This is a preprint of: “Limit cycles of discontinuous piecewise quadratic and cubic polynomial perturbations of a linear center”, Jaume Llibre, Yilei Tang, Discrete Contin. Dyn. Syst. Ser. B, vol. 24(4), 1769–1784, 2019. DOI: [10.3934/dcdsb.2018236] LIMIT CYCLES OF DISCONTINUOUS PIECEWISE QUADRATIC AND CUBIC POLYNOMIAL PERTURBATIONS OF A LINEAR CENTER JAUME LLIBRE1 AND YILEI TANG2 Abstract. We apply the averaging theory of high order for computing the limit cycles of discontinuous piecewise quadratic and cubic polynomial pertur- bations of a linear center. These discontinuous piecewise differential systems are formed by two either quadratic, or cubic polynomial differential systems separated by a straight line. We compute the maximum number of limit cycles of these discontinuous piecewise polynomial perturbations of the linear center, which can be obtained by using the averaging theory of order n for n = 1, 2, 3, 4, 5. Of course these limit cycles bifurcate from the periodic orbits of the linear center. As it was expected, using the averaging theory of the same order, the results show that the discontinuous quadratic and cubic polynomial perturbations of the linear center have more limit cycles than the ones found for continuous and discon- tinuous linear perturbations. Moreover we provide sufficient and necessary conditions for the existence of a center or a focus at infinity if the discontinuous piecewise perturbations of the linear center are general quadratic polynomials or cubic quasi–homogenous polynomials. 1. Introduction and statement of the main Results The interest on the dynamics of piecewise linear differential systems essentially started with the book of Andronov et al [1], whose Russian version appeared around the 1930’s. Due to the rich dynamics of the piecewise linear differential systems, and their applications in mechanics, electronics, economy, neuroscience, ..., these systems have been studied by researchers from many different fields, see for instance the books of Bernardo et al [4] and of Simpson [27], the survey of Makarenkov and Lamb [24], and the references mentioned in all these works. For the planar continuous piecewise linear differential systems with two zones separated by a straight line, Lum and Chua [22, 23] in 1991 conjectured that such differential systems have at most one limit cycle. In 1998 Freire, Ponce, Rodrigo and Torres [10] proved this conjecture. For other results in continuous piecewise linear differential systems see [21] and the references quoted there. While for the planar discontinuous piecewise linear differential systems with two zones separated by a straight line Han and Zhang [12] obtained differential systems 2010 Mathematics Subject Classification. 34C29, 34C25, 47H11. Key words and phrases. Periodic solution, limit cycle, discontinuous piecewise differential sys- tem, averaging theory. 1 http://www.gsd.uab.cat 2 J. LLIBRE AND Y. TANG having two limit cycles and conjectured that the maximum number of limit cycles of such class of differential systems is two. Huan and Yang [13] provided a numerical example of one of those differential system having three limit cycles. Inspired in this numerical example Llibre and Ponce [20] gave a proof of the existence of such three limit cycles in the class of these differential systems. Later on other authors also provide other discontinuous piecewise linear differential systems with two zones separated by a straight line also exhibiting three limit cycles, see [6, 7, 17]. More discussion about limit cycles of discontinuous piecewise differential systems can see references [29, 31]. Recently the averaging theory has been developed for studying the periodic so- lutions of the discontinuous piecewise differential systems. Thus Llibre, Mereu, Novaes and Teixeira [18, 19] extended the averaging theory up to order 1 and 2 for studying the periodic solutions of some discontinuous piecewise differential sys- tems using techniques of regularization. Later on Itikawa, Llibre and Novaes [14] improved the averaging theory at any order for analyzing the periodic solutions of discontinuous piecewise differential systems. We consider planar discontinuous piecewise differential systems having the line of discontinuity at y = 0 of the form ẋ = F ± (x, y, ε), (1) ẏ = G± (x, y, ε), where n X ẋ = F + (x, y, ε) =y+ εj (aj0 + aj1 x + aj2 y + aj3 x2 + aj4 xy + aj5 y 2 j=1 +aj6 x3 + aj7 x2 y + aj8 xy 2 + aj9 y 3 ), n X ẏ = G+ (x, y, ε) = −x + εj (bj0 + bj1 x + bj2 y + bj3 x2 + bj4 xy + bj5 y 2 j=1 +bj6 x3 + bj7 x2 y + bj8 xy 2 + bj9 y 3 ), if y ≥ 0, and n X ẋ = F − (x, y, ε) = y + εj (Aj0 + Aj1 x + Aj2 y + Aj3 x2 + Aj4 xy + Aj5 y 2 j=1 +Aj6 x3 + Aj7 x2 y + Aj8 xy 2 + Aj9 y 3 ), n X ẏ = G− (x, y, ε) = −x + εj (Bj0 + Bj1 x + Bj2 y + Bj3 x2 + Bj4 xy + Bj5 y 2 j=1 +Bj6 x3 + Bj7 x2 y + Bj8 xy 2 + Bj9 y 3 ), if y ≤ 0, and where n ∈ N, all parameters aji , bji , Aji , Bji , ε ∈ R, and the pertur- bation parameter |ε| is small enough. Here N is the set of positive integers and R is the set of real numbers. Notice that system (1) is a discontinuous