This is a preprint of: “The 16th Hilbert problem for discontinuous piecewise isochronous centers of degree one or two separated by a straight line”, Marina Esteban, Jaume Llibre, Clàudia Valls, Chaos, vol. 31, 043112:1–18, 2021. DOI: [10.1063/5.0023055] The 16th Hilbert problem for discontinuous piecewise isochronous centers of degree one or two separated by a straight line M. Esteban,a) J. Llibre,b) and C. Vallsc) (Dated: March 7, 2021) In this paper we deal with discontinuous piecewise differential systems formed by two differential systems separated by a straight line when these two differential systems are linear centers (which always are isochronous) or quadratic isochronous centers. It is known that there is a unique family of linear isochronous centers and four families of quadratic isochronous centers. Combining these five types of isochronous centers we obtain fifteen classes of discontinuous piecewise differential systems. We provide upper bounds for the maximum number of limit cycles that these fifteen classes of discontinuous piecewise differential systems can exhibit, so we have solved the 16th Hilbert problem for such differential systems. Moreover in seven of the classes of these discontinuous piecewise differential systems the obtained upper bound on the maximum number of limit cycles is reached. PACS numbers: 02.30.Hq a) marinaep@us.es; Dept. Matemática Aplicada II and Instituto de Matemáticas (IMUS), Escuela Técnica Superior de Ingeniería de la Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092 Sevilla (SPAIN) b) jllibre@mat.uab.cat; Dept. Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia (SPAIN) c) cvalls@math.ist.utl.pt; Dept. Matemática, Instituto Superior Técnico, Universidade Técnica de Lis- boa, Av. Rovisco Pais 1049–001, Lisboa (PORTUGAL) 1 To solve the 16th Hilbert problem, i.e. to find an upper bound for the maximum number of limit cycles that a given class of differential systems can exhibit, is in general an unsolved problem. For the classes of discontinuous piecewise differential systems here studied we can obtain the solution using the first integrals of the linear and quadratic isochronous centers. I. INTRODUCTION AND MAIN RESULTS We consider planar differential systems of the form dx dy = P (x, y), = Q(x, y), dt dt where P (x, y) and Q(x, y) are polynomial functions, and the degree of the systems is the maximum degree of such polynomials. In particular in this paper we consider discontin- uous piecewise differential systems of the form   F− (x, y) = (f − (x, y), g − (x, y)) if x < 0, (ẋ, ẏ) = F(x, y) = (1)  F+ (x, y) = (f + (x, y), g + (x, y)) if x > 0, being bi-valued on the separation line x = 0. Following9 a point (0, y) is a crossing point if f − (0, y)f + (0, y) > 0. If there exists a periodic orbit of the discontinuous differential system (1) having exactly two crossing points, then we call it a crossing periodic orbit. A crossing limit cycle is an isolated periodic orbit in the set of all crossing periodic orbits of system (1). In what follows for simplicity we shall say limit cycle instead of crossing limit cycle. The analysis of planar continuous piecewise linear systems is well established when the number of linear zones is small, see33 and the references therein. They frequently appear in many non-linear engineering devices, which are accurately modelled by piecewise linear vector fields, see7 . They appear also in mathematical biology, see6,30–32 . However when the planar vector field is discontinuous, the adaptation of the 16th Hilbert’s problem on the maximum number of existing limit cycles, is an open problem. In the last years 2 many authors have worked in this problem, trying to determine how many limit cycles can appear in planar systems separated by a straight line, see for instance1–4,8,10–14,16–18,21–29 . For details on the classical 16th Hilbert problem see for instance15,19,20 . Let p ∈ R2 be a singularity of a differential system in the plane. The singularity p is a center if there exists an open neighbourhood U of p such that all the solutions in U \ {p} are periodic. Denote by Tq the period of the periodic orbit through q ∈ U \ {p}. We say that p is an isochronous center if Tq is