This is a preprint of: “Centers of discontinuous piecewise smooth quasi–homogeneous polynomial differential systems”, Hebai Chen, Jaume Llibre, Yilei Tang, Discrete Contin. Dyn. Syst. Ser. B, vol. 24(12), 6495–6509, 2019. DOI: [10.3934/dcdsb.2019150] CENTERS OF DISCONTINUOUS PIECEWISE SMOOTH QUASI–HOMOGENEOUS POLYNOMIAL DIFFERENTIAL SYSTEMS HEBAI CHENA,B , JAUME LLIBREC AND YILEI TANGD Abstract. In this paper we investigate the center problem for the discontinuous piecewise smooth quasi–homogeneous but non–homogeneous polynomial differen- tial systems. First, we provide sufficient and necessary conditions for the existence of a center in the discontinuous piecewise smooth quasi–homogeneous polynomial differential systems. Moreover, these centers are global, and the period function of their periodic orbits is monotonic. Second, we characterize the centers of the discontinuous piecewise smooth quasi–homogeneous cubic and quartic polynomial differential systems. 1. Introduction When all the orbits of a differential system in R2 in a punctured neighborhood of an equilibrium point p of the system are periodic, we say that the p is a center of the differential system. A center p of a differential system is global when all the orbits of the system in R2 \ {p} are periodic. In the qualitative theory of planar smooth differential systems, the center problem is a classical problem, which consist in determining the existence of a center, i.e. give necessary and sufficient conditions in order that an equilibrium point of a differential system in the plane R2 be a center. The study of the centers goes back to Poincaré [29] and Dulac [9], and in the present days many questions about them remain open. It is known that there are three kind of centers for the analytic differential systems in R2 . The linear type centers or simply linear centers are the centers whose linear part have purely imaginary eigenvalues. The nilpotent centers are the centers such that the eigenvalues of their linear part are both zero, but the linear part is not identically zero. Finally, the degenerate centers are the centers whose linear part is identically zero. For more details on these three kind of centers see for instance [24] and the references quoted there. The linear centers can be studied computing their Lyapunov constants, and the study of these centers has been stimulated considerably by the use of computer algebra systems [10], but in any case these computations are in general huge and sometimes non-realistic because they need too much time of computation and too much memory of the computer. Moreover, in a neighborhood of a linear center of an analytic system always exist a local analytic first integral, as it was proved by 2010 Mathematics Subject Classification. 37J35, 34A05, 34A34, 34C14. Key words and phrases. Global center, quasi–homogeneous polynomial systems, piecewise smooth systems, dynamics. 1 2 HEBAI CHEN, JAUME LLIBRE AND YILEI TANG Poincaré [30] and Lyapunov [21]. In general do not exist local analytic first integrals in the neighborhoods of the nilpotent or degenerate centers, see for instance [24]. But any center always has a local C ∞ first integral, see [26]. For analytic and in particular for polynomial differential systems, there are some methods for studying the nilpotent centers, see [13, 17, 24, 18, 28]. But the de- termination of the degenerate centers of the analytical differential systems is more difficult and only very partial results have been obtained. The centers of the quasi–homogeneous polynomial differential systems have been classified, and all are global centers see [19] and [16]. The normal forms of the quasi-homogeneous polynomial differential systems of degree n having a center at the origin were obtained in [35]. In these last decades discontinuous piecewise smooth differential systems have been widely used in a natural way for modeling real processes and different phenom- ena, see for instance [4, 5, 25]. The study of the discontinuous piecewise differential systems started with Andronov et al in [2], they mainly analyzed the discontinu- ous piecewise linear differential systems. Recently many works paid attention to different aspects (as their limit cycles, bifurcations, integrability, ...) of the linear or nonlinear discontinuous piecewise smooth differential systems, see for instance [1, 6, 7, 8, 12, 34]. In this paper we are interested in the centers of the discontinuous piecewise smooth quasi–homogeneous polynomial differential systems. It is known that the quasi– homogeneous polynomial differential systems of even degree has no centers, see [3, 19, 32, 35]. However, as we shall see the discontinuous piecewise smooth quasi– homogeneous polynomial differential systems of even degree can have centers. A polynomial differential system (1) ẋ = P (x, y), ẏ = Q(x, y), in R2 is a quasi–homogeneous polynomial differential system if there exist constants s1 , s2 , m ∈ Z+ such that P (αs1 x, αs2 y) = αs1 +m−1 P (x, y) and Q(αs1 x, αs2 y) = αs2 +m−1 Q(x, y), for all α ∈ R+ , with P (x, y), Q(x, y) ∈ R[x, y] \ {0}. We say that the quasi– homogeneous polynomial differential system (1) or its associated vector field has weight vector w = (s1 , s2 , m), and degree n if n is the maximum of the degrees of P and Q. When s1 = s2 = 1, system (1) is a homogeneous system of degree m. Clearly a quasi–homogeneous polynomial differential system (1) has a unique minimal weight vector (MWV for short) w e = (e s2 , m̃) satisfying that s̃1 ≤ s1 , s̃2 ≤ s1 , e s2 and m̃ ≤ m for any other weight vector (s1 , s2 , m) of system (1). In what follows we assume without loss of generality that P and Q in system (1) have not a non– constant common factor. Here we shall work with discontinuous piecewise smooth polynomial differential system in R2 formed by two quasi–homogeneous polynomial differential systems separated by the straight line y = 0. More precisely, we deal with the discontinuous CENTERS OF PIECEWISE QUASI–HOMOGENEOUS SYSTEMS 3 piecewise smooth quasi–homogeneous polynomial differential system of degree n X X ẋ = P + (x, y) = a+ i,j xi j y , ẏ = Q+ (x, y) = b+ i j i,j x y , in y ≥ 0 0≤i+j≤n 0≤i+j≤n (2) X X ẋ = P − (x, y) = a− i j i,j x y , ẏ = Q− (x, y) = b− i j i,j x y , in y ≤ 0. 0≤i+j≤n 0≤i+j≤n We assume that (P + , Q+ ) and (P − , Q− ) are quasi–homogeneous polynomial vector fields with the same MWV. In this paper first we characterize the centers of the discontinuous piecewise smooth quasi–homogeneous but non–homogeneous polynomial differential systems, see Theorems 1 and 2. And after we characterize the global centers of discontinu- ous piecewise smooth quasi–homogeneous but non–homogeneous cubic and quartic polynomial vector fields. This article is organized as follows. In section 2 we present sufficient and nec- essary conditions for the existence of a center in a discontinuous piecewise smooth quasi–homogeneous polynomial differential system. These centers will be global and the period function of their periodic orbits will be monotonic. Sections 3- 4 are dedicated to analyze the center problem for discontinuous piecewise smooth quasi–homogeneous but non–homogeneous cubic and quartic polynomial differential systems, respectively. 2. Centers of piecewise smooth quasi–homogeneous polynomial differntial systems According to [14, Proposition 10], if a smooth quasi–homogeneous but non– homogeneous polynomial differential system is of degree n with the weight vector (s1 , s2 , m) and m > 1, then the system has the MWV   ς +κ κ (p − 1)ς + (n − 1)κ (3) we= , , 1+ , s s s with p ∈ {0, 1, ..., n − 1}, ς ∈ {1, 2, ..., n − p} and κ ∈ {1, . . . , n − p − ς + 1} satisfying (ς + κ)(m − 1) κ(m − 1) s1 = , s2 = , D D where D = (p − 1)ς + (n − 1)κ and s = gcd(ς, κ). Furthermore, the integers (ς + κ)/s and κ/s are coprime and at least one of them is odd, see [33, 35]. From the algorithm described in subsection 3.1 of [14] the quasi–homogeneous but non–homogeneous polynomial differential system (1) of degree n with weight vector (s1 , s2 , m) can be written as pςκ (4) Xpςκ = Xnp + Xn−ς + Vpςκ , where Xnp = (ap,n−p xp y n−p , bp−1,n−p+1xp−1 y n−p+1)T 4 HEBAI CHEN, JAUME LLIBRE AND YILEI TANG is the homogeneous part of degree n with coefficients not all zero, X pς1 κς1 Vpςκ = Xn−ς 1 , ς1 ∈ {1, . . . , n − p} \ {ς} κς1 ς = κς1 and κς1 ∈ {1, . . . , n − ς1 − p + 1} pςκ Xn−ς = (ap+κ,n−ς−p−κxp+κ y n−ς−p−κ, bp+κ−1,n−ς−p−κ+1xp+κ−1 y n−ς−p−κ+1)T , pς1 κς1 pςκ and Xn−ς 1 ’s having the same expressions as that of Xn−ς . In order that Xpςκ to be p of degree n we must have Xn 6≡ 0, and in order that it does not be an homogeneous system at least one of the monomials which are not in Xnp must be different from zero. We shall study the dynamics of the generic discontinuous piecewise smooth quasi– homogeneous polynomial differential systems (2) close to the discontinuous line, i.e., the x-axis, by using the Filippov convection method, see [4, 11, 22, 23]. The discontinuous line L := {(x, y) ∈ R2 | F (x, y) = y = 0} separates the plane into two open non–overlapping regions Y + = {(x, y) ∈ R2 | y > 0} and Y − = {(x, y) ∈ R2 | y < 0}. Suppose that σ(x, y) = (Fx , Fy ), (P + , Q+ ) · (Fx , Fy ), (P − , Q− ) , where < ·, · > denotes the standard inner product. From (2) and (4), the crossing set of system (2) is (5) Lc = {(x, y) ∈ L| σ(x, y) > 0} − = {(x, y) ∈ L| b+ p+κ−1,0 bp+κ−1,0 x 2(p+κ−1) > 0} n {(x, y) ∈ L| x 6= 0} if bp+κ−1,0 bp+κ−1,0 > 0, + − = − ∅ if b+ p+κ−1,0 bp+κ−1,0 ≤ 0. The orbits of the (2) which reach any point (x, y) ∈ Lc crosses the line of disconti- nuity L, i.e. the vectors (P + (x, y), Q+ (x, y)) and (P − (x, y), Q− (x, y)) point to the “same” direction and are transverse to Lc . The sliding set Ls is the complement of Lc in L, which is given by (6) Ls = {(x, y) ∈ L| σ(x, y) ≤ 0} − = {(x, y) ∈ L| b+ p+κ−1,0 bp+κ−1,0 x 2(p+κ−1) ≤ 0} n L + − if bp+κ−1,0 bp+κ−1,0 ≤ 0, = − {(x, y) ∈ L| x = 0} if b+ p+κ−1,0 bp+κ−1,0 > 0. Moreover, in Ls the points satisfying the equation (7) (Fx , Fy ), (P − − P + , Q− − Q+ ) = 0 are the