piecewise differential system with the discontinuity straight line y = 0. As usual the dot denotes derivative with respect to an independent real variable t. In this paper we study the limit cycles of the discontinuous piecewise quadratic (i.e. when all the cubic monomials in (1) are zero) and cubic polynomial differential system (1), which bifurcate from the periodic orbits of the linear center ẋ = y, ẏ = −x. A classical problem for smooth differential systems is the weak 16th LIMIT CYCLES OF DISCONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 3 Hilbert problem, which essentially asks for the maximal number of limit cycles that bifurcate from the periodic orbits of a center when this is perturbed inside a class of polynomial differential systems with a fixed degree, see for more details [2, 3, 15, 28]. Here we are extending this problem to the non–smooth differential system (1). We denote by L2 (n) and L3 (n) the maximum number of limit cycles of the discontinuous piecewise polynomial differential system (1) with degree 2 and 3 respectively which can be obtained using the averaging theory of order n described in section 2. Then we have the following results. Theorem 1. For n = 1, 2, 3, 4, 5 we have that L2 (n) = 2, 3, 5, 6, 8, and L3 (n) = 3, 5, 8, 11, 13, respectively. Iliev in [16] studied the maximum number of limit cycles LI (n) coming from the perturbation of the linear center ẋ = y, ẏ = −x when this center is perturbed inside the class of all polynomial differential systems of degree n. Buzzi, Pessoa and Torregrosa in [7] found the maximum number of limit cycles L1 (n), that bifurcate from the periodic orbits of the linear center ẋ = y, ẏ = −x when this center is perturbed inside the class of all discontinuous piecewise linear differential systems separated by a straight line. Their results for n = 1, 2, 3, 4, 5 together with the results of Theorem 1 are given in Table 1. Order n L1 (n) L2 (n) L3 (n) LI (n) 1 1 2 3 0 2 1 3 5 1 3 2 5 8 1 4 3 6 11 2 5 3 8 13 2 Table 1. Maximum number of limit cycles bifurcating from the periodic orbits of the linear center using averaging theory of order n. If there exists a neighborhood of the infinity in the Poincaré disc [9, Chapter 5] filled of periodic orbits, then we say that system (1) has a center at infinity. If there exists a neighborhood of the infinity in the Poincaré disc where all the orbits spiral going to or coming from the infinity, then we say that system (1) has a focus at infinity. We shall investigate the problem of the existence of a center or a focus at infinity under small perturbations, but before we need some definitions. A planar polynomial differential system (2) ẋ = P (x, y), ẏ = Q(x, y), where P (x, y) and Q(x, y) are non–zero polynomials, is quasi–homogeneous if there exist s1 , s2 , d ∈ N such that for all positive number α they satisfy P (αs1 x, αs2 y) = αs1 +d−1 P (x, y), Q(αs1 x, αs2 y) = αs2 +d−1 Q(x, y), Then as usual (s1 , s2 ) are the weight exponents, d is the weight degree with respect to the weight exponents, and w = (s1 , s2 , d) is the weight vector of the quasi– homogeneous polynomial differential system (2). 4 J. LLIBRE AND Y. TANG By Proposition 19 of Giné, Grau and Llibre [11], an irreducible quasi-homogeneous but non-homogeneous cubic ordinary polynomial differential system can be written in one of the following forms: (I) ẋ = y(a1 x + b1 y 2 ), ẏ = c1 x + d1 y 2 , with b1 c1 6= 0. (II) ẋ = a2 x2 + b2 y 3 , ẏ = c2 xy, with a2 b2 c2 6= 0. (III) ẋ = a3 y 3 , ẏ = b3 x2 with a3 b3 6= 0. (IV ) ẋ = x(a4 x + b4 y 2 ), ẏ = y(c4 x + d4 y 2 ), with a4 d4 6= 0. (V ) ẋ = a5 xy 2 , ẏ = b5 x2 + c5 y 3 , with a5 b5 c5 6= 0. (V I) ẋ = a6 xy 2 , ẏ = b6 x + c6 y 3 , with a6 b6 c6 6= 0. (V II) ẋ = a7 x + b7 y 3 , ẏ = c7 y, with a7 b7 c7 6= 0. Perturbing the linear center by discontinuous cubic quasi-homogenous but non- homogeneous polynomials, we obtain the following 7 systems: ẋ = y + εy(a1 x + b1 y 2 ), ẏ = −x + ε(c1 x + d1 y 2 ) if y ≥ 0, (3) ẋ = y + εy(A1 x + B1 y 2 ), ẏ = −x + ε(C1 x + D1 y 2 ) if y ≤ 0 where b1 c1 B1 C1 6= 0, ẋ = y + εa2 x2 + b2 y 3 , ẏ = −x + εc2 xy if y ≥ 0, (4) ẋ = y + εA2 x2 + B2 y 3 , ẏ = −x + εC2 xy if y ≤ 0, where a2 b2 c2 A2 B2 C2 6= 0, ẋ = y + εa3 y 3 , ẏ = −x + εb3 x2 if y ≥ 0, (5) ẋ = y + εA3 y 3 , ẏ = −x + εB3 x2 if y ≤ 0, where a3 b3 A3 B3 6= 0, ẋ = y + εx(a4 x + b4 y 2 ), ẏ = −x + εy(c4 x + d4 y 2 ) if y ≥ 0, (6) ẋ = y + εx(A4 x + B4 y 2 ), ẏ = −x + εy(C4 x + D4 y 2 ) if y ≤ 0, where a4 d4 A4 D4 6= 0, ẋ = y + εa5 xy 2 , ẏ = −x + εb5 x2 + c5 y 3 if y ≥ 0, (7) ẋ = y + εA5 xy 2 , ẏ = −x + εB5 x2 + C5 y 3 if y ≤ 0, where a5 b5 c5 A5 B5 C5 6= 0, ẋ = y + εa6 xy 2 , ẏ = −x + εb6 x + c6 y 3 if y ≥ 0, (8) ẋ = y + εA6 xy 2 , ẏ = −x + εB6 x + C6 y 3 if y ≤ 0, and a6 b6 c6 A6 B6 C6 6= 0, and ẋ = y + ε(a7 x + b7 y 3 ), ẏ = −x + εc7 y if y ≥ 0, (9) ẋ = y + ε(A7 x + B7 y 3 ), ẏ = −x + εC7 , if y ≤ 0, where a7 b7 c7 A7 B7 C7 6= 0. A center is called a global center when the periodic orbits surrounding the center filled the whole plain except the center itself. Theorem 2. Assume n = 1 in system (1). (i) System (1) has neither centers nor foci at infinity if the discontinuous poly- nomial perturbations are of degree 2 (i.e. if aji = bji = Aji = Bji = 0 for i = 6, ..., 9). LIMIT CYCLES OF DISCONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 5 (ii) The unique systems from (3) to (9) which can have a center or a focus at infinity are the systems (3) or (9). (iii) The infinity of system (9) is a focus. System (3) has a focus or a center at infinity if −b1 ε < 0 and −B1 ε < 0, and it has a center at infinity if −b1 ε < 0, −B1 ε < 0, a1 = −2d1 and A1 = 0 = D1 , which is a global center. 