This is a preprint of: “On the indices of singular points for planar bounded piecewise smooth polynomial vector fields”, Shimin Li, Jaume Llibre, Xianbo Sun, Nonlinear Anal. Hybrid Syst., vol. 49, 101350, 2023. DOI: [10.1016/j.nahs.2023.101350] PLANAR BOUNDED PIECEWISE SMOOTH POLYNOMIAL VECTOR FIELD SHIMIN LI1,∗ , JAUME LLIBRE2 , XIANBO SUN3 Abstract. We prove that for any piecewise-smooth bounded polynomial vec- tor field in R2 with finitely many finite H-singular points (which include sin- gular points, hyperbolic pseudo-equilibria and two fold singularities, the sum of the indices of all its finite H-singular points is 1. 1. Introduction and statement of the main result A planar polynomial differential system of the form  dx   = P (x, y), dt (1)   dy = Q(x, y), dt where P (x, y) and Q(x, y) are polynomials in the variables x and y is called a polynomial system of degree m if m is the maximum degree of the polynomials P (x, y) and Q(x, y). We denote Z(x, y) = (P (x, y), Q(x, y)) the associated vector field of system (1). In the qualitative theory of planar polynomial differential systems [10, 23], one of the most important problems is the determination and distribution of limit cy- cles, which is known as the famous Hilbert’s 16th problem. Since this problem is very difficult, mathematicians pay attention to the special forms of system (1), for Liénard systems see [4, 12], for Z2 -equivariant systems see [5, 15], for Hamiltonian systems see [21, 22]. Definition 1. A vector field (1) is said bounded when all its orbits are bounded for t ⩾ 0. Let γ(t) = (x(t, x0 ), x(t, x0 )) be the trajectory of system (1) with the initial value γ(0) = (x0 , y0 ) ∈ R2 . From [18] we know that if (1) is a bounded vector field, then the trajectory γ(t) is defined for all t ⩾ 0 and the ω−limit of any point (x0 , y0 ) is not empty and compact. In other words, Z(x, y) is a bounded vector field if for all γ(0) ∈ R2 , there exists some compact set K ⊂ R2 such that γ(t) ∈ K for each t ∈ (0, +∞). It is worth to note that most of the predator-prey systems are bounded because either the population of predator or population of prey cannot tend toward infinity in order to protect the ecological balance. 2010 Mathematics Subject Classification. 34C37, 34C07, 37G15. Key words and phrases. Bounded vector field; Index of singular points; Piecewise smooth polynomial vector field; Stereographic projection. 1 2 SHIMIN LI, JAUME LLIBRE, XIANBO SUN Σc Σa Σr 1.1 Crossing region 1.2 Attracting region 1.3 Repelling region Figure 1. Definition of the vector field following Filippov’s con- vention in the crossing, attracting and repelling regions. Bounded quadratic vector fields (1) have been studied by [8, 9, 14]. Dickson and Perko [8] established necessary and sufficient conditions in order that a quadratic vector field be bounded. The authors of [14] studied the weak focus and bifurcations of limit cycles for the bounded quadratic vector fields. Dumortier and Hersens [9] present a survey of the known results on the bounded quadratic systems. For a polynomial vector field we can define its infinite singular points using the Poincaré compactification, see for details Chapter 5 of [10]. In [6] the authors proved that: Theorem 2. Assume that a bounded polynomial vector field (1) has finitely many finite singular points and infinite singular points. Then the sum of the indices of all its finite singular points is 1. It is worth note that Theorem 2 has been generalized to the C 1 vector field in [7], and extended without the assumptions that there are finitely many infinite points in [1]. In recent years piecewise smooth differential systems [3, 11] attract more and more attention mainly due to their applications. A planar piecewise