sliding equilibrium points. The equilibrium points of a discontinuous piecewise smooth quasi–homogeneous polynomial differential system are the equilibria contained in the open half-planes {y > 0} and {y < 0}, together with the sliding equilibrium points. CENTERS OF PIECEWISE QUASI–HOMOGENEOUS SYSTEMS 5 If all periodic orbits inside the period annulus of a center have the same period, the center is isochronous. Making a quasi–homogeneous blow-up x = r s1 cos θ, y = r s2 sin θ, which was also used in [16], we transform system (2) into the system   r d H + (θ)/(s1 cos2 θ + s2 sin2 θ) rH + (θ)  = = if y ≥ 0, dr  r d−1 G+ (θ)/(s1 cos2 θ + s2 sin2 θ) G+ (θ) (8) = dθ   r d H − (θ)/(s1 cos2 θ + s2 sin2 θ) rH − (θ)  = = if y < 0, r d−1 G− (θ)/(s1 cos2 θ + s2 sin2 θ) G− (θ) where H ± (θ) = P ± (cos θ, sin θ) cos θ + Q± (cos θ, sin θ) sin θ, G± (θ) = s1 Q± (cos θ, sin θ) cos θ − s2 P ± (cos θ, sin θ) sin θ. According to [31, Chapter 2], a necessary condition for θ = θ0 to be an exceptional direction is G+ (θ) = 0 (resp. G− (θ) = 0) on the half plane y ≥ 0 (resp. y < 0), and no orbits connect with the origin O along other directions. Theorem 1. A discontinuous piecewise smooth quasi–homogeneous polynomial dif- ferential system (2) with a MWV (s1 , s2 , m) has a center at the origin O(0, 0) if and only if Lc = {(x, y) ∈ L| x 6= 0} and Z π + Z 0 − H (θ) H (θ) (9) + dθ + − = 0, 0 G (θ) −π G (θ) where G+ (θ) 6= 0 if θ ∈ [0, π] and G− (θ) 6= 0 if θ ∈ [−π, 0]. The center is global and the period function of periodic orbits is monotonic. Proof. Notice that no equilibria of piecewise smooth systems exist in the regions Y ± . From the aforementioned analysis of sliding sets and crossing sets in the discontin- uous line L, the crossing set Lc of system (2) is either ∅ or the x-axis except the origin. Thus, system (2) has a center at the origin only if Lc = {(x, y) ∈ L| x 6= 0}. When the crossing set Lc is the x-axis except the origin, let p+ (r, θ) (resp. p− (r, θ)) be the solution of piecewise smooth system (2) in polar coordinates (x, y) = (ρ cos θ, ρ sin θ) for 0 ≤ θ < π (resp. −π ≤ θ < 0), satisfying that the initial condition p+ (r, 0) = r (resp. p− (r, −π) = r) holds, which is well defined in the region R2 \ Ls . We define the positive Poincaré half-return map as Rπ H + (θ) 0 G+ (θ) dθ P + (r) := lim p+ (r, θ) = r e θ→π and the negative Poincaré half-return map as R0 H − (θ) − − dθ P (r) := lim p (r, θ) = r e −π G− (θ) θ→0 according to system (8), as shown in Figure 1. Here, without loss of generality, we suppose that the orbits surrounding the origin rotate anti-clockwise. If the direction is clockwise, we can make a time rescaling. In the next step of this proof, we will show that the orbits around the origin are spirals with a period 2π under the conditions of this theorem. 6 HEBAI CHEN, JAUME LLIBRE AND YILEI TANG r O Figure 1. Existence of closed orbits. The Poincaré return map associated to piecewise smooth system (2) is given by the composition of these two maps RπH + (θ) R0 H − (θ) dθ dθ (10) P(r) := P − (P + (r)) = r e 0 G+ (θ) e −π G− (θ) . In order to obtain the existence of a center and further a global center at the origin, we need to present P(r) − r ≡ 0 for r > 0, which is equivalent to (9). Note that the conditions G+ (θ) 6= 0 if θ ∈ [0, π] and G− (θ) 6= 0 if θ ∈ [−π, 0] can not only guarantee the integrability in (9), but also the non-existence of invariant curves passing through the origin. Otherwise, if G+ (θ0 ) = 0 for θ0 ∈ [0, π] or G− (θ0 ) = 0 for θ0 ∈ [−π, 0], we obtain the invariant (11) C1 := coss2 (θ0 ) y s1 − sins1 (θ0 ) xs2 = 0 when θ0 6= ±π/2. If θ0 = ±π/2, the half y-axis is invariant according to the expression of G± (θ) in (8) and the origin cannot be a center. Because at least one of s1 and s2 is odd, we can always solve a real branch connecting with the origin from the above equation (11), which is just an invariant curve of system (2). Actually, if p is odd and π/2 6= θ0 ∈ [0, π], from (11) we have the invariant curve sin(θ0 ) y= xs2 /s1 coss2 /s1 (θ0 ) on the half plane y ≥ 0. If p is even and π/2 6= θ0 ∈ [0, π], again from (11) we have the invariant curve cos(θ0 ) x= s1 /s2 y s1/s2 sin (θ0 ) on the half plane y ≥ 0, because s2 must be odd. On the half plane y < 0, we can discuss in a similar way and can always find an real invariant curve if G− (θ) = 0 for θ ∈ [−π, 0]. Remark that the factor y divides Q+ (x, y) or Q− (x, y) if G± (0) = 0 or G± (π) = 0, respectively. Thus, σ(x, y) = 0 for (x, y) ∈ L according to (6) and Ls = L, yielding that the origin cannot be a center in this case. Thus, the conditions G+ (θ) 6= 0 if θ ∈ [0, π] and G− (θ) 6= 0 if θ ∈ [−π, 0] are necessary in order that the origin be a center. Moreover, we actually get P(r) = r for all r > 0, implying that the solution curve of system (8) through (r, 0) is a closed orbit and the origin O is a global center. CENTERS OF PIECEWISE QUASI–HOMOGENEOUS SYSTEMS 7 Finally, we shall prove that the period annulus of the center O is monotonic with respect to the initial value