2. Averaging theory and the Descartes Theorem Using the polar coordinates (r, θ) such that x = r cos θ and y = r sin θ, the differential system (1) in these coordinates becomes ( + dr P (θ, r, ε) if 0 ≤ θ ≤ π, (10) = dθ P − (θ, r, ε) if − π ≤ θ ≤ 0, Pk j ± where P ± (θ, r, ε) = j=1 ε Pj (θ, r) + ε Q (θ, r, ε) with k ∈ N, θ ∈ S1 and k+1 ± r ∈ R+ , the functions Pj± : S1 × R+ → R for j = 1, 2, . . . , k, and Q± : S1 × R+ × (−ε0 , ε0 ) → R are analytic. Here ε0 > 0 and R+ = [0, ∞). The averaged function fj : R+ → R of order j for the differential equation (10) is defined as yj+ (π, r) − yj− (−π, r) (11) fj (r) = , j = 1, 2, . . . , k, j! where yj± for j = 1, 2, 3, 4, 5 are Z θ ± y1 (θ, r) = P1± (φ, r)dφ, 0 Z θ   y2± (θ, r) = 2P2± (φ, r) + 2∂P1± (φ, r)y1± (φ, r) dφ, 0 Z θ  y3± (θ, r) = 6P3± (φ, r) + 6∂P2± (φ, r)y1± (φ, r) 0  + 3∂ 2 P1± (φ, r)y1± (φ, r)2 + 3∂P1± (φ, r) y2± (φ, r) dφ, Z θ (12) y4± (θ, r) = 24P4± (φ, r) + 24∂P3±(φ, r)y1± (φ, r) 0 + 12∂ 2 P2± (φ, r)y1± (φ, r)2 + 12∂P2± (φ, r)y2± (φ, r) + 12∂ 2 P1± (φ, r)y1± (φ, r)y2± (φ, r)  + 4∂ 3 P1± (φ, r)y1± (φ, r)3 + 4∂P1± (φ, r)y3± (φ, r) dφ, Z θ ± y5 (θ, r) = 120P5±(φ, r) + 120∂P4±(φ, r)y1± (φ, r) 0 + 60∂ 2 P3± (φ, r)y1± (φ, r)2 + 60∂P3± (φ, r)y2± (φ, r) + 60∂ 2 P2± (φ, r)y1± (φ, r)y2± (φ, r) + 20∂ 3 P2± (φ, r)y1± (φ, r)3 + 20∂P2± (φ, r)y3± (φ, r) + 20∂ 2P1± (φ, r)y1± (φ, r)y3± (φ, r) 6 J. LLIBRE AND Y. TANG + 15∂ 2 P1± (φ, r)y2± (φ, r)2 + 30∂ 3 P1± (φ, r)y1± (φ, r)2 y2± (φ, r)  + 5∂ 4 P1± (φ, r)y1± (φ, r)4 + 5∂P1± (φ, r)y4± (φ, r) dφ. From [14] we have the following result. Theorem 3. Suppose that j is the first integer such that the averaged function fi = 0 for i = 1, 2, . . . , j − 1 and fj 6= 0. If there is r∗ ∈ R+ such that fj (r∗ ) = 0 and fj′ (r∗ ) 6= 0, then for |ε| 6= 0 small enough there is a 2π–periodic solution r(θ, ε) of (10) such that r(0, ε) → r∗ when ε → 0. Note that the simple positive zeros of the averaged function fj provides limit cycles of the differential equation (10). We shall use the following version of the Descartes Theorem as it is proved in [5]. Theorem 4 (Descartes theorem). Consider the real polynomial p(x) = ai1 xi1 + ai2 xi2 + . . .+ air xir with 0 = i1 < i2 < . . . < ir . If aij aij+1 < 0, we say that we have a variation of sign. If the number of variations of signs is m, then the polynomial p(x) has at most m positive real roots. Furthermore, always 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 We write the discontinuous piecewise cubic polynomial differential system (1) in polar coordinates, obtaining a differential system (ṙ, θ̇). After taking θ as the new independent variable we get a differential equation dr/dθ, and doing Taylor series expansion of dr/dθ with respect to the variable ε at ε = 0 we obtain the differential equation (10) associated to system (1). Since system (1) is a polynomial differential system, the functions Pj± (θ, r) and Q± j (θ, r, ε) are analytic. Moreover the differential equation dr/dθ in the form (10) is 2π−periodic because the variable θ appears through the sinus and cosinus functions. In order to apply Theorem 3 to our differential equation dr/dθ it suffices to take an open interval D = {r : 0 < r < r0 } ⊂ R+ , where the unperturbed system can have periodic orbits r(θ) such that r(0) = r with 0 < r < r0 . Here we only give the explicit expressions of  + 1 3 P1 (aij , bij , θ, r) = r 4b19 cos(2θ) − 2b18 sin(2θ) + a19 sin(4θ) − a17 sin(4θ) 8 − 4a16 cos(2θ) − 2a17 sin(2θ) + b18 sin(4θ) + a18 cos(4θ) − 2a19 sin(2θ) + b17 cos(4θ) − a16 cos(4θ) − 2b16 sin(2θ)  − b16 sin(4θ) − b19 cos(4θ) − a18 − 3a16 − 3b19 − b17  + r2 2b15 sin(3θ) − 2a14 sin(3θ) − 2b14 cos θ − 2a14 sin θ − 2b13 sin(3θ) − 6a13 cos θ + 2b14 cos(3θ) − 2a13 cos(3θ) LIMIT CYCLES OF DISCONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 7  + 2a15 cos(3θ) − 2b13 sin θ − 2a15 cos θ − 6b15 sin θ  + r − 4a12 sin(2θ) − 4a11 cos(2θ) − 4a11 + 4b12 cos(2θ)   − 4b11 sin(2θ) − 4b12 − b10 sin θ − a10 cos θ ,   P2+ (aij , bij , θ, r) = − a20 cos θ − b20 sin θ − b22 − (b22 − a21 ) cos2 θ + (a22 + b21 ) sin θ cos θ r  + − b25 sin θ − (b24 + a25 ) cos θ − (a24 + b23 − b25 ) sin θ cos2 θ   + (a25 − a23 + b24 ) cos3 θ r2 + − b29 − (a29 + b28 ) sin θ cos θ + (2b29 − a28 − b27 ) cos2 θ + (b28 + a29 − a27 − b26 ) sin θ cos3 θ   + (b27 − b29 − a26 + a28 ) cos4 θ r3 − b10 sin θ + a10 cos θ + (b12 + (b11 + a12 ) sin θ cos θ + (a11 − b12 ) cos2 θ)r + ((a15 + b14 ) cos θ + b15 sin θ + (b13 + a14 − b15 ) sin θ cos2 θ + (a13 − b14 − a15 ) cos3 θ)r2 + (b19 + (b18 + a19 ) sin θ cos θ + (a18 − 2b19 + b17 ) cos2 θ  + (b19 − a18 − b17 + a16 ) cos4 θ + (a17 + b16 − a19 − b18 ) sin θ cos3 θ)r3  b10 cos θ − a10 sin θ + (−a12 + (b12 − a11 ) sin θ cos θ + (a12 + b11 ) cos2 θ)r + (−a15 sin θ + (b15 − a14 ) cos θ + (a15 + b14 − a13 ) sin θ cos2 θ + (b13 − b15 + a14 ) cos3 θ)r2 + (−a19 + (b18 + 2a19 − a17 ) cos2 θ + (b19 − a18 ) sin θ cos θ + (b17 − a16 − b19 + a18 ) sin θ cos3 θ  + (b16 − b18 + a17 − a19 ) cos4 θ)r3 /r. We omit the explicit expressions of Pk+ (aij , bij , θ, r) for k = 3, 4, 5 because they are quite large. Moreover, we have Pk− (Aij , Bij , θ, r) =Pk+ (aij , bij , θ, r), for k = 1, 2, 3, 4, 5. From (12) we compute the functions yj+ (θ, r) and yj− (θ, r) for j = 1, . . . , 5. After we compute the averaged functions fj (r) for j = 1, . . . , 5 by using formulas (11). Thus the averaged function of first order is f1 (r) = η13 r3 + η12 r2 + η11 r + η10 , where π  η13 = − A18 − B17 − 3A16 − 3B19 − b17 − a18 − 3b19 − 3a16 , 8 2  η12 = − b13 + A14 + 2B15 − 2b15 − a14 + B13 , 3 π  η11 = − B12 − a11 − b12 − A11 , 2 η10 = − 2b10 + 2B10 . The rank of the Jacobian matrix of the function M1 = (η13 , η12 , η11 , η10 ) with respect to the parameters a1i , b1i , A1i , B1i , i = 0, 1, .., 9 is maximal, i.e. it is 4. 8 J. LLIBRE AND Y. TANG Then the coefficients η13 , η12 , η11 and η10 are linearly independent in their variables. Clearly f1 (r) = 0 has at most three solutions in D. Thus, by Theorems 3 and 4 it follows that at most 3 limit cycles can bifurcate from the periodic orbits of the linear system using the averaging theory of first order, and from the last part of Theorem 4 there are systems (1) with three limit cycles. Solving η13 for A16 , η12 for B15 , η11 for B12 and η10 for B10 , we obtain that f1 (r) ≡ 0. Applying the averaging theory of order two, we get the second averaged function f2 (r) = η25 r5 + η24 r4 + η23 r3 + η22 r2 + η21 r + η20 , where π η25 = 18πa16 a18 − 2A16 B18 − 30A16 B16 − 6B17 B18 − 10B17 B16 − 3A218 π 128 − 18B17 B19 π − 18A16 A18 π − 54A16 B19 π − 6A18 B17 π − 18A18 B19 π 2 2 + 18πa18 b19 + 6πa18 b17 − 3B17 π − 27B19 π − 27A216 π + 3πa218 + 27πa216 − 6b19 b16 + 27πb219 + 3πb217 − 2B17 A17 + 2B17 A19 + 6a18 a17 − 2a18 b16 + 6a16 a19 + 2b17 a19 − 2b17 a17 − 10b17 b16 − 6b17 b18 + 2b19 a17 + 2a18 b18 + 54πa16 b19 + 18πa16 b17 − 10b19 b18 + 10a18 a19 + 30b19 a19 − 30a16 b16 − 2a16 b18 + 10a16 a17 + 18πb17 b19 − 18A16 B17 π − 2A18 B16 + 6A18 A17 + 10A18 A19 − 6B19 B16 + 2B19 A17 + 2A18 B18 + 10A16 A17 + 6A16 A19  + 30B19 A19 − 10B19 B18 , 1 η24 = 16B14 B17 /45 − 8a17 b15 /15 + 8a18 b14 /45 + 16a19 b15 /15 − 4b13 b16 /3 2 − 8A15 A16 /15 + 8B13 B18 /15 − 16b15 b16 /15 − 16b14 b17 /45 − 8b13 b18 /15 + 4B13 B16 /3 − 8A15 B17 /45 + 8A17 B15 /15 + 8A17 B13 /15 + 8a14 a19 /15 − 4A13 B17 /45 − 28A13 A16 /15 − 8A13 A18 /9 − 8a14 b16 /15 + 4a13 b17 /45 + 4A16 B14 /15 − 32A15 B19 /15 − 16A19 B15 /15 + 16B15 B16 /15 + 8a15 a16 /15 + 8a13 b19 /5 + 4a14 a17 /15 − 8A13 B19 /5 + 8A14 B16 /15 − 8A18 B14 /45 − 8A14 A19 /15 + 16B15 B18 /15 − 32A15 A18 /45 + 32a15 a18 /45 + 28a13 a16 /15 + 8a13 a18 /9 + 8a15 b17 /45 + 32a15 b19 /15 − 4a16 b14 /15 − 16b15 b18 /15 − 8a17 b13 /15 + 5πa18 b13 /12 − 4A14 A17 /15 + 5πb15 b19 /2 + 5πb13 b17 /12 + 5πa14 a16 /4 + 5πa18 b15 /6 + 5πb13 b19 /4 + 5πb15 b17 /6 + 5πa14 a18 /12 + 5A16 B13 π/4 + 5A18 B15 π/6 + 5A14 B17 π/12 + 5A14 A18 π/12 + 5A16 B15 π/2 + 5A14 B19 π/4 + 5A14 A16 π/4 + 5B15 B19 π/2 + 5A18 B13 π/12 + 5B15 B17 π/6 + 5B13 B19 π/4 + 5B13 B17 π/12 + 5πa14 b17 /12  + 5πa14 b19 /4 + 5πa16 b13 /4 + 5πa16 b15 /2 , 1 2 η23 = 32a14 b15 /9 − 32B15 /9 − 8A214 /9 − 8B13 2 /9 + 16a14 b13 /9 − B27 π/4 2 − 32B13 B15 /9 + 32b13 b15 /9 − 16A14 B13 /9 − 32A14 B15 /9 − A28 π/4 − 3B29 π/4 − 3A26 π/4 − b27 π/4 − a28 π/4 − 3a26 π/4 + 3π 2 a16 b12 /4 LIMIT CYCLES OF DISCONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 9 − 3b29 π/4 + 32b215 /9 − πa11 b18 /16 + 3π 2 a11 b19 /4 − 11A11 B16 π/16 + π 2 a18 b12 /4 − A11 B17 π 2 /4 − 3A11 B19 π 2 /4 − 3A16 B12 π 2 /4 − 3A11 A16 π 2 /4 − 3B12 B19 π 2 /4 − B12 B17 π 2 /4 − A18 B12 π 2 /4 − A11 A18 π 2 /4 + A12 B17 π/8 + A18 B11 π/8 − B14 B13 π/4 + 8a214 /9 − A11 B18 π/16 − 7A17 B12 π/16 + 3A18 A12 π/8 + 3A11 A19 π/16 + π 2 a11 a18 /4 + π 2 a11 b17 /4 + 3a16 a12 π/8 − b17 b11 π/8 + 8b213 /9 + π 2 b12 b17 /4 + 3π 2 b12 b19 /4 + πa12 b17 /8 + πa18 b11 /8 + A14 A13 π/4 − 7πa17 b12 /16 − 11b12 b16 π/16 + 3A16 A12 π/8 + A11 A17 π/16 + A14 A15 π/4 − A13 B13 π/2 + 9b19 a12 π/8 − b14 b15 π/4 + 3b12 a19 π/16 − 11B12 B16 π/16 + 3B12 A19 π/16 − B17 B11 π/8 + 3B19 B11 π/8 − 9B12 B18 π/16 + 9B19 A12 π/8 + 3a18 a12 π/8 − 11a11 b16 π/16 + a11 a17 π/16 − 3a16 b11 π/8 + 3b19 b11 π/8 + a15 a14 π/4 − B14 B15 π/4 + a15 b15 π/2 − b14 b13 π/4 − b13 a13 π/2 + a14 a13 π/4 − 9b12 b18 π/16  − 3A16 B11 π/8 + B15 A15 π/2 + 3π 2 a11 a16 /4 + 3a11 a19 π/16 , 1 η22 = 4B23 /3 − 4a24 /3 − 4b23 /3 − 8b25 /3 + 4A24 /3 + 8B25 /3 − 4b11 b13 /3 2 − 4A12 A14 /3 − 16b12 b14 /9 + 8a12 b15 /3 − 20a13 b12 /9 − 8A12 B15 /3 + 8a11 a15 /9 − 4a11 b14 /9 − 8A15 B12 /9 − 8a17 b10 /3 + 4a11 a13 /9 − 8b10 b18 /3 − 4b10 b16 + 8a10 b19 + 4a12 a14 /3 + 8a10 a18 /3 + 4a10 b17 /3 + 8A17 B10 /3 + 16B12 B14 /9 − 4A10 B17 /3 − 8A10 B19 + 4B11 B13 /3 − 8A10 A18 /3 + 8B10 B18 /3 + 4B10 B16 + 4A11 B14 /9 + 20A13 B12 /9 − 8A11 A15 /9 − 4A11 A13 /9 − 4A10 A16 + 8a15 b12 /9 + 4a10 a16 + πa11 a14 + πa11 b13 + πa14 b12 + πb12 b13 + 3a18 b10 π/4 + 9a16 b10 π/4 + 3b10 b17 π/4 + 2πb12 b15 + 9b10 b19 π/4 + 2πa11 b15 + B12 B13 π + A14 B12 π + A11 B13 π + A11 A14 π + 3B10 B17 π/4 + 3A18 B10 π/4 + 9A16 B10 π/4 + 9B10 B19 π/4  + 2B12 B15 π + 2A11 B15 π , 1 η21 = 16b10 b15 /3 − 16B10 B15 /3 + 8b10 b13 /3 + 8a14 b10 /3 − B10 B13 8/3 2 − 8A14 B10 /3 − B22 π − A211 π 2 /4 − B12 2 2 π /4 + π 2 b212 /4 + π 2 a211 /4 − A21 π − a21 π − b22 π + 2B15 A10 π + a10 a14 π + 2a10 b15 π − b14 b10 π − 2b10 a13 π − b12 b11 π/2 + a11 a12 π/2 + b12 a12 π/2 + π 2 a11 b12 /2 − a11 b11 π/2 + A11 A12 π/2 − B14 B10 π − 2A13 B10 π + B12 A12 π/2  − B12 B11 π/2 + A14 A10 π − A11 B12 π 2 /2 − A11 B11 π/2 , η20 = − 2b20 + 2B20 − 2A10 A11 + b10 b12 π/2 + a11 b10 π/2 + A11 B10 π/2 − 2A10 B12 + 2a10 a11 + B10 B12 π/2 + 2a10 b12 + 2B10 B11 − 2b10 b11 . Because the rank of the Jacobian matrix of the function M2 = (η25 , η24 , η23 , η22 , η21 , η20 ) with respect to its variables ali , bli , Ali , Bli , l = 1, 2, i = 0, 1, .., 9 is maximal, i.e. it is 6, the functions η25 , η24 , η23 , η22 , η21 and η20 are linearly independent in 10 J. LLIBRE AND Y. TANG their variables. Hence by Theorem 4, the equation f2 (r) = 0 has at most 5 roots in D and therefore at most 5 limit cycles of system (1) can bifurcate from the periodic orbits of the linear system using the averaging theory of order two, and there are systems (1) having 5 limit cycles. Solving η25 , η24 , η23 , η22 , η21 , η20 for A21 , A24 , B20 , B27 , A17 , a17 , we get f2 (r) ≡ 0, and we can use the averaging theory of order three. Then the third averaged function is rf3 (r) = η38 r8 + η37 r7 + η36 r6 + η35 r5 + η34 r4 + η33 r3 + η32 r2 + η31 r + η30 . The functions η3j for j = 0, . . . , 8 are linearly independent in their variables, because the rank of the Jacobian matrix M3 = (η30 , . . . , η38 ) with respect to its variables is maximal, i.e. it is 9. We do not provide their explicit expressions, because they are very long. Therefore the equation f3 (r) = 0 has at most 8 zeros in D and at most 8 limit cycles of system (1) can bifurcate from the periodic orbits of the linear system using the averaging theory of order three, and again there systems (1) with 8 limit cycles. By choosing conveniently some variables to cancel the coefficients η3j for j = 0, . . . , 8 we do the third order averaged function identically zero. So we can compute the fourth averaged function f4 (r). And by doing this f4 (r) identically zero we also can compute the fifth averaged function f5 (r). These two averaged functions have the form r2 f4 (r) = η411 r11 + η410 r10 + η49 r9 + η48 r8 + η47 r7 + η46 r6 +η45 r5 + η44 r4 + η43 r3 + η42 r2 + η41 r + η40 , r2 f5 (r) = η513 r13 + η512 r12 + η511 r11 + η510 r10 + η59 r9 + η58 r8 + η57 r7 +η56 r6 + η55 r5 + η54 r4 + η53 r3 + η52 r2 + η51 r + η50 . We can prove that the coefficients ηij are linearly independent in their variables. Their expressions are very long so we do not give them here. As a result of these calculations it follows that f4 (r) = 0 (resp. f5 (r) = 0) has at most 11 (resp. 13) solutions in D, and therefore at most 11 (resp. 13) limit cycles of system (1) can bifurcate from the periodic orbits of the linear center, and there are systems (1) having 11 (resp. 13) limit cycles. Now we consider the discontinuous piecewise quadratic polynomial perturbations in system (1). Doing in the previous averaged functions fk (r) for k = 1, 2, 3, 4, 5 the coefficients aji = bji = Aji = Bji = 0 for i = 6, ..., 9, we obtain the averaged functions for the quadratic polynomial perturbations in system (1). From these averaged functions we obtain the numbers L2 (n) for n = 1, 2, 3, 4, 5 in Theorem 1. This completes the proof of Theorem 1. 4. Proof of Theorem 2 Consider system (1) having the linear center ẋ = y, ẏ = −x and being perturbed inside the class of discontinuous piecewise quadratic polynomial differential systems ẋ = y + εF1± (x, y), (13) ẏ = −x + εG± 1 (x, y), LIMIT CYCLES OF DISCONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 11 where F1± (x, y) = a± ± ± ± 2 ± ± 2 0 + a1 x + a2 y + a3 x + a4 xy + a5 y , G± ± ± ± ± 2 ± ± 2 1 (x, y) = b0 + b1 x + b2 y + b3 x + b4 xy + b5 y , are defined in the regions {y ≥ 0} and {y ≤ 0}, and all parameters a± ± j , bj ∈ R for j = 0, 1, .., 5. It is not difficult to find that systems (13) have not a center at infinity because first the equation x(b± 2 ± ± 2 ± 2 ± ± 2 3 x + b4 xy + b5 y ) − y(a3 x + a4 xy + a5 y ) = 0 has at least a real solution because it is a cubic homogeneous polynomial, and therefore system (13) has singularities at infinity. Moreover, by the analysis of the local phase portraits of the infinite singularities of quadratic systems in [8] or [26], it follows that the infinity of system (13) cannot be a center or a focus, because always some orbits have their α– or ω–limits at some