This is a preprint of: “Crossing limit cycles of planar discontinuous piecewise differential systems formed by isochronous centers”, Claudio A. Buzzi, Yagor Romano Carvalho, Jaume Llibre, Dyn. Syst., vol. 37(4), 710–728, 2022. DOI: [10.1080/14689367.2022.2122779] CROSSING LIMIT CYCLES OF PLANAR DISCONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS FORMED BY ISOCHRONOUS CENTERS CLAUDIO A. BUZZI1 , YAGOR ROMANO CARVALHO2 , AND JAUME LLIBRE3 Abstract. These last years an increasing interest appeared in studying the planar discontinuous piecewise differential systems motivated by the rich applications in mod- eling real phenomena. The understanding of the dynamics of these systems has many difficulties. One of them is the study of their limit cycles. In this paper, we study the maximum number of crossing limit cycles of some classes of planar discontinuous piecewise differential systems separated by a straight line and formed by combinations of linear centers (consequently isochronous) and cubic isochronous centers with homo- geneous nonlinearities. For these classes of planar discontinuous piecewise differential systems we solved the extension of the 16th Hilbert problem, i.e. we provide an upper bound for their maximum number of crossing limit cycles. 1. Introduction and statement of the main results A limit cycle is a periodic orbit of a differential system in R2 that is isolated in the set of all its periodic orbits. The analysis of the existence of limit cycles became important in the applications of the real world because many phenomena are related to their existence, see for instance the Van der Pol oscillator [28, 29]. The study of limit cycles began with Poincaré [23] at the end of the nineteenth century. On the other hand, the study of the continuous piecewise linear differential systems separated by a straight line has special attention from the mathematicians, mainly because these systems appear in a natural way in the control theory, see for instance the books [10, 16, 20, 21], in mechanics, electrical circuits, economy, see for instance the books [1, 25] and the surveys [19, 26]. The easiest continuous piecewise linear differential systems are formed by two linear differential systems separated by a straight line and it is known that such systems have at most one limit cycle, see [7, 13, 17, 18]. But it is also known that if both linear differential systems are linear centers, then the continuous piecewise linear differential system has no limit cycles, see for example [15]. However, if we eliminate the continuity of such systems, that is, they do not need to coincide on the line of discontinuity, then it is known that these systems can have three limit cycles as we can see in [2, 4, 8, 11, 12, 14], but it is unknown if three is the maximum number of limit cycles that they can have. When we consider planar discontinuous piecewise differential systems we can have two kinds of limit cycles: sliding limit cycles or crossing limit cycles. The first ones contain some segment of the line of discontinuity, and the second ones only contain some points of the line of discontinuity. For more details on the discontinuous piecewise differential systems see the books [1, 6, 19, 25]. In this work, we are going to study the crossing limit cycles, and in what follows sometimes when we talk about limit cycles, we are talking about crossing limit cycles. 2010 Mathematics Subject Classification. 37G15, 37D45. Key words and phrases. Limit cycles, linear centers, cubic isochronous centers with homogeneous nonlinearities, discontinuous piecewise differential systems, first integrals. 1 2 CLAUDIO A. BUZZI1 , YAGOR ROMANO CARVALHO2 , AND JAUME LLIBRE3 An isochronous center of a planar differential system is a singularity such that there is a neighborhood of it formed by periodic orbits with the same period. It is well known that the linear centers are isochronous. Llibre and Teixeira in [15] were interested in studying if a discontinuous piecewise differential system formed with only linear centers can create limit cycles. In this work, we are interested in a similar problem. The first objective is to study the maximum number of limit cycles that discontinuous piecewise differential systems separated by a straight line can have when one differential system is a linear center and the other is a cubic isochronous center with homogeneous nonlinearities. The second objective is to study the maximum number of limit cycles that discontinuous piecewise differential systems separated by a straight line can have when both differential systems are cubic isochronous centers with homogeneous nonlinearities. Without loss of generality, we can assume that the straight line of discontinuity is x = 0. Consider the polynomial differential systems of the form dx = ẋ = −y + a30 x3 + a21 x2 y + a12 xy 2 + a03 y 3 , dt (1) dy 3 2 2 3 = ẏ = x + b30 x + b21 x y + b12 xy + b03 y . dt Pleshkan in [22] classify which of these differential systems have an isochronous center at the origin of coordinates. Thus a cubic system (1) has an isochronous center at the origin if and only if the system can be transformed to one of the following four differential systems ẋ = −y + x3 − xy 2 , ẋ = −y + x3 − 3xy 2 , (S13 ) : 2 3 (S23 ) : ẏ = x + x y − y , ẏ = x + 3x2 y − y 3 , (2) 3 ẋ = −y + 3x2 y, 3 ẋ = −y − 3x2 y, (S3 ) : (S4 ) : ẏ = x − 2x3 + 9xy 2 , ẏ = x + 2x3 − 9xy 2 . doing a linear change of coordinates and a rescaling of time. The first integrals of the differential systems (2) can be founded in [3], and they are x2 + y 2 (x2 + y 2 )2 (S13 ) : H1 (x, y) = , (S23 ) : H2 (x, y) = , 1 + 2xy 1 + 4xy x2 + y 2 − 4x4 + 4x6 x2 + y 2 + 4x4 + 4x6 (S33 ) : H3 (x, y) = , (S43 ) : H4 (x, y) = , (−1 + 3x2 )3 (1 + 3x2 )3 respectively. Llibre and Teixeira proved in [15] that after doing an affine transformation and a rescaling of the independent variable any linear center can be written into the form 4A2 + ω 2 ẋ = −Ax − y + B, (Lc ) : 4D (3) ẏ = Dx + Ay + C, where A, B, C, D, ω are real numbers with D, ω > 0. This system has the first integral HL (x, y) = 4(Dx + Ay)2 + 8D(Cx − By) + y 2 ω 2 . (4) In this paper we only study discontinuous piecewise differential systems separated by a straight line and formed by two systems that can be transformed, after an affine transformation, in differential systems belonging to some of the three classes (S13 ), (S23 ) and (Lc ). Here we do not take into account a rescaling of the independent variable CROSSING LIMIT CYCLES FOR A FAMILY OF ISOCHRONOUS CENTERS 3 because such a rescaling does not change the phase portrait of the differential systems, only change the speed at which their orbits are traveled. Let A, B ∈ {(Lc ), (S13 ), (S23 )} be. Then we denote by Nclc [A, B] the maximum number of crossing limit cycles that the class of the discontinuous piecewise differential systems separated by a straight line and formed by the differential systems A and B can have. Our results are summarized in the next theorem. Theorem 1. Consider the class of discontinuous piecewise differential systems separated by a straight line and formed by two differential systems which after an affine change of variables belong to the classes (Lc ), (S13 ) or (S23 ). Then (i) Nclc [(Lc ), (S13 )] = Nclc [(S13 ), (S13 )] = 1; (ii) Nclc [(Lc ), (S23 )] = 2; (iii) 1 ≤ Nclc [(S13 ), (S23 )] ≤ 3; (iv) 1 ≤ Nclc [(S23 ), (S23 )] ≤ 9; (v) Nclc [(Lc ), (Lc )] = 0. In the following table, we summarize the results of Theorem 1, and where we have added the numbers between parentheses, which mean the maximum number of known limit cycles for the corresponding class of discontinuous piecewise differential systems. Inside the proof of Theorem 1 we will provide the corresponding discontinuous piecewise differential systems realizing the number of limit cycles that appear between parentheses. Linear (S13 ) (S23 ) Linear 0 1 (1) 2 (2) (S13 ) 1 (1) 1 (1) 3 (1) (S23 ) 2 (2) 3 (1) 9 (1) We want to stress that our contribution in Theorem 1 restricts to items (i), (ii), (iii), and (iv). As we have already said item (v) was proved, for example in [15]. Moreover, in this work, we do not work with the families of isochronous centers (S33 ) and (S43 ). The difficulties of these cases correspond to the fact that the numerator of their first integrals has a very high degree. So we still haven’t found a way to improve the upper limit of the maximum number of limit cycles. 2. Definitions and Preliminaries In this section, we review some definitions and previous results that will be used for proving our results. Let ẋ = P (x, y), ẏ = Q(x, y), (5) be a planar polynomial differential system. Then a nonconstant analytic function H = H(x, y) is a first integral of system (5) if it is constant on all solution curves (x(t), y(t)), that is, H(x(t), y(t)) = constant for all values of t for which H(x(t), y(t)) is defined. Clearly, H is the first integral of system (5) if and only if ∂H ∂H XH = P +Q = 0. ∂x ∂y 4 CLAUDIO A. BUZZI1 , YAGOR ROMANO CARVALHO2 , AND JAUME LLIBRE3 Therefore the solutions of system (5) are contained in the level curves of the function H. A differential system in the plane with a first integral is said integrable. For more details on integrable systems see Chapter 8 of [5]. Consider a planar discontinuous piecewise differential system separated by the straight line x = 0. A periodic orbit of a such system must intersect the line x = 0 exactly in two points. Since we always will work with integrable systems, let H 1 and H 2 be the two first integrals of the two differential systems forming the discontinuous piecewise differential system. If we have a limit cycle which intersect x = 0 in the two points by (0, y1 ) and (0, y2 ) with y1 < y2 , then H 1 (0, y1 ) − H 1 (0, y2 ) = 0, and H 2 (0, y1 ) − H 2 (0, y2 ) = 0. (6) Therefore our objective will be to control how many solutions the system (6) has, and to try to find discontinuous piecewise differential systems that satisfy this number of solutions. It is important to note that every solution y1 < y2 of system (6) in general does not provide a limit cycle. For instance when the system has a continuum of solutions; when the level curves of either H 1 or H 2 through the points (0, y1 ) and (0, y2 ) are disconnected; or when the two pieces of the level curves of H 1 and H 2 through the points (0, y1 ) and (0, y2 ) provide a closed curve with a non-compatible orientation. In summary, every crossing limit cycle provides a unique solution of system (6) with y1 < y2 , but a solution of the