constant for all q ∈ U \ {p}. In this paper we work with the following five types of systems which cover the classes of a linear system having a center and of all quadratic polynomial differential systems having an isochronous center. For a proof of the linear system see Lemma 1 of27 and for a proof of the quadratic systems see page 34 of5 . (I) Any linear differential system having a center can be written as ẋ = −Ax − (A2 + ω 2 )y + B, ẏ = x + Ay + C, with ω > 0, A, B, C ∈ R and A 6= 0. A first integral of this system is H1 (x, y) = (x + Ay)2 + 2(Cx − By) + y 2 ω 2 . Of course every linear center is isochronous. (II) The first family of quadratic isochronous differential systems can be obtained doing an affine transformation to the system ẋ = −y + x2 − y 2 , ẏ = x(1 + 2y), with first integral x2 + y 2 H̃2 (x, y) = . 1 + 2y (III) The second family of quadratic isochronous differential systems can be obtained 3 doing an affine transformation to the system ẋ = −y + x2 , ẏ = x(1 + y), whose first integral is x2 + y 2 H̃3 (x, y) = . (1 + y)2 (IV) The third family of quadratic isochronous differential systems can be obtained doing an affine transformation to the system 4  16  ẋ = −y + x2 , ẏ = x 1 − y , 3 3 with first integral 9(x2 + y 2 ) − 24x2 y + 16x4 H̃4 (x, y) = . −3 + 16y (V) The fourth family of quadratic isochronous differential systems can be obtained doing an affine transformation to the system 16 2 4 2  8  ẋ = −y + x − y , ẏ = x 1 + y , 3 3 3 whose first integral is 9(x2 + y 2 ) + 24y 3 + 16y 4 H̃5 (x, y) = . (3 + 8y)4 Our objective is to solve the 16th Hilbert problem for the 15 classes of discontinuous piecewise differential systems separated by a straight line and formed by two arbitrary isochronous centers of degree 1 or 2, i.e. we shall provide for all these 15 classes an upper bound on the maximum number of limit cycles that each class can exhibit. Moreover as we shall see in many cases the upper bound that we shall provide is reached. We must mention that, in general it is very difficult (many times for the moment 4 impossible) to provide an upper bound for the maximum number of limit cycles that a class of differential systems in the plane can exhibit, and of course it is even more difficult to provide the exact upper bound, see for instance15,19,20 . It was proved in Theorem 3 of27 , or in Corollary 3 of23 that discontinuous piecewise differential systems separated by a straight line and formed by two arbitrary linear centers has no limit cycles. So this case is not considered here. Our first main result is to provide the maximum number of limit cycles that can exist for discontinuous piecewise differential systems of the form (1), where in x < 0 there is an arbitrary linear differential center (I), and for x > 0 there is one of the four quadratic isochronous differential systems (II), (III), (IV) or (V), after an arbitrary affine change of variables. Theorem 1. Consider discontinuous piecewise differential systems separated by the straight line x = 0 and formed by a linear differential center (I) after an affine change of variables in x < 0, and by a quadratic isochronous system of type either (II), or (III), or (IV), or (V) after an affine change of variables in x > 0. The maximum number of limit cycles of these discontinuous piecewise differential systems are (a) at most one for systems of type (I)-(II), and there are systems of this type with exactly one limit cycle, see Figure 1; (b) at most one for systems of type (I)-(III), and there are systems of this type with exactly one limit cycle, see Figure 2; (c) at most two for systems of type (I)-(IV), and there are systems of this type with exactly one limit cycle, see Figure 3; (d) at most two for systems of type (I)-(V), and there are systems of this type with exactly two limit cycles, see Figure 4. Note that for all systems of type (I)-(k) with k ∈ {II, III, V } the upper bound on the maximum number of limit cycles is reached. The proof of Theorem 1 is given in section III. 5 10 5 -10 -5 5 10 -5 Figure 1: The unique limit cycle that exists for system (8)-(9) of class (I)–(II). It is travelled in counter-clockwise sense. 