infinite singularity. Hence statement (i) of Theorem 2 is proved. Perturbing the linear center by discontinuous cubic quasi-homogenous but non- homogeneous polynomials, we obtain ẋ = y + εF2± (x, y), (14) ẏ = −x + εG± 2 (x, y), where F2± (x, y) and G±2 (x, y) belonging to one of systems (I)− (V II) in the Section 1, are defined in the regions {y ≥ 0} and {y ≤ 0}. Notice that system (14) has not a center or a focus at infinity if F2± (x, y) and G± 2 (x, y) have the forms (III) − (V I) because one of singularities at infinity of these systems is a saddle, node, saddle- node or a nilpotent equilibrium by Poincaré transformations (15) x = 1/z, y = u/z, and x = v/z, y = 1/z together with the time variables dτ = dt/z 2 . For simplicity, we only give the compactification systems for (14) when F2± (x, y) and G± 2 (x, y) belonging to system (IV ). System ẋ = y + εx(a4 x + b4 y 2 ), (16) ẏ = −x + εy(c4 x + d4 y 2 ), around the equator of the Poincaré sphere can be written respectively in u̇ = −ε(a4 − c4 )uz − z 2 − ε(b4 − d4 )u3 − u2 z 2 , ż = −z(εa4 z + εb4 u2 + uz 2 ), and u̇ = ε(b4 − d4 )v + z 2 + ε(a4 − c4 )v 2 z + v 2 z 2 , (17) ż = z(−εd4 − εc4 vz + vz 2 ), after changes (15), where a4 d4 6= 0. Notice that the origin of (17), which is located at the end of the y–axis and is a singularity at infinity of system (16) of hyperbolic type if b4 − d4 6= 0 or of semi-hyperbolic type if b4 − d4 = 0. Then by Theorems 2.15 and 2.19 of [9] this singularity can only be a saddle, node or saddle-node. Hence there exist no centers or foci at infinity of systems (16). When F2± (x, y) and G± 2 (x, y) have the forms (I), (II), or (V II), systems (14) becomes (18) ẋ = y + εy(a1 x + b1 y 2 ), ẏ = −x + ε(c1 x + d1 y 2 ), 12 J. LLIBRE AND Y. TANG (19) ẋ = y + ε(a2 x2 + b2 y 3 ), ẏ = −x + εc2 xy, (20) ẋ = y + ε(a7 x + b7 y 3 ), ẏ = −x + εc7 y, respectively, which have no singularities at infinity at the endpoints of the y-axis from simple calculations. By the first change of (15), systems (18)-(20) can be transformed into u̇ = (c1 ε − 1)z 2 − u2 z 2 − ε(a1 − d1 )u2 z − b1 εu4 , (21) ż = −uz(a1εz + εb1 u2 + z 2 ), u̇ = −ε(a2 − c2 )uz − z 2 − u2 z 2 − b2 εu4 , (22) ż = −z(εa2 z + εb2 u3 + uz 2 ), and u̇ = −z 2 − ε(a7 − c7 )uz 2 − εb7 u4 − u2 z 2 , (23) ż = −z(εa7 z 2 + εb7 u3 + uz 2 ), respectively. Note that all systems (21)-(23) have a unique singularity at the end points of the x-axis, which corresponds to the origin denoted by Cj = (0, 0) for j = I, II, V II. We will analyze the local phase portrait of the singularity Cj . First we consider the properties of CI for system (21). Notice that the vector field of (21) is invariant under the change of variables (u, z, t) → (−u, z, −t). Therefore, system (21) is symmetric with respect to the z-axis. Thus, we only need to consider the right half-plane u ≥ 0 for studying the local phase portrait of CI . Using the change u1 = u2 , z1 = z, system (21) becomes u̇1 = −2εb1 u21 − 2ε(a1 − d1 )u1 z + 2(c1 ε − 1)z 2 − u1 z 2 , (24) ż = −z(εb1 u1 + εa1 z + z 2 ). We need to use the following notions. Consider the analytic differential system ẋ = Xm (x, y) + Φm (x, y) := X(x, y), (25) ẏ = Ym (x, y) + Ψm (x, y) := Y (x, y), where Xm (x, y) and Ym (x, y) are homogeneous polynomials of degree m ≥ 1 such m that p simultaneously do not vanish, and Φm (x, y), Ψm (x, y) = o(r ) when r = 2 2 x + y → 0. Let the origin O be an isolated singularity of (25). In order to see when there exist orbits connecting with O, by Lemmas 1 and 3 in [25, Chapter 2] we only need to discuss the orbits along exceptional directions of system (25) at O. Applying the polar coordinate changes x = r cos θ and y = r sin θ, system (25) can be written in 1 dr Hm (θ) + o(1) (26) = , as r → 0, r dθ Gm (θ) + o(1) where Gm (θ) = cos θYm (cos θ, sin θ) − sin θXm (cos θ, sin θ). Hm (θ) = sin θYm (cos θ, sin θ) + cos θXm (cos θ, sin θ). Hence a necessary condition for θ = θ0 to be an exceptional direction is Gm (θ0 ) = 0. For our system (24) we calculate   G2 (θ) = sin θ b1 ε cos2 θ + (−a1 ε + 2ε(a1 − d1 )) sin θ cos θ + (−2c1 ε + 2) sin2 θ . LIMIT CYCLES OF DISCONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 13 When −b1 ε < 0, the equation G2 (θ) = 0 has only two zeros 0 and π if θ ∈ [0, 2π). When θ → 0 equation (26) has the form 1 dr H2 (θ) + o(1) 2 = = − + O(1). r dθ G2 (θ) + o(1) θ Rθ − 2 +O(1) Then r = r1 e θ0 θ dθ → +∞ as θ → 0. Thus, by a similar proof of Theorem 10.1 and Theorem 10.5 in [30], we obtain that the u-axis is the only orbit connecting with the origin of system (24) if −b1 ε < 0. Therefore the discontinuous piecewise cubic polynomial differential system (3) has a center or a focus at infinity when −b1 ε < 0 and −B1 ε < 0, where b1 c1 B1 C1 6= 0. Second we consider the local phase portrait of CII = (0, 0) for system (22). By the blow-up u = u2 z along the u-axis together with a time scaling dt = dt1 /z, system (22) becomes u̇ = −1 + c2 εu, (27) ż = −z(a2 ε + uz 2 + b2 εu3 z 2 ), where we still write u2 as u for simplicity. The u-axis system (27) has only one singularity (1/(c2 ε), 0), which is a saddle or node if a2 c2 