system (6) does not necessarily provide a crossing limit cycle. Since the system (6) for our discontinuous piecewise differential systems always can be reduced to a polynomial system, a good tool that we will use for estimating its number of solutions will be the Bezout Theorem. This theorem says that if a polynomial system has a finite number of solutions, then the number of its solutions is at most the product of the degrees of the polynomials that appear in the system, for more details see [24, 27]. We also must take into account that if we have a polynomial system F (y1 , y2 ) = 0, G(y1 , y2 ) = 0, satisfying that F (y1 , y2 ) = F (y2 , y1 ) and G(y1 , y2 ) = G(y2 , y1 ), then (y1 , y2 ) and (y2 , y1 ) are solutions. Since we only are interested in the solutions y1 < y2 . So the number of solutions of system (6) must be divided by two, to obtain an upper bound for the number of limit cycles. Moreover, whenever we have a symmetric polynomial system (invariant by permuta- tions of its variables), as our system (6) sometimes it will be convenient to do the change of variables (y1 , y2 ) → (z, w) given by z = y1 + y2 and w = y1 y2 , in order to study its solutions. For more details see Section III.4 of [9]. Doing the general affine change of variables x = aX + bY + c, y = αX + βY + γ, with aβ − bα 6= 0, the first integral of the system (S13 ) is transformed into (c + aX + bY )2 + (Xα + Y β + γ)2 H1c (X, Y ) = , 1 + 2(c + aX + bY ) + (Xα + Y β + γ) and the first integral of the system (S23 ) is transformed into ((c + aX + bY )2 + (Xα + Y β + γ)2 )2 H2c (X, Y ) = . 1 + 4(c + aX + bY )(Xα + Y β + γ) The normal form of a general linear center and its first integral are given in (3) and (4), respectively. CROSSING LIMIT CYCLES FOR A FAMILY OF ISOCHRONOUS CENTERS 5 3. Proof of Theorem 1 Figure 1. The limit cycle of the discontinuous piecewise differential sys- tem (7)-(8) of Proposition 1. The proof that Nclc [(Lc ), (S13 )] = 1 is given in the next proposition. Proposition 1. Consider the class of discontinuous piecewise differential systems sep- arated by the straight line x = 0 and formed by a linear center and a cubic isochronous center (S13 ) after an arbitrary affine change of variables. Then these differential sys- tems can have at most one limit cycle. Moreover, the discontinuous piecewise differential system in this class is formed by the differential system y ẋ = 2 − , ẏ = x, (7) 4 in x ≥ 0; and by the differential system 1 ẋ = (23 + 38x + 7y + 16x2 + 16xy − 3y 2 + 8x2 y − 2xy 2 ), 2 (8) ẏ = 12 + 12x + 13y + 16xy + 4xy 2 − y 3 , in x ≤ 0, has one limit cycle, reaching the maximum upper bound. See Figure 1. Proof. Under the assumptions of the proposition if such discontinuous piecewise differ- ential systems have a limit cycle intersecting the discontinuity straight line x = 0 in the two points (0, y1 ) and (0, y2 ), we have that y1 and y2 must satisfy that H1c (0, y1 ) − H1c (0, y2 ) = 0, and HL (0, y1 ) − HL (0, y2 ) = 0, or equivalently (y1 − y2 )(8BD − 4A2 y1 − 4A2 y2 − y1 ω 2 − y2 ω 2 ) = 0, (y1 − y2 )(−2bc − b2 y1 − b2 y2 + 2c3 β + 2bc2 y1 β + 2bc2 y2 β + 2b2 cy1 y2 β − y1 β 2 − y2 β 2 − 2cy1 y2 β 3 − 2bc2 γ − 2b2 cy1 γ − 2b2 cy2 γ − 2b3 y1 y2 γ − 2βγ − 2cy1 β 2 γ − 2cy2 β 2 γ + 2by1 y2 β 2 γ − 2cβγ 2 + 2by1 βγ 2 + 2by2 βγ 2 + 2bγ 3 )/((1 + 2cy1 β + 2by12 β + 2cγ + 2by1 γ)(1 + 2cy2 β + 2by22 β + 2cγ + 2by2 γ)) = 0. 6 CLAUDIO A. BUZZI1 , YAGOR ROMANO CARVALHO2 , AND JAUME LLIBRE3 Since (1 + 2cy1 β + 2by12 β + 2cγ + 2by1 γ)(1 + 2cy2 β + 2by22 β + 2cγ + 2by2 γ) 6= 0 and we have y1 6= y2 , then we obtain the equivalent system F11 (y1 , y2 ) = 8BD − 4A2 y1 − 4A2 y2 − y1 ω 2 − y2 ω 2 = 0, F21 (y1 , y2 ) = −2bc − b2 y1 − b2 y2 + 2c3 β + 2bc2 y1 β + 2bc2 y2 β + 2b2 cy1 y2 β − y1 β 2 − y2 β 2 − 2cy1 y2 β 3 − 2bc2 γ − 2b2 cy1 γ − 2b2 cy2 γ − 2b3 y1 y2 γ − 2βγ (9) 2 2 2 2 2 2 − 2cy1 β γ − 2cy2 β γ + 2by1 y2 β γ − 2cβγ + 2by1 βγ + 2by2 βγ + 2bγ 3 = 0. So in system (9) we have that F11 (y1 , y2 ) is a polynomial of degree 1 and F21 (y1 , y2 ) is a polynomial of degree 2. Eventually, system (9) could have a continuum of solutions (y1 , y2 ), but then the possible periodic solutions would not be limit cycles. Therefore we assume that this system has finitely many solutions. Then by Bezout Theorem, this system has at most two solutions. Since if (y1∗ , y2∗ ) is a solution, also (y2∗ , y1∗ ) is a solution, but we are interested in the solutions whose first component is smaller than the second one, so it follows that the discontinuous piecewise differential system has at most one limit cycle. Now we shall prove that the discontinuous piecewise differential system is separated by the straight line x = 0 and defined by the differential systems (7) and (8) has one limit cycle. For x ≥ 0 the system (7) is a linear center and for x ≤ 0 the system is the cubic isochronous center (S13 ) after the affine change of variables (x, y) → (3 + 2x, 1 + 2x − y). The first integrals of this piecewise differential system are (3 + 2x)2 + (1 + 2x − y)2 HL1 (x, y) = 4x2 − 16y + y 2 and H1c1 (x, y) = . 