and then it cannot be isochronous. Assuming that Γr0 is the closed trajectory through (r0 , 0) inside the periodic annulusR of the center O, we can define the positive half-period function as T + (r0 ) := Γ+r dt and the R 0 negative half-period function as T − (r0 ) := Γ−r dt, where r0 > 0, Γ+ r0 = {(ρ, θ) ∈ 0 R2 | ρ = p+ (r0 , θ)} and Γ− r0 = {(ρ, θ) ∈ R | ρ = p (r0 , θ)}. Thus, the complete 2 − period function associated to system (8) is given by the sum of these two functions I T (r0 ) = dt = T + (r0 ) + T − (r0 ) Γr0 Z Z s1 cos2 θ + s2 sin2 θ s1 cos2 θ + s2 sin2 θ (12) = dθ + dθ Γ+ r0 r m−1 G+ (θ) Γ− r0 r m−1 G− (θ) Z π Z 0 s1 cos2 θ + s2 sin2 θ s1 cos2 θ + s2 sin2 θ = dθ + dθ. 0 (p+ (r0 , θ))m−1 G+ (θ) −π (p− (r0 , θ))m−1 G− (θ) ± ± Notice that the functions p (r0 , θ) and G (θ) are bounded and nonzero with respect to θ for arbitrary fixed r0 > 0, because the origin is a center and p− (p+ (r0 , θ), θ) has period 2π in θ. Therefore, the two integrand functions in (12) are integrable. By (8), RθH + (θ) Rθ H − (θ) + dθ − −π G− (θ) dθ p (r0 , θ) = r0 e 0 G+ (θ) and p (r0 , θ) = r0 e , it follows that dT (r0 ) (1 − m) = T (r0 ) < 0, dr0 r0 which implies that the period T (r0 ) of the periodic orbits inside the period annulus of the center O is monotonic in r0 . Clearly, the center cannot be isochronous. This completes the proof of the theorem.  Remark that we can also obtain the results of Theorem 1 using the properties of quasi-homogeneous functions η ± (x, y) = s1 xQ± (x, y) − s2 yP ± (x, y). According to [27, Proposition 4], if η + (0, 1) = 0 (or η − (0, 1) = 0), then the half y-axis is invariant for the flow of the vector (P + , Q+ ) (or (P − , Q− )). Moreover, if there are real values λ± such that η ± (1, λ± ) = 0 respectively, then there is an invariant curve from the real branch of the curve y s1 − λs±1 xs2 = 0 for system (2), where λ+ ≥ 0 and λ− ≤ 0 if s2 is even. Thus, the origin cannot be a center. Notice that the smooth quasi-homogeneous polynomial differential system (1) with a MWV (s1 , s2 , d) is Liouvillian integrable. Moreover, we can find its inverse integrating factors s1 xQ(x, y) − s2 yP (x, y) or (s1 xQ(x, y) − s2 yP (x, y))/H̃(x, y) ac- cording to [19], where 1 R arcT n(y s1 /xs2 ) F (θ) H̃(x, y) = (s1 x2s2 + s2 y 2s1 ) 2s1 s2 e− 0 G(θ) dθ is a first integral of system (1), F (θ) = Cs2s2 −1 θ P (Csθ, Snθ) + Sn2s1 −1 θ Q(Csθ, Snθ), G(θ) = s1 Csθ Q(Csθ, Snθ) − s2 Snθ P (Csθ, Snθ), 8 HEBAI CHEN, JAUME LLIBRE AND YILEI TANG Sns1 θ T n(θ) = Css2 θ and Csθ and Snθ are the (s1 , s2 )-trigonometric functions. Thus, we can always suppose that system (2) has a first integral H+ (x, y) for y ≥ 0 and a first integral H− (x, y) for y < 0. With the analysis of Poincaré return map, we can present another sufficient and necessary conditions for the existence of center as follows, which is equivalent to the conditions in Theorem 1. Theorem 2. Piecewise smooth quasi–homogeneous polynomial differential system (2) with a MWV (s1 , s2 , m) has a center at the origin O(0, 0) if and only if Lc = {(x, y) ∈ L| x 6= 0}, η ± (0, 1) 6= 0, η ± (1, λ) have not real zeros for odd s2 and η + (1, λ) (resp. η − (1, λ)) has not non-negative (resp. non-positive) zeros for even s2 , and there exists r1 < 0 such that H+ (r0 , 0) = H+ (r1 , 0) and H− (r0 , 0) = H− (r1 , 0) for arbitrary r0 > 0. The center is global and the period function of periodic orbits is monotonic. 3. Global center of piecewise smooth cubic quasi–homogeneous systems Due to Proposition 19 of Garcı́a, Llibre and Pérez del Rı́o [14], a smooth quasi- homogeneous but non-homogeneous cubic polynomial differential system without common factors has one of the following seven forms: (i) : ẋ = y(a11 x + a12 y 2 ), ẏ = b11 x + b12 y 2 with MWV (2, 1, 2) and a12 b11 6= 0, a11 b12 − a12 b11 6= 0; (ii) : ẋ = a21 x2 + a22 y 3, ẏ = b2 xy with MWV (3, 2, 4) and a21 a22 b2 6= 0; 3 2 (iii) : ẋ = a3 y , ẏ = b3 x with MWV (4, 3, 6) and a3 b3 6= 0; (iv) : ẋ = x(a41 x + a42 y 2 ), ẏ = y(b41 x + b42 y 2 ) with MWV (2, 1, 3) and a41 b42 6= 0, a41 b42 − a42 b41 6= 0; (v) : ẋ = a5 xy 2 , ẏ = b51 x2 + b52 y 3 with MWV (3, 2, 5) and a5 b51 b52 6= 0; (vi) : ẋ = a6 xy 2 , ẏ = b61 x + b62 y 3 with MWV (3, 1, 3) and a6 b61 b62 6= 0; (vii) : ẋ = a71 x + a72 y 3 , ẏ = b7 y with MWV (3, 1, 1) and a71 a72 b7 6= 0. We have the following piecewise smooth quasi-homogeneous but non-homogeneous cubic polynomial differential systems directly from the smooth systems (i)-(vii). CENTERS OF PIECEWISE QUASI–HOMOGENEOUS SYSTEMS 9 Lemma 3. Every planar piecewise smooth quasi-homogeneous but non-homogeneous cubic polynomial differential system is one of the following seven systems: (I) : ẋ = y(a11 x + a12 y 2 ), ẏ = b11 x + b12 y 2 if y ≥ 0, ẋ = y(ã11 x + ã12 y 2 ), ẏ = b̃11 x + b̃12 y 2 if y < 0; (II) : ẋ = a21 x2 + a22 y 3, ẏ = b2 xy if y ≥ 0, ẋ = ã21 x2 + ã22 y 3, ẏ = b̃2 xy if y < 0; (III) : ẋ = a3 y 3 , ẏ = b3 x2 if y ≥ 0, ẋ = ã3 y 3 , ẏ = b̃3 x2 if y < 0; (IV ) : ẋ = x(a41 x + a42 y 2 ), ẏ = y(b41 x + b42 y 2) if y ≥ 0, ẋ = x(ã41 x + ã42 