smooth (PWS) differential system is defined by   dx ±   dt = P (x, y), (2) dy   = Q± (x, y),  dt where the vector field (P + (x, y), Q+ (x, y)) is defined in y ≥ 0, and the vector field (P − (x, y), Q− (x, y)) is defined in y ≤ 0. Note that the whole plane R2 is partitioned into two open zones Σ± : {(x, y)|±y > 0} by the discontinuous boundary Σ = {(x, y)|y = 0}. Similarly, let Z ± (x, y) = (P ± (x, y), Q± )(x, y) be the piecewise smooth vector filed associated to the piecewise differential system (2). Definition 3. The discontinuous boundary Σ can be divided into three open regions: (i) Crossing region Σc = {p ∈ Σ|Q+ (p)Q− (p) > 0}, see Figure 1.1; (ii) Attracting region Σa = {p ∈ Σ|Q+ (p) < 0, Q− (p) > 0}, see Figure 1.2; (iii) Repelling region Σr = {p ∈ Σ|Q+ (p) > 0, Q− (p) < 0}, see Figure 1.3. In the crossing region Σc the trajectories of Z + (x, y) and Z − (x, y) can be con- catenated naturally. However in the attracting region Σa (resp. repelling region Σr ), the trajectories must be continued through Σa (resp. Σr ) and they slid in Σa PLANAR BOUNDED PIECEWISE SMOOTH POLYNOMIAL VECTOR FIELD 3 2.1 Pseudo-saddle 2.2 Stable pseudo-node 2.3 Unstable pseudo-node Figure 2. Definition of the hyperbolic pseudo-equilibrium of a sliding vector field (3). Σ+ Σ+ Σ− − Σ + 3.1 visible fold of Z 3.2 invisible fold of Z + Figure 3. Definition of the fold singularities. (resp. Σr ) in forward (resp. backward) time. Thus both the attracting region and the repelling region are called sliding regions, that is Σs = Σa ∪ Σr . Following the Filippov’s convex method we construct the sliding vector filed in the form (3) Zs (x, y) = λZ + (x, y) + (1 − λ)Z − (x, y), where λ ∈ (0, 1) is such that Zs (x, y) is tangent to Σs . Then Q− (x, y) (4) λ= . Q− (x, y) − Q+ (x, y) Definition 4. The point p := (x, y) ∈ Σs is a hyperbolic pseudo-equilibrium if ′ Zs (p) = 0 and Zs (p) ̸= 0, where the derivative is taken along the tangent direction of the Σ. Moreover, we will call stable pseudo-node to any point p ∈ Σa such that ′ Zs (p) = 0 and Zs (p) < 0, unstable pseudo-node to any point p ∈ Σr such that ′ Zs (p) = 0 and Zs (p) > 0, pseudo-saddle to any point p ∈ Σa such that Zs (p) = 0 ′ ′ and Zs (p) > 0 or p ∈ Σr such that Zs (p) = 0 and Zs (p) < 0, see Figure 2. The boundaries of the above three regions Σc , Σa and Σr are singularities called tangencies, that is Σt = {p ∈ Σ|Q+ (p)Q− (p) = 0}. Definition 5. A point p ∈ Σt is a fold singularity of Z + (x, y) if Q+ (p) = 0, Q+ + + + + x (p)P (p) + Qy (p)Q (p) ̸= 0. Moreover, the fold singularity p ∈ Z (x, y) is + + + + visible (resp. invisible) if Qx (p)P (p) + Qy (p)Q (p) > 0 (resp. < 0). Simi- larly, a point p ∈ Σt is a fold singularity of Z − (x, y) if Q− (p) = 0, Q− − x (p)P (p) + − − − Qy (p)Q (p) ̸= 0. Moreover, the fold singularity p ∈ Z (x, y) is visible (resp. invisible) if Q+ + + + x (p)P (p) + Qy (p)Q (p) < 0 (resp. > 0). See Figure 3. Definition 6. A point p ∈ Σ is a two fold singularity when it is a fold singularity in both Z + (x, y) and Z − (x, y). The six different types of two fold singularity are shown in Figure 4. A point p ∈ Σ is a fold-regular singularity when p is a fold singularity of Z + (x, y) (resp. Z − (x, y)) and a regular point of Z − (x, y) (resp. Z + (x, y)). 