10 8 6 4 2 -10 -8 -6 -4 -2 2 -2 Figure 2: The unique limit cycle that exists for system (8)-(11) of class (I)-(III). It is travelled in counter-clockwise sense. The second main result of the paper is to give the maximum number of limit cycles that can appear in discontinuous piecewise differential systems of the form (1) such that in a half-plane there is a general quadratic isochronous differential system of type (II), 6 4 2 -6 -4 -2 2 4 -2 -4 Figure 3: The existing limit cycle for system (13)–(14) of class (I)-(IV). It is travelled in counter-clockwise sense. . 1.0 0.8 1.0 0.6 0.4 0.8 0.2 0.6 0.0 0.4 0.2 -0.5 -0.4 -0.3 -0.2 -0.1 Figure 4: The pair of limit cycles that exist for system (16)-(17) of class (I)-(V). They are travelled in counter-clockwise sense. and in the other one there is a general quadratic isochronous differential system of type 7 (II), (III), (IV) or (V) after an arbitrary affine change of variables. Theorem 2. Consider discontinuous piecewise differential systems separated by the straight line x = 0 and formed by a quadratic isochronous center of type (II) after an affine change of variables in x < 0, and by a quadratic isochronous system of type either (II), or (III), or (IV), or (V) after an affine change of variables in x > 0. The maximum number of limit cycles of these discontinuous piecewise differential systems are (a) at most one for systems of type (II)-(II), and there are systems of this type with exactly one limit cycle, see Figure 5; (b) at most one for systems of type (II)-(III), and there are systems of this type with exactly one limit cycle, see Figure 6; (c) at most three for systems of type (II)-(IV), and there are systems of this type with exactly two limit cycles, see Figure 7; (d) at most three for systems of type (II)-(V), and there are systems of this type with exactly two limit cycles, see Figure 8. Note that for systems of type (II)-(II) and (II)–(III) after an affine change of variables the upper bound on the maximum number of limit cycles is reached. The proof of Theorem 2 is given in section IV. The third main result of the paper is to give the maximum number of limit cycles that can appear in discontinuous piecewise differential systems of the form (1) such that in a half-plane there is a general quadratic isochronous differential system of type (III), and in the other one a general quadratic isochronous differential system of type (III), (IV) or (V) after an arbitrary affine change of variables. Theorem 3. Consider discontinuous piecewise differential systems separated by the straight line x = 0 and formed by a quadratic isochronous center of type (III) after an affine change of variables in x < 0, and by a quadratic isochronous system of type either (III), or (IV), or (V) after an affine change of variables in x > 0. The maximum number of limit cycles of these discontinuous piecewise differential systems are 8 2 1 -3 -2 -1 1 2 -1 -2 Figure 5: The unique limit cycle that exists for system (20)-(21) of type (II)-(II). It is travelled in counter-clockwise sense. 1.0 0.5 -2.5 -2.0 -1.5 -1.0 -0.5 0.5 1.0 -0.5 -1.0 -1.5 Figure 6: The unique limit cycle that exists for system (20)-(23) of type (II)-(III). It is travelled in counter-clockwise sense. (a) at most one for systems of type (III)-(III), and there are systems of this type with exactly one limit cycle, see Figure 9; 9 40 30 20 10 -120 -100 -80 -60 -40 -20 Figure 7: The two limit cycles that exist for system (25)-(26) of type (II)-(IV). They are travelled in counter-clockwise sense. 1.5 1.0 0.5 -4 -3 -2 -1 -0.5 -1.0 Figure 8: The two limit cycles existing for system (29)-(30) of type (II)-(V). They are travelled in counter-clockwise sense. (b) at most three for systems of type (III)-(IV), and there are systems of this type with exactly two limit cycles, see Figure 10; 10 (c) at most three for systems of type (III)-(V), and there are systems of this type with exactly two limit cycles, see Figure 9; Note that for systems of type (III)-(III) after an affine change of variables the upper bound on the maximum number of limit cycles is reached. The proof of Theorem 3 is given in section V. 0.5 -3 -2 -1 1 -0.5 -1.0 -1.5 -2.0 Figure 9: The unique limit cycle that exists for system (23)-(34) of type (III)-(III). It