ! L yj (φ, r)bj  dφ 0 K ∂r j=1 l=1 Sl (14)   Z θ 0 −1 X iX Yl i0 ! Fei0 (φ, r) + 1 = (2m+1)i −1 Pi −l (φ, r) ỹj (φ, r)bj  dφ. r 0 0 K 0 j=1 l=1 Sl Ql With the properties of Pi0 −l (θ, r) and j=1 ỹj (θ, r)bj obtained above, i.e. l Y Pi0 −l (θ, r) ỹj (θ, r)bj j=1 is a polynomial of degree i0 n (resp. (2m + 1)(i0 − L − 1) + (L + 1)n) in r if n ≥ 2m + 1 (resp. n ≤ 2m). Moreover, from Lemma 4 we know that Fei0 (θ, r) is a polynomial of degree i0 n (resp. (2m + 1)(i0 − 1) + n) in the variable r if n ≥ 2m + 1 (resp. n ≤ 2m). Consequently, if n ≥ 2m + 1 we obtain that ỹi0 (θ, r) = yi0 (θ, r)r(2m+1)i0 −1 is a polynomial of degree i0 n in the variable r from (14), while if n ≤ 2m, we have (2m + 1)(i0 − L − 1) + (L + 1)n < (2m + 1)(i0 − 1) + n and then ỹi0 (θ, r) is a polynomial of degree (2m + 1)(i0 − 1) + n in the variable r by (14) again from (14). This completes the proof of the claim. LIMIT CYCLES OF PIECEWISE POLYNOMIAL DIFFERENTIAL SYSTEMS 7 According to the definitions (7) and (11), the averaged function of order N associated to system (4) satisfying (5) and (6) is yN (2π, r) ỹN (2π, r) FN (r) = = . N! N !r(2m+1)N −1 Thus FN (r) and ỹN (2π, r) have the same simple positive real zeros. By the above claim, ỹN (2π, r) is a polynomial of degree N n (resp. (2m + 1)(N − 1) + n) if n ≥ 2m + 1 (resp. n ≤ 2m), so that Proposition 5 follows.  Proposition 5 gives a unified upper bound for the maximum number of simple positive real zeros of the averaged function of order N . Note that the upper bound for N = 1 is always n whatever n ≥ 2m + 1 or n ≤ 2m. But for N = 2 and n ≤ m − 1 analyzing the second order averaged function we can reduce the upper bound as it is stated in the next proposition. Proposition 6. If n ≤ m − 1, the second order averaged function associated to system (4) satisfying (5) and (6) has at most 3n + 1 simple positive real zeros. Proof. Since n ≤ m − 1 ≤ 2m, by Lemma 4 and the proof of Proposition 5 we know that (a) P1 (θ, r) = r2m+1 ∂F1 (θ, r)/∂r is a polynomial of degree n in the variable r, (b) ỹ1 (θ, r) = y(θ, r)r2m is a polynomial of degree n in the variable r, (c) Fe2 (θ, r) = F2 (θ, r)r4m+1 is a polynomial of degree 2m + n + 1 in the variable r. On the other hand, it follows from (6) that Fe2k (θ, r) = F2k (θ, r)r4m+1 = 2m r2m+1 pk2 (θ, r) + 22m pk1 (θ, r)q1k (θ, r), for k = 1, 2, 3, 4, where pk1 , q1k and pk2 are given in (3). Since n ≤ m − 1 and both pk1 (θ, r) and q1k (θ, r) are polynomial of degree n in the variable r as stated in the proof of Lemma 4, all the terms of Fe2k (θ, r) from r2n+1 to r2m vanish for k = 1, 2, 3, 4, i.e. Fe2k (θ, r) has at most 3n + 2 terms. Because of (5), hence all the terms of Fe2 (θ, r) from r2n+1 to r2m also vanish. Finally, using (7) and statements (a), (b) and (c) we have Z 2π Z 2π ∂F1 (θ, r) 1 F2 (r) = F2 (θ, r) + y1 (θ, r)dθ = 4m+1 Fe2 (θ, r) + P1 (θ, r)ỹ1 (θ, r)dθ, 0 ∂r r 0 and then F2 (r)r4m+1 is a polynomial of degree 2m + n + 1 having at most 3n + 2 terms. Thus F2 (r)r4m+1 has at most 3n + 1 simple positive real zeros due to Theorem 3. This concludes the proof of Proposition 6 because F2 (r) and F2 (r)r4m+1 have the same simple positive real zeros.  Next we study the realization of the upper bounds obtained in Propositions 5 and 6. Proposition 7. Consider the discontinuous piecewise polynomial system   2 m Xn  2 m   x + y2 i x + y2   ẋ = −y +ε ai y , ẏ = x if (x, y) ∈ R1 , 2 i=0 2 (15)  2 m  2 m   x + y2 x + y2  ẋ = −y , ẏ = x if (x, y) ∈ R2 ∪R3 ∪R4 . 2 2 For any given m ≥ 0 and n ≥ 1 there exists a choice of parameters ai , i = 0, 1, 2, ..., n, such that the corresponding first order averaged function has exactly n simple positive real zeros. 8 TAO LI AND JAUME LLIBRE Proof. Writing system (15) into the form (4) in the polar coordinates (r, θ), we get  m n  2 X π   2m cos θ ai sini θri if θ ∈ [0, ], F1 (θ, r) = r i=0 2   π 0 if θ ∈ [ , 2π]. 2 Thus the first order averaged function is Z π Z π Xn n 2 2 2m i 2m X 1 F1 (r) = F11 (θ, r)dθ = cos θ ai sin θr i dθ = ai ri 0 0 r2m i=0 r 2m i=0 i + 1 by (7) and (8). Clearly, F1 (r) and F1 (r)r2m have the same simple positive real zeros and F1 (r)r2m is a complete polynomial of degree n, i.e. all coefficients of F1 (r)r2m can be chosen arbitrarily. Consequently, there exists a choice of parameters ai , i = 0, 1, 2, ..., n, such that F1 (r) has n simple positive real zeros. This ends the proof of Proposition 7.  