4 SHIMIN LI, JAUME LLIBRE, XIANBO SUN 4.1 II1 4.2 VI1 4.3 VV1 4.4 II2 4.5 VI2 4.6 VV2 Figure 4. Two fold singularities. σ(t) φ̄(t) σ(I) q r Figure 5. Definition of φ(t). Definition 7. A point p of the vector field Z ± (x, y) defined in (2) is called H- singular point if one of the following conditions hold: •p∈ / Σ is either a singular point of Z + (x, y) with y > 0, or a singular point of − Z (x, y) with y < 0. • p ∈ Σs is a hyperbolic pseudo-equilibrium of Zs (x, y), see Definition 4. • p ∈ Σt is a two fold singularity of Z ± (x, y), see Definition 6. Definition 8. A piecewise-smooth vector field (2) is said bounded when all its orbits are bounded for t ⩾ 0. In [16] the authors proved that the fact that there exist bounded vector fields Z + (x, y) and Z − (x, y), such that Z ± (x, y) is not bounded, and then they obtained the sufficient and necessary conditions for the piecewise-smooth quadratic system (2) to be bounded. A path in the plane R2 is a continuous map from the interval I = [0, 1] to R2 (σ : I → R2 ); that is, we assign to every t ∈ [0, 1] the point σ(t) = (σ1 (t), σ2 (t)) in the plane, such that σi : I → R are continuous maps. We say that the path σ is closed if σ(0) = σ(1). Let q be a point of R2 which does not belong to σ(I) and let r be a ray with origin at q. For every point σ(t) we denote by φ̄(t) the angle formed by the rays r and qσ(t). The angle φ̄(t) is an element of the circle R/2πZ. The function φ̄ : I → R/2πZ is continuous with respect to the parameter t; see Figure 5. Let q be a point of R2 which does not belong to σ(I). Then the difference φ(1) − φ(0) is a multiple of 2π, independent of the chosen ray r. We define the PLANAR BOUNDED PIECEWISE SMOOTH POLYNOMIAL VECTOR FIELD 5 quotient φ(1) − φ(0) i(q, σ) = , 2π which is an integer called the (topological) index of the closed path σ around the point q. Given an isolated singularity p of a vector field Z(x, y) in R2 , there is a neigh- borhood V of p on which there is no other singularity of Z(x, y). Consider now a closed path σ : I → V \{p} such that σ(I) is a small circle surrounding p. We define the (topological) index of p equal to i(Z ◦ σ, p); it is equal to the number of turns of the closed path Z ◦ σ around the origin of coordinates of R2 in a counter–clockwise sense. The index of p is independent of the chosen closed path σ, see for more details Chapter 6 of [10]. It is easy to check from the definition of index of a singular point that the index of a hyperbolic pseudo-saddle and a hyperbolic pseudo-node are −1 and 1, respectively. While the indices of the two fold singularities II1 , II2 , V I1 , V V1 , V I2 and V V2 are 1, 0, 0, −1, 0 and 0, respectively. Note that the index of two fold singularities II2 , V I1 , V V2 are 0 because the regularisation of Z ± (x, y) has no singular points, for the proof see statement (ii) of Proposition 17. Motivated by the works [1, 6, 7], the purpose of this paper is to extend Theorem 2 to the piecewise smooth vector field (2). The main result of this manuscript can be stated as follows: Theorem 9. For any bounded piecewise-smooth vector field (2) with finitely many finite H-singular points such that (5) F (x, y) := P + (x, y)Q− (x, y) − P − (x, y)Q+ (x, y) ̸= 0 if |y| is sufficiently small, the sum of the indices for all finite H-singular points is 1. 2. Proof of Theorem 9 The proof of Theorem 9 will be completed doing the following four steps. Step 1. We approximate the piecewise smooth vector field (2) by an one-parameter family of continuous vector field, which is known as regularization. We refer readers to [19] for more detail on regularization. Definition 10. A φε -regularization of a piecewise smooth vector field (2) is an one parameter family of vector field Zε (x, y) defined by (6) Zε (x, y) := (F1 (x, y), F2 (x, y)) = (1 − φε (y)) Z − (x, y) + φε (y)Z + (x, y),  where φε (y) = φ yε , and φ : R → R is an analytic transition function satisfying ′ φ(t) = 0 if y ⩽ −ε, φ(t) = 1 if y ⩾ ε and φ (t) > 0 if y ∈ (−ε, ε). It is obvious that (7) Zε (x, y) = Z ± (x, y), |y| ≥ ε. 6 SHIMIN LI, JAUME LLIBRE, XIANBO SUN The strip with y ∈ (−ε, ε) is the region where the piecewise smooth vector field is regularized. Let y = εu in (6), then we obtain the so called slow-fast system  dx   = (1 − φ(u))P − (x, εu) + φ(u)P + (x, εu) := F1 (x, εu), dt (8)   ε du = (1 − φ(u))Q− (x, εu) + φ(u)Q+ (x, εu) := F (x, εu). 