ε 6= 0. Therefore there exist neither centers nor foci at infinity of systems (4). Finally we consider the local phase portrait of CV II = (0, 0) for system (23). By the blow-up u = u7 z along the u-axis together with a time scaling dt = dt7 /z, system (23) becomes u̇ = −1 + c7 εuz, (28) ż = −z 2 (a7 ε + uz + b7 εu3 z), where we still write u7 as u for simplicity. The u-axis system (28) has no singu- larities. Notice that the u-axis is an orbit of system (28) and no other orbits can connect with the u-axis. Furthermore for system (23) we calculate G̃2 (θ) = sin3 θ in (26), implying that all possible orbits connecting with CV II must be along the direction of u-axis. Therefore we obtain that no orbits can go to or come from the singularities at infinity of system (20), consequently no orbits can go to or come from the singularities at infinity of system (9) This completes the proof of statement (ii) in Theorem 2. Now we shall study the existence or not of a center or a focus at infinity for the discontinuous piecewise cubic polynomial differential systems (3) and (9), for this we shall use the averaging theory for proving statement (iii) in Theorem 2. We claim that the infinity is a focus for system (9). In fact, from the formula (11), we can compute the averaged functions of order ≤ 4 for system (9). The averaged function of first order is f˜1 (r) = −rπ(a7 + c7 + A7 + C7 )/2. Then we solve a7 = −c7 − A7 − C7 from f˜1 (r) ≡ 0. Applying the averaging theory of order two, we get the second averaged function f˜2 (r) = 3πr3 (B7 − b7 )(A7 + C7 )/32. 14 J. LLIBRE AND Y. TANG Solving f˜2 (r) ≡ 0, we obtain b7 = B7 or A7 = −C7 . We use the averaging theory of order three and get the third averaged function  35 3 1  f˜3 (r) = π(A7 +C7 )r − (B72 −b27 )r4 + π(B7 −b7 )(A7 +C7 )r2 + (C7 +c7 )(A7 +c7 ) . 512 32 4 When A7 = −C7 we have f3 (r) ≡ 0, and when b7 = B7 we have f˜3 = π(C7 + ˜ c7 )(A7 + c7 )(A7 + C7 )r/4. Hence we get (C7 + c7 )(A7 + c7 )(A7 + C7 ) = 0 from f˜3 (r) ≡ 0. We compute the averaged function of order 4 for system (9) and we get  3 33 2  f˜41 (r) = B7 πr3 − (2A7 + 3C7 )(−C7 + A7 )2 − B7 (2A7 + 5C7 )r4 128 1024 if b7 = B7 and A7 + c7 = 0, or 3   f˜42 (r) = B7 πr3 − (16A37 − 8A27 C7 − 32A7 C72 + 24C73 )− (22A7 B72 + 55B72 C7 )r4 1024 if b7 = B7 and C7 + c7 = 0, or  99 2 4 3  f˜43 (r) = B7 C7 πr3 − B7 r − C72 1024 32 if A7 = −C7 . Therefore, the functions f˜41 (r), f˜42 (r) and f˜43 (r) cannot be identically zero, otherwise we have a contradiction with the fact that B7 C7 6= 0. So there are no periodic orbits, and we can obtain some isolated spiral orbit as close as we want to infinity. Therefore the infinity is a focus for system (9) and the claim is proved. For system (3) we can compute the averaged functions of order ≤ 5. Here we omit the tedious calculations and only show the results. Doing all the averaged functions of order less than 5 identically zero, we obtain a1 = −2d1 and A1 = 0 = D1 . Then system (3) becomes ẋ = y + εy(−2d1 x + b1 y 2 ), ẏ = −x + ε(c1 x + d1 y 2 ), if y ≥ 0, (29) ẋ = y + εB1 y 3 , ẏ = −x + εC1 x, if y ≤ 0, where −b1 ε < 0, −B1 ε < 0 and b1 c1 B1 C1 6= 0. System (29) has the polynomial first integral y2 εb1 4 (1 − εc1 )x2 H1 (x, y) = + y + − εd1 xy 2 2 4 2 if y ≥ 0, and the first integral y2 εB1 4 (1 − εC1 )x2 H2 (x, y) = + y + 2 4 2 if y ≤ 0. Let arbitrary |γ| p > 0 and R0 > 0 such that H1 (γ, 0) = R0 . From this last equality we get γ± = ± 2R0 /(1 − εc1 ). Substituting x = γ± , y = 0 into H2 (x, y), we have that H2 (γ+ , 0) = H2 (γ− , 0), as shown in Figure 1. Notice that the origin O is the unique singularity of system (29) if |ε| is small enough. Therefore we have a global center at the origin and consequently the infinity is a center if −b1 ε < 0, −B1 ε < 0 and b1 c1 B1 C1 6= 0. Statement (iii) is proved and the proof of Theorem 2 is completed. We can illustrate the existence of a global center at the origin in Theorem 2 by taking ε = 0.1, b1 = 2, c1 = 0.6, d1 = −1, B1 = 1 and C1 = −0.6, as it is shown in Figure 2. LIMIT CYCLES OF DISCONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 15 y x Figure 1. Existence of closed orbits for system (29). x Figure 2. Existence of global center for system (29). From the proof of statement (iii) in Theorem 2 and the averaging theory, we have the following results because f˜41 (r) or f˜42 (r) has at most two positive zeros. Proposition 5. At most 2 limit cycles of system (9) can bifurcate from the periodic orbits of the linear system using the averaging theory of order four, and there are systems (9) with 2 limit cycles. Acknowledgements The first author is partially supported by a MINECO grants MTM2016-77278-P and MTM2013-40998-P, an AGAUR grant number 2014SGR-568, and the grant FP7-PEOPLE-2012-IRSES 318999, and from the recruitment program of high–end foreign experts of China. The second author is partially supported by NNSF of China grant 11431008, by RFDP of Higher Education of China grant 20130073110074, and by the MARIE SkLodowska-CURIE ACTIONS 655212 - UBPDS - H2020 - MSCA - IF-2014. 