1 + 2(3 + 2x)(1 + 2x − y) Then the system (9) for this piecewise differential system is − 16 + y1 + y2 = 0, −(−46 − 7y1 − 7y2 + 6y1 y2 ) = 0, (−7 + 6y1 )(−7 + 6y2 )) √ √  with the solution (y11∗ , y21∗ ) = 31 24 − 339, 24 + 339 , and observe that we have y11∗ < y21∗ . The solution (x11 (t), y11 (t)) in x ≥ 0 of system (7) such that (x11 (0), y11 (0)) = (0, y11∗ ) is contained in the level curve = HL1 (0, y11∗ ) = − 793 = HL1 (0, y21∗ ), i.e. in the curve 79 HL1 (x, y) = 4x2 − 16y + y 2 = − . 3 The solution (x12 (t), y21 (t)) in x ≤ 0 of system (8) such that (x12 (0), y21 (0)) = (0, y21∗ ) is 7 contained in the level curve H1c1 (0, y11∗ ) = − = H1c1 (0, y21∗ ), i.e. in the curve 3 (3 + 2x) + (1 + 2x − y)2 2 7 H1c1 (x, y) = =− . 1 + 2(3 + 2x)(1 + 2x − y) 3 Drawing the orbits (x1k (t), yk1 (t)), k = 1, 2, we obtain the limit cycle of Figure 1.  The proof that Nclc [(Lc ), (S23 )] = 2 is given in the next proposition. Proposition 2. Consider the class of discontinuous piecewise differential systems sep- arated by the straight line x = 0 and formed by a linear center and a cubic isochronous center (S23 ) after an arbitrary affine change of variables. Then these differential systems CROSSING LIMIT CYCLES FOR A FAMILY OF ISOCHRONOUS CENTERS 7 Figure 2. The two limit cycles of the discontinuous piecewise differential system (10)-(11) of Proposition 2. The figure on the right shows that there is no intersection between the two limit cycles. can have at most two limit cycles. Moreover, the discontinuous piecewise differential system in this class is formed by the differential system 5 4 ẋ = 1 + x − y, ẏ = + x − y. (10) 4 5 in x ≥ 0, and the differential system 1 ẋ = (−1 + 30x + 25y + 24x2 − 15y 2 − 48x3 − 120x2 y − 90xy 2 − 20y 3 ), 2 (11) ẏ = −20x − 15y + 24xy + 18y 2 + 32x3 + 72x2 y + 48xy 2 + 9y 3 , in x ≤ 0, has two limit cycles, reaching the maximum upper bound. See Figure 2. Proof. Under the assumptions of the proposition if such discontinuous piecewise differ- ential systems have a limit cycle intersecting the discontinuity straight line x = 0 in the two points (0, y1 ) and (0, y2 ), then y1 and y2 must satisfy that H2c (0, y1 ) − H2c (0, y2 ) = 0, and HL (0, y1 ) − HL (0, y2 ) = 0, (12) or equivalently G21 (y1 , y2 ) = −(y1 − y2 )(8BD − 4A2 y1 − 4A2 y2 − y1 ω 2 − y2 ω 2 ) = 0, G22 (y1 , y2 ) = −((y1 − y2 )P22 (y1 , y2 )/((1 + 4cy1 β + 4by12 β + 4cγ + 4by1 γ) (1 + 4cy2 β + 4by22 β + 4cγ + 4by2 γ))) = 0, where P22 (y1 , y2 ) is a polynomial of degree 5. Since (1 + 4cy1 β + 4by12 β + 4cγ + 4by1 γ)(1 + 4cy2 β + 4by22 β + 4cγ + 4by2 γ) 6= 0 and y1 < y2 , we can remove these terms to solve the system, and we get the equivalent system F12 (y1 , y2 ) = 8BD − 4A2 y1 − 4A2 y2 − y1 ω 2 − y2 ω 2 = 0, (13) F22 (y1 , y2 ) = P22 (y1 , y2 ) = 0. Eventually, system (13) could have a continuum of solutions (y1 , y2 ), but then the possible periodic solutions would not be limit cycles. Therefore we assume that this system has finitely many solutions. From F12 (y1 , y2 ) = 0 we obtain that 8BD − 4A2 Y − Y ω 2 y1 = , (14) 4A2 + ω 2 8 CLAUDIO A. BUZZI1 , YAGOR ROMANO CARVALHO2 , AND JAUME LLIBRE3 with 4A2 + ω 2 6= 0 because ω > 0. So if we substitute (14) in F22 (y1 , y2 ) = 0, then we get a polynomial p2 (y2 ) of degree 4 in the variable y2 , and p2 (y2 ) has at most four roots. Therefore the system (13) has at most four solutions, and consequently the discontinuous piecewise differential system can have at most two limit cycles. Now we shall prove that the discontinuous piecewise differential system separated by the straight line x = 0 and formed by the linear center (10) and the cubic isochronous center (11) obtained from (S23 ) after the affine change of variables (x, y) → (−1 − 2x − y, 1 − 2x − 2y), has two limit cycles. The first integrals of this piecewise differential system are   2 4 2 ((−1 − 2x − y)2 + (1 − 2x − 2y)2 )2 HL2 (x, y) = 4(x−y) +8 x − y +y , H2c1 (x, y) = . 5 1 + 4(−1 − 2x − y)(1 − 2x − 2y) So system (12) is equivalent to − 8 + 5y1 + 5y2 = 0, 8 − 104y1 + 60y12 − 75y13 − 104y2 + 220y1 y2 − 155y12 y2 + 100y13 y2 + 60y22 − − 155y1 y22 − 60y12 y22 + 200y13 y22 − 75y23 + 100y1 y23 + 200y12 y23 = 0, with the solutions   q   q  2∗ 2∗ 1 √ 1 √ (y11 , y12 ) = 8 − 2(227 − 5 2041) , 8 + 2(227 − 5 2041) , 10 10   q   q  2∗ 2∗ 1 √ 1 √ (y21 , y22 ) = 8 − 2(227 + 5 2041) , 8 + 2(227 + 5 2041) . 10 10 2∗ 2∗ 2∗ 2∗ Observe that we have y11 < y12 and y21 < y22 . 2 The solution (x211 (t), y11 2 (t)) in x ≥ 0 of system (10) such that (x211 (0), y11 (0)) = (0, y11 2∗ ) 2∗ 1 √  2∗ is contained in the level curve HL2 (0, y12 )= 39 − 2041 = HL2 (0, y11 ), i.e. in the 2 curve   2 4 1 √  HL2 (x, y) = 4(x − y) + 8 x − y + y2 = 39 − 2041 . 5 2 The solution (x21 (t), y21 (t)) in x ≤ 0 of system (11) such that (x221 (0), y21 2 2 2 (0)) = (0, y12 2∗ ) 2∗ 1 √  2∗ is contained in the the level curve H2c1 (0, y12 )= 263 − 5 2041 = H2c1 (0, y11 ), i.e. 14 in the curve ((−1 − 2x − y)2 + (1 − 2x − 2y)2 )2 1  √  H2c1 (x, y) = = 263 − 5 2041 . 1 + 4(−1 − 2x − y)(1 − 2x − 2y) 14 The solution (x212 (t), y12 2 (t)) in x ≥ 0 of system (10) such that (x211 (0), y11 2 2∗ (0)) = (0, y21 ) 1 √  is contained in the level HL2 (0, Y222∗ ) = 39 + 2041 = HL2 (0, y21 2∗ ), i.e. in the curve 2   2 4 2 1 √  HL2 (x, y) = 4(x − y) + 8 x−y +y = 39 + 2041 . 5 2 The solution (x222 (t), y22 2 (t)) in x ≤ 0 of system (11) such that (x222 (0), y22 2 (0)) = (0, Y222∗ ) 1 √  is contained in the level curve H2c1 (0, Y222∗ ) = 263 + 5 2041 = H2c1 (0, y21 2∗ ), i.e. in 14 the curve ((−1 − 2x − y)2 + (1 − 2x − 2y)2 )2 1  √  H2c1 (x, y) = = 263 + 5 2041 . 