y 2 ), ẏ = y(b̃41 x + b̃42 y 2) if y < 0; (V ) : ẋ = a5 xy 2 , ẏ = b51 x2 + b52 y 3 if y ≥ 0, ẋ = ã5 xy 2 , ẏ = b̃51 x2 + b̃52 y 3 if y < 0; (V I) : ẋ = a6 xy 2 , ẏ = b61 x + b62 y 3 if y ≥ 0, ẋ = ã6 xy 2 , ẏ = b̃61 x + b̃62 y 3 if y < 0; (V II) : ẋ = a71 x + a72 y 3 , ẏ = b7 y if y ≥ 0, ẋ = ã71 x + ã72 y 3 , ẏ = b̃7 y if y < 0; where all parameters satisfy the same inequality conditions as in the smooth systems (i)-(vii). The piecewise smooth systems (II), (IV ) and (V II) have the same sliding set L, yielding that their origins cannot be a center. Moreover, both systems (V ) and (V I) have the invariant straight line x = 0, so their origins cannot be a center. Thus, only the piecewise smooth systems (I) and (III) may have a center at the origin. After taking linear changes together with scalings (x, y, dt) → (b11 x/a11 , y, a11 dt) if y ≥ 0, and (x, y, dt) → (a11 x/ã12 , y, a11 dt) if y < 0, system (I) becomes ẋ = y(x + a12 y 2), ẏ = x + b12 y 2 if y ≥ 0, (13) ẋ = y(ã11 x + y 2), ẏ = b̃11 x + b̃12 y 2 if y < 0. 1 2 1 Moreover, with the scaling (x, y, dt) → ((b3 /a3 ) 3 x, y, a33 b33 dt), system (III) becomes ẋ = y 3 , ẏ = x2 if y ≥ 0, (14) ẋ = ã3 y 3 , ẏ = b̃3 x2 if y < 0. Here, we still write the new parameters as aij , bij , ãij and b̃ij for simpler notations. From the definitions (5) and (6) we compute for system (13) its crossing and sliding sets in L: n {(x, y) ∈ L| x 6= 0} if b̃11 > 0, (15) LIc = {(x, y) ∈ L| b̃11 x2 > 0} = ∅ if b̃11 < 0, and n {(x, y) ∈ L| x = 0} if b̃11 > 0, (16) LIs = {(x, y) ∈ L| b̃11 x2 ≤ 0} = L if b̃11 < 0, respectively. Then we find that the only solution of (7) for system (13) in LIs is the origin, which is a singular sliding point and also a boundary equilibrium because the vector fields vanish at the origin. 10 HEBAI CHEN, JAUME LLIBRE AND YILEI TANG By an analogous analysis of system (13) we obtain for piecewise smooth system (14) the crossing and the sliding sets in L: n {(x, y) ∈ L| x 6= 0} if b̃3 > 0, (17) LIII c = {(x, y) ∈ L| b̃3 x2 > 0} = ∅ if b̃3 < 0, n {(x, y) ∈ L| x = 0} if b̃3 > 0, (18) LIII s = {(x, y) ∈ L| b̃3 x2 ≤ 0} = L if b̃3 < 0. We also find that the origin of system (14) in LIII s is a unique singular sliding point, which is a boundary equilibrium. Theorem 4. Piecewise smooth quasi–homogeneous cubic polynomial differential sys- tems (II)-(V II) have no centers. Reduced piecewise smooth quasi–homogeneous cu- bic polynomial differential system (13) from (I) has a center at the origin if and only if b̃11 > 0, (2b12 − 1)2 + 8a12 < 0 (or a12 > 0, −1 + 2b12 < 0, √ √ or a12 < 0, 2 −8a12 = 2b12 − 1), (A11 − 2B12 ) + 8B11 < 0 (or A11 − 2B12 ≥ −8B11 ) and B11 < 0. The center of system (13) is global and the period function of the periodic orbits is monotonic. Proof. Obviously, systems (II), (IV )-(V II) have no centers. We consider the re- duced systems (13) and (14) in place of (I) and (III). From the above mentioned analysis of the sliding sets and singular sliding points on the discontinuous line L, and by Theorem 1 systems (13) (respectively (14)) have a center at the origin only if b̃11 > 0 (respectively b̃3 > 0). For system (14) we calculate that G+ + + III (θ) = s1 Q (cos θ, sin θ) cos θ − s2 P (cos θ, sin θ) sin θ = 4 cos3 θ − 3 sin4 θ. Then G+ III (θ) has real roots for θ ∈ [0, π] satisfying cos(θ) ≈ 0.639697327225298 or cos(θ) ≈ 2.16002341801127. Thus the origin of system (14) cannot be a center according to Theorem 1. For system (13) with y ≥ 0 we compute ηI+ (0, 1) := s1 xQ+ (0, 1) − s2 yP + (0, 1) = −a12 6= 0, and ηI+ (1, λ) = s1 xQ+ (1, λ) − s2 yP + (1, λ) = 2 + (2b12 − 1)y 2 − a12 y 4. Clearly, ηI+ (0, 1) 6= 0. Besides ηI+ (1, λ) has not real 2 pzeros if and only if (2b12 − 1) + 2 2 8a12 < 0, or (2bp 12 −1) +8a12 ≥ 0, a12 (−1+2b12 + 4b12 + 8a12 − 4b12 + 21)) < 0 and a12 (2b12 −1− (2b12 − 1)2 + 8a12 ) < 0,√ which can be reduced to (2b12 −1) +8a12 < 0, or a12 > 0, −1 + 2b12 < 0, or a12 < 0, −8a12 = 2b12 − 1. For system (13) with y < 0 we compute ηI− (0, 1) := s1 xQ− (0, 1) − s2 yP − (0, 1) = −1 6= 0 and ηI− (1, λ) = s1 xQ− (1, λ) − s2 yP −(1, λ) = 2B11 + (−A11 + 2B12 )y 2 − y 4. Clearly ηI− (1, λ) has not real zeros if andponly if (A11 − 2B12 )2 + 8B11 < 0; or (A11 − 2B12 )2 + 8B11 ≥ 0, −A11 + 2B12 + (A11 − 2B12 )2 + 8B11 < 0 and −A11 + CENTERS OF PIECEWISE QUASI–HOMOGENEOUS SYSTEMS 11 p 2 2B12 − (A11 − 2B√ 2 12 ) + 8B11 < 0, which can be reduced to (A11 − 2B12 ) + 8B11 < 0; or A11 − 2B12 ≥ −8B11 and B11 < 0. According to Theorem 2, if η + (1, λ) or η − (1, λ) has a real zero, there is an invariant curve passing through the origin of system (13) and the origin cannot be a center. Note that system (13) has a first integral   2a12 y 2 −2b12 x+x  −2arctanh √  (1+2b12 ) x 4b2 12 +8a12 −4b12 +1 √ 2 H̃1+ (x, y) = (a12 y 4 − 2b12 xy 2 + xy 2 − 2x2 ) e 4b +8a12 −4b12 +1 12 if y ≥ 0, and a first integral   A11 x−2xB12 +2y 2  2arctan √  (A11 +2B12 ) x −A +4A11 B12 −4B 2 −8B11 2 √ 11 12 −A2 2 H̃1− (x, y) = (A11 xy 2 − 2B12 xy 2 + y 4 − 2B11 x2 ) e 11 +4A11 B12 −4B12 −8B11 if y < 0. In addition we have    −2arctanh √ 2 −2b12 +1  (1+2b12 ) 4b +8a12 −4b12 +1 √ 212 H̃1+ (x, 0) = −2x2 e 4b +8a12 −4b12 +1 12 , and    2arctan √ 2 A11 −2B12  (A11 +2B12 ) −A +4A11 B12 −4B 2 −8B11 √ 2 11 12 2 H̃1− (x, 0) = −2B11 x2 e −A +4A11 B12 −4B −8B11 11 12 , yielding that H̃1+ (r0 , 0) = H̃1+ (r1 , 0) and H̃1− (r0 , 0) = H̃1− (r1 , 0) for arbitrary r1 = −r0 < 0. Therefore the center of the piecewise smooth quasi–homogeneous cubic polynomial differential system (13) at the origin is global. This completes the proof of the theorem for the system (13). In the case a1 < 0, b1 > 0 and ã1 > 0, we prove that the center O1 at the origin of piecewise smooth system (I) is not isochronous. For r0 > 0 we assume that Γr0 is the closed trajectory through (r0 , 0) inside the periodic annulus R of the center O1 , we + can define the positive half-period function as T (r0 ) := Γ+r dt and the negative R 0 half-period function as T − (r0 ) := Γ−r dt, where 0   3b  13  1 Γ+ r0 = (ρ, θ) ∈ R | ρ = p (r0 , θ)} = {(x, y) ∈ R | y = 2 + 2 2 2 (x − r0 ) 2a1 and   3  31  Γ− r0 − = (ρ, θ) ∈ R | ρ = p (r0 , θ)} = {(x, y) ∈ R | y = 2 2 2 2 (x − r0 ) . 2ã1 12 HEBAI CHEN, JAUME LLIBRE AND YILEI TANG Thus the complete period function associated to the piecewise smooth system (I) is given by the sum of these two functions I Z Z + − dx dx TI (r0 ) = dt = T (r0 ) + T (r0 ) = 2 + Γr0 Γ+r0 a1 y Γ− r0 ã1 y 2 Z −r0 Z r0 dx dx =   32 +   32 r0 3b1 2 − r2) −r0 3 2 − r2) a1 2a 1 (x 0 ã1 2ã1 (x 0 −1 = β0 r0 3 , where 5 3√ ( 23 ) 3 π 2 3 − 13 − 23 − 13 β0 = (−a1 b1 + ã1 )>0 Γ( 23 )Γ( 56 ) R∞ and Γ(z) = 0 e−s sz−1 ds is the Gamma function. Clearly the period TI (r0 ) of the periodic orbits inside the period annulus of the center O1 is monotonic in r0 . This completes the proof of the theorem.  Notice that all smooth quasi–homogeneous quadratic polynomial differential sys- tems (i) − (iii) have no centers, because there exists an invariant straight line or an invariant curve passing through the origin for such systems. However for the piece- wise smooth quasi–homogeneous quadratic polynomial differential systems we will find the existence of a center at the origin under some conditions on the parameter. 4. Global center for piecewise smooth quasi–homogeneous quartic polynomial differential systems According to Proposition 1 of [20] there exist 17 quasi-homogeneous but non- homogeneous quartic polynomial differential systems. Fourteen of them either have the invariant straight line x = 0 or y = 0. From the proof of Theorem 1 these four- teen classes of piecewise smooth quasi-homogeneous but non-homogeneous quartic polynomial differential systems cannot have a center at the origin. We consider the following three remainder piecewise smooth quartic polynomial differential systems ẋ = c1 y 4 , ẏ = x3 if y ≥ 0, (19) with c1 C1 D1 6= 0, MW V = (5, 4, 12), ẋ = C1 y 4 , ẏ = D1 x3 if y < 0, ẋ = c2 y 4 , ẏ = x if y ≥ 0, (20) with c2 C2 D2 6= 0, MW V = (5, 2, 4), ẋ = C2 y 4 , ẏ = D2 x if y < 0, and ẋ = c3 y 4, ẏ = x2 if y ≥ 0, (21) with c3 C3 D3 6= 0, MW V = (5, 3, 8). ẋ = C3 y 4 , ẏ = D2 x2 if y < 0, After time rescalings and linear transformations (x, y, dt) → (x, y, d1dt), (x, y, dt) → (x, y, d2dt) and (x, y, dt) → ((d3 c23 )1/3 x/c3 , y, d3dt/c21 ), respectively. Theorem 5. Only two classes of piecewise smooth quasi–homogeneous quartic poly- nomial differential systems have a center at the origin. In particular reduced piece- wise smooth quasi–homogeneous quartic polynomial differential system (19) has a center at the origin if and only if D1 > 0, c1 < 0 and C1 > 0. Reduced piecewise CENTERS OF PIECEWISE QUASI–HOMOGENEOUS SYSTEMS 13 smooth quartic quasi–homogeneous system (20) has a center at the origin if and only if D2 > 0, c2 < 0 and C2 > 0. Both centers of system (19) are global and the period function of periodic orbits is monotonic. Proof. Obviously we only need to consider reduced systems (19)-(21) for the exis- tence of centers at the origin. For system (21) we compute G+ 3 5 3 (θ) = s1 ẏ cos θ − s2 ẋ sin θ = 5 cos θ − 3 sin θ, according to (8). It follows that G+ 3 (θ) has real roots θ ≈ 0.9394265760 and θ ≈ 2.201353768 for θ ∈ [0, π]. Therefore, by Theorem 1 the origin of system (21) cannot be a center. From above mentioned analysis of sliding sets and singular sliding points and Theorem 1, on the discontinuous line L systems (19) (or (20)) have a center at the origin only if D1 > 0 (or D2 > 0). For system (19) and y ≥ 0 we compute η1+ (0, 1) := s1 xẏ − s2 y ẋ|(x,y)=(0,1) = −4c1 6= 0 and η1+ (1, λ) = s1 xẏ − s2 y ẋ|(x,y)=(1,λ) = −4c1 y 5 + 5. Clearly, η1+ (0, 1) 6= 0. Besides η1+ (1, λ) only has negative zeros if and only if c1 < 0. For system (19) and y < 0 we compute η1− (0, 1) := s1 xẏ − s2 y ẋ|(x,y)=(0,1) = −4C1 6= 0 and η1− (1, λ) := s1 xẏ − s2 y ẋ|(x,y)=(1,λ) = −4C1 y 5 + 5D1 . Clearly η1− (1, λ) only has positive zeros if and only if C1 D1 > 0. From Theorem 2 if η1+ (1, λ) has a non-negative zero or η1− (1, λ) has a non-positive zero, there is an invariant curve passing through the origin of system (19) and