16 J. LLIBRE AND Y. TANG References [1] A. Andronov, A. Vitt and S. Khaikin, Theory of Oscillations, Pergamon Press, Oxford, 1966. [2] V.I. Arnold, Loss of stability of self–oscillations close to resonance and versal deformations of equivariant vector fields, Funct. Anal. Appl. 11 (1977), 85–92. [3] V.I. Arnold, Ten problems, Adv. Soviet Math. 1 (1990), 1–8. [4] M. di Bernardo, C.J. Budd, A.R. Champneys and P. Kowalczyk, Piecewise-Smooth Dynamical Systems. Theory and Applications, Appl. Math. Sci. Series, 163, Springer-Verlag London, Ltd., London, 2008. [5] I.S. Berezin and N.P. Zhidkov, Computing Methods, Volume II, Pergamon Press, Oxford, 1964. [6] D.C. Braga and L.F. Mello, Limit cycles in a family of discontinuous piecewise linear differential systems with two zones in the plane, Nonlinear Dynam. 73 (2013), 1283–1288. [7] C. Buzzi, C. Pessoa and J. Torregrosa, Piecewise linear perturbations of a linear center, Discrete and Continuous Dynamical Systems 33 (2013), 3915–3936. [8] B. Coll, Qualitative study of some vector fields in the plane (Ph. Thesis in Catalan), Uni- versitat Autnoma de Barcelona, 1987. [9] F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Sys- tems, Springer–Verlag, Berlin, 2006. [10] E. Freire, E. Ponce, F. Rodrigo and F. Torres, Bifurcation sets of continuous piecewise linear systems with two zones, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 8 (1998), 2073– 2097. [11] B. Garcı́a, J. Llibre and J. S. Pérez del Rı́o, Planar quasihomogeneous polynomial dif- ferential systems and their integrability, J. Differential Equation 255 (2013), 3185–3204. [12] M. Han and W. Zhang, On Hopf bifurcation in non-smooth planar systems, J. Differential Equation 248 (2010), 2399–2416. [13] S. Huan and X. Yang, The number of limit cycles in general planar piecewise linear systems, Discrete and Continuous Dynamical Systems 32 (2012), 2147–2164. [14] J. Itikawa, J. Llibre and D.D. Novaes, A new result on averaging theory for a class of discontinuous planar differential systems with applications, to appear in Revista Matemática Iberoamericana. [15] A.G. Khovansky, Real analytic manifolds with finiteness properties and complex Abelian integrals, Funct. Anal. Appl. 18 (1984), 119–128. [16] I.D. Iliev, The number of limit cycles due to polynomial perturbations of the harmonic oscillator, Math. Proc. Cambridge Philos. Soc. 127 (1999), 317–322. [17] L. Li, Three crossing limit cycles in planar piecewise linear systems with saddle-focus type, Electron. J. Qual. Theory Differ. Equ. 2014, no. 70, pp. 14. [18] J. Llibre, A.C. Mereu and D.D. Novaes, Averaging theory for discontinuous piecewise differential systems, J. Differential Equation 258 (2015), 4007–4032. [19] J. Llibre, D.D. Novaes and M.A. Teixeira, On the birth of limit cycles for non-smooth dynamical systems, Bull. Sci. Math. 139 (2015) 229–244. [20] J. Llibre and E. Ponce, Three nested limit cycles in discontinuous piecewise linear dif- ferential systems with two zones, Dynam. Contin. Discrete Impuls. Systems. Ser. B Appl. Algorithms 19 (2011), 325–335. [21] J. Llibre, E. Ponce and C. Valls, Uniqueness and non-uniqueness of limit cycles of piece- wise linear differential systems with three zones and no symmetry, J. Nonlinear Science 25 (2015), 861–887. [22] R. Lum and L.O. Chua, Global properties of continuous piecewise-linear vector fields. Part I: Simplest case in R2 , Internat. J. Circuit Theory Appl. 19 (1991), 251–307. [23] R. Lum and L.O. Chua, Global properties of continuous piecewise–linear vector fields. Part II: simplest symmetric in R2 , Internat. J. Circuit Theory Appl. 20 (1992), 9–46. [24] O. Makarenkov and J.S.W. Lamb, Dynamics and bifurcations of nonsmooth systems: A survey, Physica D 241 (2012), 1826–1844. [25] G. Sansone and R. Conti, Non-Linear Differential Equations, 2nd edition, Pergamon Press, New York, 1964. [26] D. Schlomiuk and N. Vulpe, Geometry of quadratic differential systems in the neighborhood of infinity, J. Differential Equations 215 (2005), 357–400. LIMIT CYCLES OF DISCONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 17 [27] D.J.W. Simpson, Bifurcations in Piecewise–Smooth Continuous Systems, World Scientific Series on Nonlinear Science A, vol 69, World Scientific, Singapore, 2010. [28] A.N. Varchenko, An estimate of the number of zeros of an Abelian integral depending on a parameter and limiting cycles, Funct. Anal. Appl. 18 (1984), 98–108. [29] L. Wei and X. Zhang, Limit cycle bifurcations near generalized homoclinic loop in piecewise smooth differential systems, Discrete and Continuous Dynamical Systems 36 (2016), 2803– 2825. [30] Y.Q. Ye, Theory of Limit Cycles, Trans. Math. Monographs 66, Amer. Math. Soc., Provi- dence, RI, 1986. [31] Y. Zou, T. Kupper and W.J. Beyn, Generalized Hopf bifurcation for planar Filippov sys- tems continuous at the origin, J. Nonlinear Science 16 (2006), 159–177. 1 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bel- laterra, Barcelona, Catalonia, Spain E-mail address: jllibre@mat.uab.cat 2 School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, 200240, P. R. China E-mail address: mathtyl@sjtu.edu.cn