1 + 4(−1 − 2x − y)(1 − 2x − 2y) 14 CROSSING LIMIT CYCLES FOR A FAMILY OF ISOCHRONOUS CENTERS 9 2 Drawing the orbits (x2kj (t), ykj (t)), k, j = 1, 2, we obtain the two limit cyles of Figure 2.  The proof that Nclc [(S13 ), (S13 )] = 1 is given in the next proposition. Figure 3. The limit cycle of the discontinuous piecewise differential sys- tem (15)-(16) of Proposition 3. Proposition 3. Consider the class of discontinuous piecewise differential systems sepa- rated by the straight line x = 0 and formed by two distincts cubic isochronous center (S13 ) after an arbitrary affine change of variables. Then these differential systems can have at most one limit cycle. Moreover, the discontinuous piecewise differential system in this class is formed by the differential system 1 ẋ = − (−1 − 4x + 3y + 12x2 − 4xy − y 2 − 8x3 + 2xy 2 ), 2 (15) ẏ = 1 − 6x + 2y + 4x2 + 4xy − 3y 2 − 4x2 y + y 3 , in x ≥ 0 and by the differential system 1 ẋ = (23 + 38x + 7y + 16x2 + 16xy − 3y 2 + 8x2 y − 2xy 2 ), 2 (16) ẏ = 12 + 12x + 13y + 16xy + 4xy 2 − y 3 , in x ≤ 0 has one limit cycle, reaching the maximum upper bound. See Figure 3. Proof. Under the assumptions of the proposition if such discontinuous piecewise differ- ential systems have a limit cycle intersecting the discontinuity straight line x = 0 in the two points (0, y1 ) and (0, y2 ), then y1 and y2 must satisfy that H3c (0, y1 ) − H3c (0, y2 ) = 0, and H31 c (0, y1 ) − H31 c (0, y2 ) = 0. (17) where (c + aX + bY )2 + (Xα + Y β + γ)2 H3c (X, Y ) = , 1 + 2(c + aX + bY ) + (Xα + Y β + γ) and (c1 + a1 X + b1 Y )2 + (Xα1 + Y β1 + γ1 )2 H31 c (X, Y ) = . 1 + 2(c1 + a1 X + b1 Y ) + (Xα1 + Y β1 + γ1 ) 10 CLAUDIO A. BUZZI1 , YAGOR ROMANO CARVALHO2 , AND JAUME LLIBRE3 System (17) is G31 (y1 , y2 ) = −(y1 − y2 )(−2bc − b2 y1 − b2 y2 + 2c3 β + 2bc2 y1 β + 2bc2 y2 β + 2b2 cy1 y2 β − y1 β 2 − y2 β 2 − 2cy1 y2 β 3 − 2bc2 γ − 2b2 cy1 γ − 2b2 cy2 γ − 2b3 y1 y2 γ − 2βγ − 2cy1 β 2 γ − 2cy2 β 2 γ + 2by1 y2 β 2 γ − 2cβγ 2 + 2by1 βγ 2 + 2by2 βγ 2 + 2bγ 3 )/((1 + 2cy1 β + 2by12 β + 2cγ + 2by1 γ) (1 + 2cy2 β + 2by22 β + 2cγ + 2by2 γ)) = 0, G32 (y1 , y2 ) = −(y1 − y2 )(−2b1 c1 − b21 y1 − b21 y2 + 2c31 β1 + 2b1 c21 y1 β1 + 2b1 c21 y2 β1 + 2b21 c1 y1 y2 β1 − y1 β12 − y2 β12 − 2c1 y1 y2 β13 − 2b1 c21 γ1 − 2b21 c1 y1 γ1 − 2b21 c1 y2 γ1 − 2b31 y1 y2 γ1 − 2β1 γ1 − 2c1 y1 β12 γ1 − 2c1 y2 β12 γ1 + 2b1 y1 y2 β12 γ1 − 2c1 β1 γ12 + 2b1 y1 β1 γ12 + 2b1 y2 β1 γ12 + 2b1 γ13 ))/ ((1 + 2c1 y1 β1 + 2b1 y12 β1 + 2c1 γ1 + 2b1 y1 γ1 ) (1 + 2c1 y2 β1 + 2b1 y22 β1 + 2c1 γ1 + 2b1 y2 γ1 )) = 0, Since the denominators in the previous system cannot be zero and y1 < y2 , we obtain the equivalent system F13 (y1 , y2 ) = −2bc − b2 y1 − b2 y2 + 2c3 β + 2bc2 y1 β + 2bc2 y2 β + 2b2 cy1 y2 β − y1 β 2 − y2 β 2 − 2cy1 y2 β 3 − 2bc2 γ − 2b2 cy1 γ − 2b2 cy2 γ − 2b3 y1 y2 γ − 2βγ − 2cy1 β 2 γ − 2cy2 β 2 γ + 2by1 y2 β 2 γ − 2cβγ 2 + 2by1 βγ 2 + 2by2 βγ 2 + 2bγ 3 = 0, (18) F23 (y1 , y2 ) = −2b1 c1 − b21 y1 − b21 y2 + 2c31 β1 + 2b1 c21 y1 β1 + 2b1 c21 y2 β1 + 2b21 c1 y1 y2 β1 − y1 β12 − y2 β12 − 2c1 y1 y2 β13 − 2b1 c21 γ1 − 2b21 c1 y1 γ1 − 2b21 c1 y2 γ1 − 2b31 y1 y2 γ1 − 2β1 γ1 − 2c1 y1 β12 γ1 − 2c1 y2 β12 γ1 + 2b1 y1 y2 β12 γ1 − 2c1 β1 γ12 + 2b1 y1 β1 γ12 + 2b1 y2 β1 γ12 + 2b1 γ13 = 0. The system (18) could have a continuum of solutions (y1 , y2 ), but then the possible periodic solutions would not be limit cycles. Therefore we assume that this system has finitely many solutions. Assume that q31 (y1 ) 6= 0. From the equation F14 (y1 , y2 ) = 0 we obtain p31 (y1 ) y2 = , (19) q31 (y1 ) where p31 (y1 ) = −2bc + 2c3 β − 2bc2 γ − 2βγ − 2cβγ 2 + 2bγ 3 + + y1 (−b2 + 2bc2 β − β 2 − 2b2 cγ − 2cβ 2 γ + 2bβγ 2 ), q31 (y1 ) = b2 − 2bc2 β + β 2 + 2b2 cγ + 2cβ 2 γ − 2bβγ 2 + + y1 (−2b2 cβ + 2cβ 3 + 2b3 γ − 2bβ 2 γ). We substitute (19) in F23 (y1 , y2 ) = 0, and we get a rational function. The polynomial p3 (y1 ) in the numerator has degree 2. Therefore p3 (y1 ) has at most two real roots, and consequently system (18) has at most two solutions. Hence the discontinuous piecewise differential system has at most one limit cycle. Now assume that q31 (y1 ) = 0. Then also p31 (y1 ) = 0, and then at most one solution for system (18). CROSSING LIMIT CYCLES FOR A FAMILY OF ISOCHRONOUS CENTERS 11 In summary, the discontinuous piecewise differential systems of this proposition have at most one limit cycle. Now we shall prove that the discontinuous piecewise differential system formed by the differential systems (15) and (16) has one limit cycle. In x ≥ 0 the differential system (15) comes from the cubic isochronous center (S13 ) after doing the affine change of variables (x, y) → (1 − y, 1 − 2x), and in x ≤ 0 the differential system (16) also comes from the cubic isochronous center (S13 ) after doing the affine change of variables (x, y) → (3 + 2x, 1 + 2x − y). The first integrals of these two differential systems are (1 − y)2 + (1 − 2x)2 (3 + 2x)2 + (1 + 2x − y)2 H3c1 (x, y) = , and H31 c1 (x, y) = , 1 + 2(1 − y)(1 − 2x) 1 + 2(3 + 2x)(1 + 2x − y) respectively. Then system (18) for this piecewise differential system becomes 2 − 3y1 − 3y2 + 2y1 y2 = 0, − 46 − 7y1 − 7y2 + 6y1 y2 = 0. √ √  Its solution is (y13∗ , y23∗ ) = 13 − 131, 13 + 131 , observe that we have y13∗ < y23∗ . The solution (x31 (t), y13 (t)) in x ≥ 0 of system (15) such that (x31 (0), y13 (0)) = (0, y13∗ ) is contained in the level curve H3c1 (0, y23∗ ) = −12 = H3c1 (0, y13∗ ), i.e. in the curve (1 − y)2 + (1 − 2x)2 H3c1 (x, y) = = −12. 