the origin cannot be a center. Note that system (19) has the first integral φ+ 5 1 (x, y) = 4c1 y − 5x 4 if y ≥ 0, and the first integral φ− 5 1 (x, y) = 4C1 y − 5D1 x 4 if y < 0. Consequently we have φ+ 1 (x, 0) = −5x 4 and φ− 4 1 (x, 0) = −5D1 x , − − yielding that φ+ + 1 (r0 , 0) = φ1 (r1 , 0) and φ1 (r0 , 0) = φ1 (r1 , 0) for arbitrary r1 = −r0 < 0. Therefore system (19) has a center at the origin if and only if D1 > 0, c1 < 0 and C1 > 0, and this center is global. In the case D1 > 0, c1 < 0 and C1 > 0 we prove that the center O1 at the origin of system (19) is not isochronous and the period function of its periodic orbits is monotonic. Assuming that Γr0 is the closed trajectory through (r0 , 0) inside the 14 HEBAI CHEN, JAUME LLIBRE AND YILEI TANG periodic annulus R of the center O1 , we can define the positive half-period R function as T1+ (r0 ) := Γ+r dt and the negative half-period function as T1− (r0 ) := Γ−r dt, with 0 0 r0 > 0. Thus we have   5  15  Γr0 = (ρ, θ) ∈ R | ρ = p (r0 , θ)} = {(x, y) ∈ R | y = + 2 + 2 4 4 (x − r0 ) 4c1 and   5D  15  1 Γ− r0 = − (ρ, θ) ∈ R | ρ = p (r0 , θ)} = {(x, y) ∈ R | y = 2 2 4 (x − r04 ) . 4C1 Then the complete period function associated to system (19) is given by the sum of these two functions I Z Z + − dx dx T1 (r0 ) = dt = T1 (r0 ) + T1 (r0 ) = 4 + Γr0 Γ+r0 c1 y Γ− r0 C1 y 4 Z −r0 Z r0 dx dx =   45 +   45 r0 5 4 4 −r0 5D1 4 4 c1 4c1 (x − r0 ) C1 4C1 (x − r0 ) Z −1 Z 1 dx dx = 4 11 4 + 4 11 4 1 c1 ( 4c51 ) 5 r05 (x4 − 1) 5 −1 C ( 5D1 ) 5 r 5 (x4 − 1) 5 1 4C1 0 − 11 = β1 r0 5 , where √ 2π csc( π5 ) sin( 9π 20 ) Γ( 11 20 ) 1 1  β1 = 4 − > 0, 2Γ( 45 )Γ( 43 ) 4 C1 ( 5D 4C1 ) 1 5 c1 ( 4c51 ) 5 R∞ where Γ(z) = 0 e−s sz−1 ds is the Gamma function. Clearly the period T1 (r0 ) of the periodic orbits inside the period annulus of the center O1 is monotonic in r0 and then it cannot be isochronous. For system (20) we compute η2− (0, 1) := s1 xẏ − s2 y ẋ|(x,y)=(0,1) = −2c2 6= 0, η2− (1, λ) := s1 xẏ − s2 y ẋ|(x,y)=(1,λ) = 5 − 2c2 y 5, if y ≥ 0, and η2− (0, 1) := s1 xẏ − s2 y ẋ|(x,y)=(0,1) = −2C2 6= 0, η2− (1, λ) := s1 xẏ − s2 y ẋ|(x,y)=(1,λ) = 5D2 − 2C2 y 5 , if y < 0. Using a similar analysis as the one done for system (19), we have that η2+ (1, λ) only has negative zeros if and only if c2 < 0, and η2− (1, λ) only has positive zeros if and only if C2 D2 > 0. Moreover system (19) has the first integral φ+ 2 (x, y) = 2c2 y 5 − 5x2 if y ≥ 0, and the first integral φ− 2 (x, y) = 2C 2 y 5 − 5D 2 x2 if y < 0, − satisfying that φ+ 2 (x, 0) = −5x 2 and φ 2 (x, 0) = −5D 2 x2 , yielding that φ + 2 (r 0 , 0) = + − − φ2 (r1 , 0) and φ2 (r0 , 0) = φ2 (r1 , 0) for arbitrary r1 = −r0 < 0. Therefore system (20) has a global center at the origin if and only if D2 > 0, c2 < 0 and C2 > 0. In the case D2 > 0, c2 < 0 and C2 > 0 we assume that ζr0 is the closed trajectory through (r0 , 0) inside the periodic annulus of the center O2 at the origin of system CENTERS OF PIECEWISE QUASI–HOMOGENEOUS SYSTEMS 15 R (20), then we define the positive half-period function as T2+ (r0 ) := ζr+ dt and the R 0 negative half-period function as T2− (r0 ) := ζr− dt with r0 > 0. Then we have 0   5  15  ζr0 = (ρ, θ) ∈ R | ρ = p (r0 , θ)} = {(x, y) ∈ R | y = + 2 + 2 2 2 (x − r0 ) 2c2 and   5D  15  2 ζr−0 − = (ρ, θ) ∈ R | ρ = p (r0 , θ)} = {(x, y) ∈ R | y = 2 2 2 2 (x − r0 ) . 2C2 Therefore the complete period function associated to system (20) is given by the sum of these two functions I Z Z + − dx dx T2 (r0 ) = dt = T2 (r0 ) + T2 (r0 ) = 4 + 4 ζr0 ζr+0 c2 y ζr−0 C2 y Z −r0 Z r0 dx dx =   54 +   45 r0 −r0 c2 2c52 (x2 − r02 ) C2 5D2C2 2 (x 2 − r2) 0 Z −1 Z 1 dx dx = 4 3 4 + 4 3 4 1 c2 ( 2c52 ) 5 r05 (x2 − 1) 5 −1 C ( 5D2 ) 5 r 5 (x2 − 1) 5 2 2C2 0 −3 = β2 r0 5 , where π 2 csc( π5 )   3 1 1 β2 = 4 7 4 − 5 45 > 0. Γ( 5 )Γ( 10 ) C2 ( 5D 2C2 ) 2 5 c 2 ( 2c2 ) Clearly the period T2 (r0 ) of the periodic orbits inside the period annulus of the center O2 is monotonic in r0 . This completes the proof of the theorem.  Acknowledgments The first author is supported by the National Natural Science Foundation of China (No. 11801079). The second author is supported by the Ministerio de Economı́a, Industria y Com- petitividad, Agencia Estatal de Investigación grants MTM2016-77278-P (FEDER) and MDM-2014-0445, the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017- 777911. The third author is supported by the National Natural Science Foundations of China (No. 11871041, 11431008). References [1] A. Algaba, E. Gamero, C. Garcı́a, The integrability problem for a class of planar systems, Nonlinearity 22 (2009), 396–420. [2] A. Andronov, A. Vitt, S. Khaikin, Theory of Oscillations, Pergamon Press, Oxford, 1966. [3] W. Aziz, J. Llibre, C. Pantazi, Centers of quasi–homogeneous polynomial differential equations of degree three, Adv. Math. 254 (2014), 233–250. 