Proposition 8. Consider the discontinuous piecewise polynomial system   2 m Xn  m n X   x + y2 x2 + y 2   ẋ = −y + ε ai y i , ẏ = x +ε bi xi if (x, y) ∈ R1 ,   2 2   i=0 i=0    2  2 m  m   x + y x2 + y 2 ẋ = −y  , ẏ = x if (x, y) ∈ R2 ,  2 2 (16)  2 m Xn  m   x + y2 x2 + y 2    ẋ = −y + ε ci y i , ẏ = x if (x, y) ∈ R3 ,   2 2   i=0    2  2 m Xn  m   x +y 2 x2 + y 2   ẋ = −y + ε di y i , ẏ = x if (x, y) ∈ R4 , 2 i=0 2 with ci = (−1)i (ai + bi ) for i = 0, 1, 2, ..., n. For any given m ≥ 0 and n ≥ 1 the corresponding first order averaged function vanishes, and there exists a choice of parameters ai , bi and di such that the corresponding second order averaged function has exactly 2n (resp. 2m + n + 1, 3n + 1) simple positive real zeros if n ≥ 2m + 1 (resp. m ≤ n ≤ 2m, n ≤ m − 1). Proof. Writing system (16) into the form (4) in the polar coordinates (r, θ) and using the condition ci = (−1)i (ai + bi ) for i = 0, 1, 2, ..., n, we get  n   2m X π    (ai cos θ sini θ+bi sin θ cosi θ)ri if θ ∈ [0, ],  r2m i=0  2     π  0 if θ ∈ [ , π], 2 F1 (θ, r) = n    2m X 3π   2m (−1)i (ai + bi ) cos θ sini θri if θ ∈ [π, ],   r 2   i=0   3π  0 if θ ∈ [ , 2π], 2 LIMIT CYCLES OF PIECEWISE POLYNOMIAL DIFFERENTIAL SYSTEMS 9 and  2m n n  2 X X π   (a cos θ sin i θ+b sin θ cos i θ)r i (ai sini+1 θ − bi cosi+1 θ)ri if θ ∈ [0, ],   r4m+1 i=0 i i 2   i=0     π   0 if θ ∈ [ , π],  2 F2 (θ, r) = 22m X n Xn 3π   (−1) i (a + b ) cos θ sin i θr i (−1)i (ai + bi ) sini+1 θri if θ ∈ [π, ],   r 4m+1 i i 2     i=0 i=0  m X  n   2 i i 3π   r2m di cos θ sin θr if θ ∈ [ , 2π]. i=0 2 Therefore, by (7) and (8) the first order averaged function is Z π n X 2m 2 F1 (r) = 2m (ai cos θ sini θ + bi sin θ cosi θ)ri dθ r 0 i=0 Z 3π n X 2m 2 + 2m (−1)i (ai + bi ) cos θ sini θri dθ = 0, r π i=0 and the second order averaged function is (17) Z 2π n X 2m F2 (r) = di cos θ sini θri dθ r2m 3π i=0 2 Z π Xn n X 22m 2 + (ai cos θ sini θ+bi sin θ cosi θ)ri (ai sini+1 θ−bi cosi+1 θ)ri dθ r4m+1 0 i=0 i=0 2m Z 3π Xn n X 2 2 + (−1)i (ai + bi ) cos θ sini θri (−1)i (ai + bi ) sini+1 θri dθ r4m+1 π i=0 i=0 2m Z π Xn Xn 2 2 ai sini+1 θ−bi cosi+1 θ+bi i + (ai cos θ sini θ+bi sin θ cosi θ)(i − 2m)ri r dθ r4m+1 0 i=0 i=0 i+1 2m Z 3π Xn Xn 2 2 (−1)i sini+1 θ + 1 i + (−1)i (ai +bi ) cos θ sini θ(i − 2m)ri (ai +bi ) r dθ. r4m+1 π i=0 i=0 i+1 We can write F2 (r) as F2 (r) = Fe2 (r)/r4m+1 where n X (−1)i Fe2 (r) =2m di ri+2m+1 i=0 i+1   2n X X X   i + 22m  αi1 ,i2 ai1 ai2 + (βi1 ,i2 bi1 bi2 + γi1 ,i2 ai1 bi2 )  r . i=0 i1 +i2 =i i1 +i2 =i 0≤i1 ≤i2 ≤n 0≤i1 ,i2 ≤n 10 TAO LI AND JAUME LLIBRE To determine the number of simple positive real zeros of Fe2 (r), or equivalently F2 (r), we next compute