2 dt Using the geometry singular perturbation theory and the blow up technique to (8), we can obtain the dynamics of piecewise smooth vector field (2) near the discontinuous boundary Σ, see for instance [13, 17, 20]. Lemma 11. If the piecewise smooth vector field (2) is bounded, then the regularized continuous vector field (6) is bounded. Proof. From (6) it follows that when ε → 0 the vector field (5) tends to the vector field (2) in R2 \ {y = 0}. So the vector field (2) is bounded in R2 \ {y = 0}. On y = 0 cannot be a sliding orbit escaping to infinity, otherwise from the definition of a sliding orbit in its neighborhood would be orbits in R2 \ {y = 0} escaping to infinity. Hence we obtain that also on the straight line y = 0 the orbits are bounded, and the lemma is proved. □ Step 2. We shall prove that the vector field Zε (x, y) has finitely many finite singular points. Lemma 12. If the piecewise smooth vector field Z ± (x, y) defined in (2) has finitely many finite H−singular points, then the regularized continuous vector field Zε (x, y) defined in (6) has finitely many finite singular points. In fact if a piecewise smooth vector field (2) has an isolated finite singular point / Σ, then we can choose ε0 > 0 sufficiently small such that |y0 | ⩾ ε0 . p0 = (x0 , y0 ) ∈ Thus p0 is a singular point of Zε (x, y) for 0 < ε ⩽ ε0 . Now we just need to consider the case p ∈ Σ. Without loss of generality we assume that p = (0, 0) ∈ Σ. Lemma 12 can be proved from the following six propositions. Proposition 13. Let p ∈ Σc be a crossing point of Z ± (x, y). Then there exist a neighborhood V of p in Σ and ε0 such that for every 0 < ε ⩽ ε0 , Zε (x, y) has no singular points in V . Proof. For the proof see Proposition 6 of [19]. □ Proposition 14. Let p ∈ Σs := Σa ∪ Σr with Zs (p) ̸= 0. Then there exist a neighborhood V of p in Σ and ε0 such that for every 0 < ε ⩽ ε0 , Zε (x, y) has no singular points in V . Proof. For the proof see Proposition 6 of [19]. □ s a r Proposition 15. Let p ∈ Σ = Σ ∪ Σ be hyperbolic pseudo-equilibrium of Zs (p). Then there exist a neighborhood V of p in Σ and ε0 such that for every 0 < ε ⩽ ε0 , Zε (x, y) has a unique singular point in V . PLANAR BOUNDED PIECEWISE SMOOTH POLYNOMIAL VECTOR FIELD 7 Proof. For the proof see Proposition 8 of [19]. □ Proposition 16. Let p ∈ Σ be a fold-regular point of Z ± (x, y). Then there exist a neighborhood V of p in Σ and an ε0 > 0 such that for every 0 < ε ⩽ ε0 , Zε (x, y) has no singular points in V . Proof. For the proof see Proposition 9 of [19]. □ Proposition 17. Let p ∈ Σ be a two fold singularity of Z ± (x, y), the following statements hold. (i) If p ∈ II1 , V I2 , V V1 , then there exist a neighborhood V of p in Σ and an ε0 > 0 such that for every 0 < ε ⩽ ε0 , Zε (x, y) has a unique singular point in V . (ii) If p ∈ II2 , V V2 , V I1 , there exist a neighborhood V of p in Σ and an ε0 > 0 such that for every 0 < ε ⩽ ε0 , Zε (x, y) has no singular points in V . Proof. Without loss of generality we assume that p = (0, 0) is a two fold singu- larity of Z ± (x, y), we have Q+ (0, 0) = Q− (0, 0) = 0, P + (0, 0)Q+ x (0, 0) ̸= 0 and P − (0, 0)Q− x (0, 0) ̸ = 0. − − In the case II1 we have P + (0, 0)Q+ x (0, 0) < 0 and P (0, 0)Qx (0, 0) > 0. With- out loss of generality we assume that P + (0, 0) > 0 and P − (0, 0) < 0, see Figure 4.1. Recall that F (x, y) = P + (x, y)Q− (x, y) − P − (x, y)Q+ (x, y), it is obvious that − F (0, 0) = 0, and Fx (0, 0) = P + (0, 0)Q+ + x (0, 0) − P (0, 0)Qx (0, 0) > 0. Then there exists a neighborhood V of (0, 0), in which F (x, y) = 0 has a unique solution