1 + 2(1 − y)(1 − 2x) The solution (x32 (t), y23 (t)) in x ≤ 0 of system (16) such that (x32 (0), y23 (0)) = (0, y23∗ ) is contained in the level curve H31 c1 (0, y23∗ ) = −4 = H31 c1 (0, y13∗ ), i.e. in the curve (3 + 2x)2 + (1 + 2x − y)2 H31 c1 (x, y) = = −4. 1 + 2(3 + 2x)(1 + 2x − y) Drawing the orbits (x3k (t), yk3 (t)), k = 1, 2, we obtain the limit cycle of Figure 3.  Figure 4. The limit cycle of the discontinuous piecewise differential sys- tem (20)-(21) of Proposition 4. The proof that Nclc [(S13 ), (S23 )] ≤ 3 is given in the next proposition. Proposition 4. Consider the class of discontinuous piecewise differential systems sepa- rated by the straight line x = 0 and formed by two cubic isochronous centers of type (S13 ) and (S23 ) after an arbitrary affine change of variables. Then these differential systems can 12 CLAUDIO A. BUZZI1 , YAGOR ROMANO CARVALHO2 , AND JAUME LLIBRE3 have at most three limit cycles. Moreover, the discontinuous piecewise differential system in this class is formed by the differential system ẋ = −1 − x + 3y + 3x2 + 4xy − y 2 + x3 − xy 2 , , (20) ẏ = 7 − 3x − 11y − 2x2 + 2xy + 6y 2 + x2 y − y 3 , in x ≥ 0, and by the differential system ẋ = y − x3 + 3xy 2 , ẏ = −x − 3x2 y + y 3 , (21) in x ≤ 0, has one limit cycle. Proof. Under the assumptions of the proposition if such discontinuous piecewise differ- ential systems have a limit cycle intersecting the discontinuity straight line x = 0 in the two points (0, y1 ) and (0, y2 ), then y1 and y2 must satisfy that H1c (0, y1 ) − H1c (0, y2 ) = 0, and H21 c (0, y1 ) − H42 c (0, y2 ) = 0 (22) where (c + ax + by)2 + (xα + yβ + γ)2 H1c (x, y) = , 1 + 2(c + ax + by)(xα + yβ + γ) and ((c1 + a1 X + b1 Y )2 + (Xα1 + Y β1 + γ1 )2 )2 H21 c (x, y) = . 1 + 4(c1 + a1 X + b1 Y )(Xα1 + Y β1 + γ1 ) Then system (22) becomes G41 (y1 , y2 ) = −(y1 − y2 )(−2bc − b2 y1 − b2 y2 + 2c3 β + 2bc2 y1 β + 2bc2 y2 β + 2b2 cy1 y2 β − y1 β 2 − y2 β 2 − 2cy1 y2 β 3 − 2bc2 γ − 2b2 cy1 γ − 2b2 cy2 γ − 2b3 y1 y2 γ − 2βγ − 2cy1 β 2 γ − 2cy2 β 2 γ + 2by1 y2 β 2 γ − 2cβγ 2 + 2by1 βγ 2 + 2by2 βγ 2 + 2bγ 3 )/((1 + 2cy1 β + 2by12 β + 2cγ + 2by1 γ) (1 + 2cy2 β + 2by22 β + 2cγ + 2by2 γ))) = 0, G42 (y1 , y2 ) = −(((y1 − y2 )(G422 (y1 , y2 ))))/ ((1 + 4c1 y1 β1 + 4b1 y12 β1 + 4c1 γ1 + 4b1 y1 γ1 ) (1 + 4c1 y2 β1 + 4b1 y22 β1 + 4c1 γ1 + 4b1 y2 γ1 )) = 0, where G422 (y1 , y2 ) ı́s a polynomial of degree 5. Since the denominators in the previous system cannot be zero and y1 < y2 , the previous system reduces to the system F14 (y1 , y2 ) = −2bc − b2 y1 − b2 y2 + 2c3 β + 2bc2 y1 β + 2bc2 y2 β + 2b2 cy1 y2 β − y1 β 2 − y2 β 2 − 2cy1 y2 β 3 − 2bc2 γ − 2b2 cy1 γ − 2b2 cy2 γ − 2b3 y1 y2 γ − 2βγ − 2cy1 β 2 γ − 2cy2 β 2 γ + 2by1 y2 β 2 γ − 2cβγ 2 (23) 2 2 3 + 2by1 βγ + 2by2 βγ + 2bγ , F24 (y1 , y2 ) = G42 (y1 , y2 ) = 0. Eventually, the system (23) could have a continuum of solutions (y1 , y2 ), but then the possible periodic solutions would not be limit cycles. Therefore we assume that this system has finitely many solutions. Assume that q41 (y1 ) 6= 0. From equation F14 (y1 , y2 ) = 0 we get p41 (y1 ) y2 = , (24) q41 (y1 ) CROSSING LIMIT CYCLES FOR A FAMILY OF ISOCHRONOUS CENTERS 13 where p41 (y1 ) = −2bc + 2c3 β − 2bc2 γ − 2βγ − 2cβγ 2 + 2bγ 3 + + y1 (−b2 + 2bc2 β − β 2 − 2b2 cγ − 2cβ 2 γ + 2bβγ 2 ), q41 (y1 ) = b2 − 2bc2 β + β 2 + 2b2 cγ + 2cβ 2 γ − 2bβγ 2 + + y(−2b2 cβ + 2cβ 3 + 2b3 γ − 2bβ 2 γ). We substitute (24) in F24 (y1 , y2 ) = 0, and we get a rational function. The polynomial p4 (y1 ) of the numerator has degree 6. Therefore p4 (y1 ) has at most six roots, and con- sequently system (23) has at most six solutions. Therefore the discontinuous piecewise differential system has at most three limit cycles. Assume that q41 (y1 ) = 0. Then p41 (y1 ) = 0, and there is at most one solution for y1 , and consequently the discontinuous piecewise differential system has at most one limit cycle in this case. In summary, we have proved these discontinuous piecewise differential systems have at most three limit cycles. Now we shall prove that the differential system formed by the differential systems (20) and (21) has one limit cycle. In x ≥ 0 the differential system (20) comes from the cubic isochronous center (S13 ) after doing the affine change of variables (x, y) → (1 + x, −2 + y), and in x ≤ 0 the differential system (21) comes from the cubic isochronous center (S23 ) after reversing the independent variable of sign. The first integrals of these two differential systems are (1 + x)2 + (−2 + y)2 (x2 + y 2 )2 H1c (x, y) = , and H2c (x, y) = , 1 + 2(1 + x)(−2 + y) 1 + 4xy respectively. Therefore system (22) becomes 2 − 3y1 − 3y2 + 2y1 y2 = 0, (y1 + y2 )(y12 + y22 ) = 0. Its solution is (y14∗ , y24∗ ) = (−1, 1), note that y14∗ < y24∗ . The solution (x41 (t), y14 (t)) in x ≥ 0 of system (20) such that (x41 (0), y14 (0)) = (0, y14∗ ) is contained in the level curve H2c (0, y24∗ ) = 1 = H2c (0, y14∗ ), i.e. in the curve (x2 + y 2 )2 H2c (x, y) = = 1. 