αi1 ,i2 , βi1 ,i2 and γi1 ,i2 . Collecting all terms of ai1 ai2 from (17), we obtain Z π 2 Z 3π 2 i1 i2 +1 αi1 ,i2 = cos θ sin θ sin θdθ + (−1)i1 +i2 cos θ sini1 θ sini2 +1 θdθ 0 π Z π i2 +1 2 sin θ + (i1 − 2m) cos θ sini1 θ dθ 0 i2 + 1 Z 3π 2 (−1)i2 sini2 +1 θ + 1 + (−1)i1 (i1 − 2m) cos θ sini1 θ dθ π i2 + 1 Z π 2 Z 3π 2 + cos θ sini2 θ sini1 +1 θdθ + (−1)i1 +i2 cos θ sini2 θ sini1 +1 θdθ 0 π Z π 2 sini1 +1 θi2 + (i2 − 2m) cos θ sin θ dθ 0 i1 + 1 Z 3π 2 (−1)i1 sini1 +1 θ + 1 + (−1)i2 (i2 − 2m) cos θ sini2 θ dθ π i1 + 1 1 1 = + i1 + 1 i2 + 1 if i1 < i2 , while if i1 = i2 = 2i , Z π 2 Z 3π 2 i i i i α 2i , 2i = cos θ sin 2 θ sin 2 +1 θdθ + (−1)i cos θ sin 2 θ sin 2 +1 θdθ 0 π Z  π  i i 2 i sin 2 +1 θ + − 2m cos θ sin 2 θ i dθ 0 2 2 +1 Z 3π   i i 2 i i i (−1) 2 sin 2 +1 θ + 1 + (−1) 2 − 2m cos θ sin 2 θ i dθ π 2 2 +1 1 =i . 2 +1 Collecting all terms of bi1 bi2 and ai1 bi2 from (17) respectively, we obtain Z π 2 Z 3π 2 βi1 ,i2 = sin θ cosi1 θ(− cosi2 +1 θ)dθ + (−1)i1 +i2 cos θ sini1 θ sini2 +1 θdθ 0 π Z π i2 +1 2 1 − cos θ + (i1 − 2m) sin θ cosi1 θ dθ 0 i2 + 1 Z 3π 2 (−1)i2 sini2 +1 θ + 1 + (−1)i1 (i1 − 2m) cos θ sini1 θ dθ π i2 + 1 =0, LIMIT CYCLES OF PIECEWISE POLYNOMIAL DIFFERENTIAL SYSTEMS 11 and Z π 2 γi1 ,i2 = cos θ sini1 θ(− cosi2 +1 θ) + sin θ cosi2 θ sini1 +1 θdθ 0 Z 3π 2 + (−1)i1 +i2 cos θ sini1 θ sini2 +1 θ + (−1)i1 +i2 cos θ sini2 θ sini1 +1 θdθ π Z π 2 1 − cosi2 +1 θ sini1 +1 θ + (i1 − 2m) cos θ sini1 θ + (i2 − 2m) sin θ cosi2 θ dθ 0 i2 + 1 i1 + 1 Z 3π 2 (−1)i2 sini2 +1 θ + 1 + (−1)i1 (i1 − 2m) cos θ sini1 θ π i2 + 1 (−1)i1 sini1 +1 θ + 1 + (−1)i2 (i2 − 2m) cos θ sini2 θ dθ i1 + 1 Z π2 Z π2 = − sini1 θ cosi2 +2 θ + cosi2 θ sini1 +2 θdθ + 2 cos θ sini1 +i2 +1 θdθ 0 0 Z π 2 1 − cosi2 +1 θ sini1 +1 θ + (i1 − 2m) cos θ sini1 θ + (i2 − 2m) sin θ cosi2 θ dθ 0 i2 + 1 i1 + 1 Z π 2 sini2 +1 θ − 1 sini1 +1 θ − 1 + (i1 − 2m) cos θ sini1 θ + (i2 − 2m) cos θ sini2 θ dθ 0 i2 + 1 i1 + 1 Z π 2 i1 + i2 + 1 i1 + i2 + 1 1 = − sini1 θ cosi2 +2 θ + cosi2 θ sini1 +2 θdθ + 0 i2 + 1 i1 + 1 i2 + 1 1 = , i2 + 1 where we have used the formula Z π2 Z π i1 i2 + 1 2 sin θ cos i2 +2 θdθ = cosi2 θ sini1 +2 θdθ. 0 i 1 + 1 0 The above computations reduce Fe2 (r) to n X (−1)i Fe2 (r) =2m di ri+2m+1 i=0 i+1   2n X X X  1 1 1 1  i + 22m  ai1 bi2   i + 1 a 2i a 2i + ( + )ai ai + i1 + 1 i2 + 1 1 2 i2 + 1 r i=0 2 i1 +i2 =i i1 +i2 =i 0≤i1