x = α(y), y ∈ JV , a neighborhood of y = 0, i.e. F (α(y), y) ≡ 0 in JV according to the implicit function theorem. Similar treatment as that in the proof of Proposition 15, we can conclude that Zε (x, y) has a unique singular point in V . − − In the case V I2 we have P + (0, 0)Q+x (0, 0) < 0 and P (0, 0)Qx (0, 0) < 0. With- + − out loss of generality we assume that P (0, 0) > 0 and P (0, 0) < 0, see Figure 4.5. Since Fx (0, 0) ̸= 0, following the proof of II1 , we ensure that Zε (x, y) has a unique singular point in V . − − In the case V V1 we have P + (0, 0)Q+ x (0, 0) > 0 and P (0, 0)Qx (0, 0) < 0. We + − assume that P (0, 0) > 0 and P (0, 0) < 0, see Figure 4.3. Similar to the case II1 , we can prove that Zε (x, y) has a unique singular point in V . − − In the case V I1 we have P + (0, 0)Q+x (0, 0) < 0 and P (0, 0)Qx (0, 0) < 0. We + − assume without loss of generality that P (0, 0) > 0 and P (0, 0) > 0, see Figure 4.5. We claim that Zε (x, y) cannot have singular points in any small neighbor- hood V of (0, 0). Indeed, if P − (0, 0) > P + (0, 0), then from (6) we have that P − (0, 0) P − (x, y) ϕε (−ε) = − + > 1, and hence − > 1 in a small P (0, 0) − P (0, 0) P (x, y) − P + (x, y) neighborhood of (0, 0) by continuity, which is in contradiction with φε (y) < 1. If P − (0, 0) P − (x, y) P − (0, 0) < P + (0, 0), then − + < 0, and − <0 P (0, 0) − P (0, 0) P (x, y) − P + (x, y) in a small neighborhood of (0, 0) by continuity. Thus φε (y) does not satisfy the requirement that φε (y) ∈ [0, 1]. If P − (0, 0) = P + (0, 0), then we can choose a P − (x, y) suitable neighborhood V of (0, 0), such that | − | > 1, and so P (x, y) − P + (x, y) 8 SHIMIN LI, JAUME LLIBRE, XIANBO SUN P − (x, y) φε (y) = also does not satisfy the requirement in any small P − (x, y) − P + (x, y) neighborhood of (0, 0). − − In the case II2 we have P + (0, 0)Q+ x (0, 0) < 0 and P (0, 0)Qx (0, 0) > 0. Without + − loss of generality we assume that P (0, 0) > 0 and P (0, 0) > 0, see Figure 4.4. Similar to the case V I1 , we can conclude that Zε (x, y) has no unique singular points in V . − − In the case V V2 we have P + (0, 0)Q+x (0, 0) > 0 and P (0, 0)Qx (0, 0) < 0. With- + − out loss of generality we assume that P (0, 0) > 0 and P (0, 0) > 0, see Figure 4.6. Similar to the case V I1 , we can conclude that Zε (x, y) has no singular points in V . The proposition follows. □ Step 3. We introduce the stereographic projection, which project the continuous vector field Zε (x, y) := (F1 (x, y), F2 (x, y)) defined in (6) in the plane R2 to the tangent vector field Vε (u, v, w) := (V1 , V2 , V3 ) on the sphere S2 = {(u, v, w)|u2 + v 2 + (w − 1/2)2 = 1/4}. More precisely, the stereographic projection ρ : R2 7→ S2 is the intersection of the straight line passing through the point (x, y, 0) of the plane R2 and the north pole (0, 0, 1) with the sphere S2 . Thus the sphere S2 is tangent to R2 at the south pole point (0, 0, 0), see Figure 6. Now we consider the projection of the vector field Zε (x, y) from R2 to S2 via stereographic projection which assigns to each point p := (x, y) ∈ R2 the point p̄ := (u, v, w) ∈ S2 through the relations u v x= ,y = , then we have 1−w 1−w     u v u v V1 (u, v, w) = (1 − w − u2 )F1 , − uvF2 , , 1−w 1−w 1−w 1−w     u v u v (9) V2 (u, v, w) = −uvF1 , + (1 − w − v 2 )F2 , , 1−w 1−w 1−w 1−w     u v u v V3 (u, v, w) = u(1 − w)F1 , + v(1 − w)F2 , . 