1 + 4xy The solution (x42 (t), y24 (t)) in x ≤ 0 of system (21) with (x42 (0), y24 (0)) = (0, y24∗ ) is con- tained in level curve H1c (0, y24∗ ) = −2 = H1c (0, y14∗ ), i.e. in the curve (1 + x)2 + (−2 + y)2 H1c (x, y) = = −2. 1 + 2(1 + x)(−2 + y) that is, Drawing the orbits (x4k (t), yk4 (t)), k = 1, 2, we obtain the limit cycle of Figure 4.  The problem if the upper bound of Proposition 4 for the maximum number of limit cycles of that class of discontinuous piecewise differential systems is reached or not remains open. The proof that Nclc [(S23 ), (S23 )] ≤ 9 is given in the next proposition. Proposition 5. Consider the class of discontinuous piecewise differential systems sepa- rated by the straight line x = 0 and formed by two distinct cubics isochronous center (S23 ) after an arbitrary affine change of variables. Then these differential systems can have at 14 CLAUDIO A. BUZZI1 , YAGOR ROMANO CARVALHO2 , AND JAUME LLIBRE3 Figure 5. The limit cycle of the discontinuous piecewise differential sys- tem (25)-(26) of Proposition 5. most nine limit cycles. Moreover, the discontinuous piecewise differential system in this class is formed by the differential system ẋ = −y + x3 − 3xy 2 , ẏ = x + 3x2 y − y 3 , (25) in x ≥ 0, and the differential system ẋ = −1 − 2x3 − 5y − 3y 2 + x2 (6 + 6y) + x(5 − 3y 2 ), (26) ẏ = −4x3 − 5y + 6x2 y − 6y 2 − y 3 + x(10 + 12y), in x ≤ 0 has one limit cycle. Proof. Under the assumptions of the proposition if such discontinuous piecewise differ- ential systems have a limit cycle intersecting the discontinuity straight line x = 0 in the two points (0, y1 ) and (0, y2 ), then y1 and y2 must satisfy that H2c (0, y1 ) − H2c (0, y2 ) = 0, and H21 c (0, y1 ) − H21 c (0, y2 ) = 0, (27) where ((c + ax + by)2 + (xα + yβ + γ)2 )2 H2c (x, y) = , 1 + 4(c + axX + by)(xα + yβ + γ) and ((c1 + a1 x + b1 y)2 + (xα1 + yβ1 + γ1 )2 )2 H21 c (x, y) = . 1 + 4(c1 + a1 x + b1 y)(xα1 + yβ1 + γ1 ) Then system (27) becomes G51 (y1 , y2 ) = −(((y1 − y2 )A51 (y1 , y2 )))/ ((1 + 4cy1 β + 4by12 β + 4cγ + 4by1 γ) (1 + 4cy2 β + 4by22 β + 4cγ + 4by2 γ))) = 0, G52 (y1 , y2 ) = −(((y1 − y2 )A52 (y1 , y2 )))/ ((1 + 4c1 y1 β1 + 4b1 y12 β1 + 4c1 γ1 + 4b1 y1 γ1 ) (1 + 4c1 y2 β1 + 4b1 y22 β1 + 4c1 γ1 + 4b1 y2 γ1 ))) = 0, where A5i (y1 , y2 ) for i = 1, 2 are polynomials of degree 5. Since the denominators of the previous system cannot be zero and y1 < y2 , this system is equivalent to the system A51 (y1 , y2 ) = 0, A52 (y1 , y2 ) = 0. CROSSING LIMIT CYCLES FOR A FAMILY OF ISOCHRONOUS CENTERS 15 where A51 (y1 , y2 ) = A0 + A1 (y1 + y2 ) + A2 (y12 + y22 ) + A3 y1 y2 + A4 (y13 + y23 ) + A5 (y12 y2 + y1 y22 ) + A6 (y13 y2 + y1 y23 ) + A7 y12 y22 + A8 (y13 y22 + y12 y23 ), (28) A52 (y1 , y2 ) = B0 + B1 (y1 + y2 ) + B2 (y12 + y22 ) + B3 y1 y2 + B4 (y13 + y23 ) + B5 (y12 y2 + y1 y22 ) + B6 (y13 y2 + y1 y23 ) + B7 y12 y22 + B8 (y13 y22 + y12 y23 ). The above polynomials can be rewritten in the variables z and w, where z = y1 + y2 and w = y1 y2 . For example, y12 + y22 = (y1 + y2 )2 − 2y1 y2 = z 2 − 2w. In these new variables system (28) becomes P15 (z, w) = A0 + A1 z + (−2A2 + A3 )w + A4 z 3 + (A5 − 3A4 )zw + A2 z 2 + A6 z 2 w+ + (−2A6 + A7 )w2 + A8 zw2 = 0, (29) P15 (z, w) = B0 + B1 z + (−2B2 + B3 )w + B4 z 3 + (B5 − 3B4 )zw + B2 z 2 + B6 z 2 w+ + (−2B6 + B7 )w2 + B8 zw2 = 0. Eventually, system (29) could have a continuum of solutions (z, w), but then the pos- sible periodic solutions would not be limit cycles. So we assume that this system has a finite number of solutions. The two equations in system (29) are polynomials of degree 3, then by Bezout Theorem, the discontinuous piecewise differential system has a maximum of 9 limit cycles. Now we will prove that the discontinuous piecewise differential system formed by the differential systems (25) and (26) has one limit cycle. In x ≥ 0 the differential system (25) is the cubic isochronous center (S23 ), and in x ≤ 0 the differential system (26) comes from the cubic isochronous center (S23 ) after the affine change of variables (x, y) → (1+x, −1+x−y). The first integrals of these two differential systems are (x2 + y 2 )2 ((1 + x)2 + (−1 + x − y)2 )2 H2c (x, y) = , and H21 c (x, y) = , 1 + 4xy 1 + 4(1 + x)(−1 + x − y) respectively. So system (27) becomes (y1 + y2 )(y12 + y22 ) = 0, − 8 − 24y1 − 12y12 − 3y13 − 24y2 − 44y1 y2 − 19y12 y2 − 4y13 y2 − 12y22 − 19y1 y22 − − 4y12 y22 − 3y23 − 4y1 y23 = 0.  q q  √ √ Its solution is (y15∗ , y25∗ ) = − (−5 + 33)/2, (−5 + 33)/2 , note that y15∗ < y25∗ . The solution (x51 (t), y15 (t)) in x ≥ 0 of the system√(25) such that (x51 (0), y15 (0)) = (0, y15∗ ) is contained in the level curve H2c (0, y25∗ ) = (−5 + 33)2 /4 = H2c (0, y15∗ ), i.e. in the curve √ (x2 + y 2 )2 (−5 + 33)2 H2c (x, y) = = . 1 + 4xy 4 The solution (x52 (t), y25 (t)) in x ≤ 0 of the system √ (26) with (x52 (0), y25 (0)) = (0, y25∗ ) is contained in the level curve H21 c (0, y25∗ ) = (1 − 33)/2 = H21 c (0, y15∗ ), i. e. in the curve √ ((1 + x)2 + (−1 + x − y)2 )2 (1 − 33) H21 c (x, y) = = . 1 + 4(1 + x)(−1 + x − y) 2 Drawing the orbits (x5k (t), yk5 (t)), k = 1, 2, we obtain the limit cycle of Figure 5.  