16 HEBAI CHEN, JAUME LLIBRE AND YILEI TANG [4] M. di Bernardo, C.J. Budd, A.R. Champneys, P. Kowalczyk, Piecewise-smooth Dynamical Systems: Theory and Applications, Springer-Verlag, London, 2008. [5] M. di Bernardo, C.J. Budd, A.R. Champneys, P. Kowalczyk, A. Nordmark, G. Tost, P. Piiroinen, Bifurcations in nonsmooth dynamical systems, SIAM Review 50 (2008), 629-701. [6] X. Cen, C. Liu, L. Yang, M. Zhang, Limit cycles by perturbing quadratic isochronous centers inside piecewise polynomial differential systems, J. Differential Equations 265 (2018), 6083– 6126. [7] H. Chen, S. Duan, Y. Tang, J. Xie, Global dynamics of a mechanical system with fry friction, J. Differential Equations 265 (2018), 5490–5519. [8] H. Chen, Y. Tang, At most two limit cycles in a piecewise linear differential system with three zones and asymmetry, Physica D, https://doi.org/10.1016/j.physd.2018.08.004. [9] H. Dulac, Détermination et integration dune certaine classe déquations différentielle ayant par point singulier un centre, Bull. Sci. Math. Sér. (2) 32 (1908), 230–252. [10] F. Dumortier, J. Llibre, J.C. Artés, Qualititive theory of planar differential systems, Springer– Verlag, Berlin, 2006. [11] A.F. Filippov, Differential Equations with Discontinuous Right-Hand Sides, Kluwer Academic, Dordrecht, 1988. [12] E. Freire, E. Ponce, F. Torres, Canonical discontinuous planar piecewise linear system, SIAM J. Appl. Dyn. Syst. 11 (2012), 181-211. [13] I. Garcia, H. Giacomini, J. Giné, J. Llibre, Analytic nilpotent centers as limits of nondegen- erate centers revisited, J. Math. Anal. and Appl. 441 (2016), 893–899. [14] B. Garcı́a, J. Llibre, J.S. Pérez del Rı́o, Planar quasihomogeneous polynomial differential systems and their integrability, J. Differential Equations 255 (2013), 3185–3204. [15] J. Giné, On the centers of planar analytic differential systems, Int. J. Bifurcation and Chaos 17 (2007), 3061–3070. [16] J. Giné, M. Grau, J. Llibre, Limit cycles bifurcating from planar polynomial quasi- homogeneous centers, J. Differential Equations 259 (2015), 7135–7160. [17] J. Giné, J. Llibre, A method for characterizing nilpotent centers, J. Math. Anal. Appl. 413 (2014), 537–545. [18] R. Moussu, Symetrie et forme normale des centres et foyers degeneres, Ergodic Theory Dynam. Systems 2 (1982), 241–251. [19] W. Li, J. Llibre, J. Yang, Z. Zhang, Limit cycles bifurcating from the period annulus of quasi–homegeneous centers, J. Dyn. Diff. Equat. 21 (2009), 133–152. [20] H. Liang, J. Huang, Y. Zhao, Classification of global phase portraits of planar quartic quasi– homogeneous polynomial differential systems, Nonlinear Dynam. 78 (2014), 1659–1681. [21] M. A. Lyapunov, Probléme Général de la Stabilité du Mouvement, Ann. of Math. Stud., Vol. 17 , Princeton University Press, 1947 [22] M. Kunze, Non-Smooth Dynamical Systems, Springer-Verlag, Berlin-Heidelberg, 2000. [23] Yu.A. Kuznetsov, S. Rinaldi, A. Gragnani, One-parameter bifurcations in planar Filippov systems, Internat. J. Bifur. Chaos 13 (2003), 2157-2188. [24] J. Llibre, Centers: their integrability and relations with the divergence, Appl. Math. and Nonl. Sci. 1 (2016), 79–86. [25] O. Makarenkov, J.S.W. Lamb, Dynamics and bifurcations of nonsmooth systems: A survey, Physica D 241 (2012), 1826–1844. [26] L. Mazzi, M. Sabatini, A characterization of centres via first integrals, J. Differential Equations 76 (1988), 222–237. [27] R. Oliveira, Y. Zhao, Structural stability of planar quasihomogeneous vector fields, Qual. Theory Dyn. Syst., 13 (2014), 39–72. [28] J.M. Pearson, N.G. Lloyd, C.J. Christopher, Algorithmic derivation of centre conditions, SIAM Rev. 38 (1996), 619–636. [29] H. Poincaré, Mémoire sur les courbes définies par les équations différentielles, J. de Mathématiques 37 (1881), 375–422; Oeuvres de Henri Poincaré, vol. I, Gauthier-Villars, Paris, 1951, pp 3–84. CENTERS OF PIECEWISE QUASI–HOMOGENEOUS SYSTEMS 17 [30] H. Poincaré, Sur l’intégration des équations différentielles du premier ordre et du premier degré I and II, Rendiconti del Circolo Matematico di Palermo 5 (1891), 161–191; 11 (1897), 193–239. [31] G. Sansone, R. Conti, Non-Linear Differential Equations, Pergamon Press, New York, 1964. [32] Y. Tang, L. Wang, X. Zhang, Center of planar quintic quasi–homogeneous polynomial differ- ential systems, Discrete Contin. Dyn. Syst. 35 (2015), 2177–2191. [33] Y. Tang, L. Wang, X. Zhang, Global dynamics of planar quasi–homogeneous differential sys- tems, preprint. [34] L. Wei, X. Zhang, Limit cycle bifurcations near generalized homoclinic loop in piecewise smooth differential systems, Discrete Contin. Dyn. Syst. 36 (2016), 2803–2825. [35] J. Yu and L. Zhang, Center of planar quasi-homogeneous polynomial differential systems, Bull. Sci. Math. 147 (2018), 7–25. a College of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian 350108, P. R. China E-mail address: chen hebai@sina.com b Key Laboratory of Operations Research and Control of University in Fujian, Fuzhou University, Fuzhou, Fujian 350116, P. R. China c Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bel- laterra, Barcelona, Catalonia, Spain E-mail address: jllibre@mat.uab.cat d School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, 200240, P.R. China E-mail address: mathtyl@sjtu.edu.cn