1−w 1−w 1−w 1−w Note that system (9) is not defined at the north pole point p = (0, 0, 1), thus we extend it to S2 by a change of time scale t = (1 − w)n τ , where n is the maximum of the degrees of the polynomials F1 and F2 , and then we obtain the Ṽε = (1 − w)m Vε . We shall call the stereographic compactification of Zε (x, y), S(Ṽε ) the induced vector field in the sphere S2 . It is obvious that the dynamics of the orbits of Zε (x, y) near infinity are determined by the dynamics of the orbits of S(Ṽε ) near p = (0, 0, 1). Lemma 18. If Zε (x, y) has finitely many finite singular points, then S(Ṽε ) is a continuous tangent vector field in the sphere S2 has finitely many singular points. Proof. S(Ṽε ) is a continuous tangent vector field because Zε (x, y) is continuous. Suppose that (x∗ , y ∗ ) is a singular point of Zε (x, y) = (F1 (x, y), F1 (x, y)), that is F1 (x∗ , y ∗ ) = F2 (x∗ , y ∗ ) = 0. From (9), we have   ∗ ∗ ∗ x∗ y∗ (x∗ )2 + (y ∗ )2 (u , v , w ) = , , 1 + (x∗ )2 + (y ∗ )2 1 + (x∗ )2 + (y ∗ )2 1 + (x∗ )2 + (y ∗ )2 PLANAR BOUNDED PIECEWISE SMOOTH POLYNOMIAL VECTOR FIELD 9 w N p̄ y/v O p x/u Figure 6. Stereographic projection. O = (0, 0, 0), N = (0, 0, 1), p = (x, y, 0), p̄ = (u, v, w), is a singular point of S(Ṽε ). Since Zε (x, y) have finitely many finite singular points, we can conclude that S(Ṽε ) have finitely many singular points. □ Step 4. According to Lemma 18, S(Ṽε ) is continuous on S2 with finitely many singular points, the sum of the indices IS(Ṽε ) = 2 by Poincaré-Hopf Theorem (see for instance Theorem 6.30 of [10]). From Lemma 11, we know that Zε (x, y) is a bounded vector field, and then there are no orbits of S(Ṽε ) whose ω-limit is N = (0, 0, 1). Thus we have both the number of elliptic sectors e(N ) = 0 and the number of hyperbolic sector h(N ) = 0. According to the Poincaré index formula (see for more details Proposition 6.32 of [10]), the index of the north pole point iN = 1 + (e(N ) − h(N ))/2 = 1. From the above analysis, we can conclude that the sum of the indices at all the finite singular points of Zε (x, y) is 1 because the north pole locally is an unstable node due to the fact that the vector field is bounded. From (6) the phase portraits of the vector field Zε (x, y) tends in a continuous way to the phase portrait of the vector field Z ± (x, y). Since there are finitely many finite singular points of Zε (x, y) and of Z ± (x, y) the sum of their indices are preserved by continuous deformations (see for instance [2]), hence the sum of the indices of the H-singular points of Z ± (x, y) is also 1. This completes the proof of Theorem 9. 3. Acknowledgements The first author is supported by NNSF of China (No. 12071091) and the Natural Science Foundation of Guangdong Province, PR China (No. 2019A1515011885). 10 SHIMIN LI, JAUME LLIBRE, XIANBO SUN The second author is partially supported by the Agencia Estatal de Investigación grant PID2019-104658GB-I00, and the H2020 European Research Council grant MSCA-RISE-2017-777911. The third author is supported by the NNSFs of China (No. 11861009, 12001121) and the Natural Science Foundation of Guangxi Province (No. 2020GXMSFAA). References [1] L. Barreira, J. Llibre, C. Valls, Bounded polynomial vector fields in R2 and Rn , J. Diff. Equations 268 (2020) 4416–4422. [2] R.F. 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Xiao, The maximal number of limit cycles bifurcationg from a Hamil- tonian triangle in quadratic systems, J. Diff. Equations, 280 (2021) 139–178. [23] Z. Zhang, T. Ding, W. Huang, Z. Dong, Qualitative Theory of Differential Equation, Transl. Math. Monogr., Providence, Rhode Island, 1992. PLANAR BOUNDED PIECEWISE SMOOTH POLYNOMIAL VECTOR FIELD 11 1 School of Mathematics and Statistics, Guangdong University of Finance and Eco- nomics, Guangzhou, 510320, P.R. China Email address: lism1983@126.com 2 Departament de Matemàtiques, Universitat Autònoma de Barcelona, Barcelona, 08193, Spain Email address: jllibre@mat.uab.cat 3 Department of Applied Mathematics, Guangxi University of Finance and Econom- ics, Nanning, 530003, P.R. China Email address: xianbo01@126.com