16 CLAUDIO A. BUZZI1 , YAGOR ROMANO CARVALHO2 , AND JAUME LLIBRE3 The problem in which the upper bound of Proposition 5 for the maximum number of limit cycles of that class of discontinuous piecewise differential systems is reached or not remains open. Acknowlegements This work has received funding from the Agencia Estatal de Investigación grant PID2019- 104658GB-I00, the H2020 European Research Council grant MSCA-RISE-2017-777911, CNPq 304798/2019-3 grant, and São Paulo Paulo Research Foundation (FAPESP) grants 2021/14695-7, 2019/10269-3, 2018/05098-2, and 2016/00242-2. References [1] Bernardo, M., Budd, C., Champneys, A. R., and Kowalczyk, P. Piecewise-smooth dynam- ical systems: theory and applications, vol. 163. Springer Science & Business Media, 2008. [2] Buzzi, C., Pessoa, C., and Torregrosa, J. Piecewise linear perturbations of a linear center. Discrete Contin. Dyn. Syst. 33, 9 (2013), 3915–3936. [3] Chavarriga, J., and Sabatini, M. A survey of isochronous centers. Qualitative theory of dynam- ical systems 1, 1 (1999), 1. [4] de Carvalho Braga, D., and Mello, L. F. Limit cycles in a family of discontinuous piecewise linear differential systems with two zones in the plane. Nonlinear Dynam. 73, 3 (2013), 1283–1288. [5] Dumortier, F., Llibre, J., and Artés, J. C. Qualitative theory of planar differential systems. Universitext. Springer-Verlag, Berlin, 2006. [6] Filippov, A. F. Differential equations with discontinuous righthand sides, vol. 18 of Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht, 1988. [7] Freire, E., Ponce, E., Rodrigo, F., and Torres, F. Bifurcation sets of continuous piecewise linear systems with two zones. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 28, 11 (1998), 2073–2097. [8] Freire, E., Ponce, E., and Torres, F. A general mechanism to generate three limit cycles in planar Filippov systems with two zones. Nonlinear Dynam. 78, 1 (2014), 251–263. [9] Garcia, A., and Lequain, Y. Elementos de álgebra. Instituto de Matemática Pura e Aplicada, 2006. [10] Henson, M. A., and Seborg, D. E. Nonlinear process control. Prentice Hall PTR Upper Saddle River, New Jersey, 1997. [11] Huan, S.-M., and Yang, X.-S. On the number of limit cycles in general planar piecewise linear systems. Discrete Contin. Dyn. Syst. 32, 6 (2012), 2147–2164. [12] Li, L. Three crossing limit cycles in planar piecewise linear systems with saddle-focus type. Electron. J. Qual. Theory Differ. Equ. (2014), No. 70, 14. [13] Llibre, J., Ordóñez, M., and Ponce, E. On the existence and uniqueness of limit cycles in planar continuous piecewise linear systems without symmetry. Nonlinear Anal. Real World Appl. 14, 5 (2013), 2002–2012. [14] Llibre, J., and Ponce, E. Three nested limit cycles in discontinuous piecewise linear differential systems with two zones. Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms 19, 3 (2012), 325–335. [15] Llibre, J., and Teixeira, M. A. Piecewise linear differential systems with only centers can create limit cycles? Nonlinear Dynam. 91, 1 (2018), 249–255. [16] Llibre, J., and Teruel, A. E. Introduction to the qualitative theory of differential systems. Springer, 2014. [17] Lum, R., and Chua, L. O. Global properties of continuous piecewise linear vector fields. part i: Simplest case in r2. International Journal of Circuit Theory and Applications 19, 3 (1991), 251–307. [18] Lum, R., and Chua, L. O. Global properties of continuous piecewise linear vector fields. part ii: Simplest symmetric case in r2. International journal of circuit theory and applications 20, 1 (1992), 9–46. [19] Makarenkov, O., and Lamb, J. S. W. Dynamics and bifurcations of nonsmooth systems: a survey. Phys. D 241, 22 (2012), 1826–1844. [20] Narendra, K. S. Frequency domain criteria for absolute stability. Elsevier, 2014. [21] Ogata, K. Modern control engineering, prectice-hall, 1990. CROSSING LIMIT CYCLES FOR A FAMILY OF ISOCHRONOUS CENTERS 17 [22] Pleshkan, I. A new method of investigating the isochronicity of a system of two differential equa- tions. Differential Equations 5, 4 (1969), 796–802. [23] Poincaré, M. Sur l’intégration algébrique des équations différentielles du premier ordre et du premier degré. Rendiconti del Circolo Matematico di Palermo (1884-1940) 11, 1 (1897), 193–239. [24] Shafarevich, I. R. Basic algebraic geometry. Springer-Verlag, New York-Heidelberg, 1974. Trans- lated from the Russian by K. A. Hirsch, Die Grundlehren der mathematischen Wissenschaften, Band 213. [25] Simpson, D. J. W. Bifurcations in piecewise-smooth continuous systems, vol. 70 of World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2010. [26] Teixeira, M. A. Perturbation theory for non-smooth systems. In Mathematics of complexity and dynamical systems. Vols. 1–3. Springer, New York, 2012, pp. 1325–1336. [27] Vainsencher, I. Introdução às curvas algébricas planas. Instituto de Matemática Pura e Aplicada, 1996. [28] van der Pol, B. A theory of the amplitude of free and forced triode vibrations. Radio Rev. 1 (1920), 701–710. [29] van der Pol, B. On relaxation-oscillations. Lond. Edinb. Dublin Philos. Mag. J. Sci. 2, 7 (1926), 978–992. 1 Mathematics Department, Universidade Estadual Paulista Julio de Mesquita Filho, 15054-000 São José do Rio Preto, São Paulo, Brazil Email address: claudio.buzzi@unesp.br 2 Mathematics Department, Universidade de São Paulo, 13566-590 São Carlos, São Paulo, Brazil Email address: yagor.carvalho@usp.br 3 Mathematics Department, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain Email address: jllibre@mat.uab.cat