is travelled in counter-clockwise sense. The following result gives the maximum number of limit cycles that can appear in discontinuous piecewise differential systems of the form (1) such that in a half-plane there is a general quadratic isochronous differential system of type (IV), and in the other one a general quadratic isochronous differential system of type (IV) or (V) after an arbitrary affine change of variables. Theorem 4. Consider discontinuous piecewise differential systems separated by the straight line x = 0 and formed by a quadratic isochronous center of type (IV) after an affine change of variables in x < 0, and by a quadratic isochronous system of type either (IV), or (V) after an affine change of variables in x > 0. The maximum number of limit cycles of these discontinuous piecewise differential systems are 11 3 2 1 -1.0 -0.5 0.5 1.0 1.5 2.0 -1 -2 Figure 10: The pair of limit cycles that exists for system (36)-(26) of type (III)-(IV). They are travelled in counter-clockwise sense. 2 1 -200 -150 -100 -50 1.5 -1 1.0 -2 -3 0.5 -4 0.02 0.04 0.06 0.08 0.10 0.12 -5 Figure 11: The two limit cycles existing for system (38)-(30) of type (III)-(V). They are travelled in counter-clockwise sense. (a) at most three for systems of type (IV)-(IV), and there are systems of this type with 12 exactly two limit cycles, see Figure 12; (b) at most three for systems of type (IV)-(V), and there are systems of this type with exactly two limit cycle, see Figure 13. 3 2 1 -2.0 -1.5 -1.0 -0.5 0.5 1.0 -1 -2 Figure 12: The pair of limit cycles that exists for system (41)-(43) of type (IV)-(IV). They are travelled in counter-clockwise sense. The proof of Theorem 4 is given in section VI. The last main result gives the maximum number of limit cycles that can appear in discontinuous piecewise differential systems of the form (1) such that in both half-planes there is a quadratic isochronous differential system of type (V) after an affine change of variables. Theorem 5. The maximum number of limit cycles for discontinuous piecewise isochronous quadratic differential systems formed by two systems of type (V) separated by the straight line x = 0 after an affine change of variables is at most twelve, and there are systems of this type with exactly two limit cycles, see Figure 14. 13 3 2 1 -2.0 -1.5 -1.0 -0.5 -1 -2 -3 Figure 13: The two limit cycles existing for system (41)-(45) of type (IV)-(V). They are travelled in counter-clockwise sense. 1.5 1.0 0.5 -2.0 -1.5 -1.0 -0.5 -0.5 Figure 14: The two limit cycles existing for system (48)-(49) of type (V)-(V). They are travelled in counter-clockwise sense. The proof of Theorem 5 is given in section VII. See the remark at the end of the proof of Theorem 5 related with this theorem. 14 II. THE QUADRATIC ISOCHRONOUS DIFFERENTIAL SYSTEMS (II), (III), (IV) AND (V) AFTER AN AFFINE CHANGE OF VARIABLES In this section we show the expressions for the quadratic isochronous systems (II), (III), (IV) and (V) and their first integrals, after doing the general affine change of variables of the form (x, y) → (ax + by + c, αx + βy + γ), (2) with bα − aβ 6= 0. Thus the differential system (II) after this affine change of variables becomes 1 ẋ = βγ 2 + 2bγc + bc + βγ − βc2 + (2abγ + 2αβγ + ab + αβ bα − aβ −2aβc + 2αbc)x + (2γ + 1)(b2 + β 2 )y + (−a2 β + α2 β + 2αab)x2  +2α(b2 + β 2 )xy + β(b2 + β 2 )y 2 , 1 (3) ẏ = −αγ 2 − 2aγc − ac − αγ + αc2 − (2γ + 1)(a2 + α2 )x bα − aβ +(−2abγ − 2αβγ − ab − αβ − 2aβc + 2αbc)y − α(a2 + α2 )x2  −2β(a2 + α2 )xy − (αβ 2 + 2aβb − αb2 )y 2 , whose first integral is (c + ax + by)2 + (xα + yβ + γ)2 H2 (x, y) = . 1 + 2(xα + yβ + γ) The differential system (III) becomes 1 ẋ = −bγc − bc − βγ + βc2 + (−abγ − ab − αβ + 2aβc αb − aβ −αbc)x − (b2 γ + b2 + β 2 + βbc)y + a(aβ − αb)x2  −b(αb − aβ)xy , 1 (4) ẏ = −aγc − ac − αγ + αc2 − (a2 γ + a2 + α2 − αac)x αb − aβ +(−abγ − ab − αβ − aβc + 2αbc)y − a(aβ − αb)xy  +b(αb − aβ)y 2 15 whose first integral is (ax + by + c)2 + (γ + αx + βy)2 H3 (x, y) = . (γ + αx + βy + 1)2 The differential system (IV) becomes, 1 ẋ = −16bγc + 3bc + 3βγ + 4βc2 + (−16abγ + 3ab 3(αb − aβ) +3αβ + 8aβc − 16αbc)x + (−16b2 γ + 3b2 + 3β 2 − 8βbc)y  +4a(aβ − 4αb)x2 − 8b(aβ + 2αb)xy − 12b2 βy 2 , 1 (5) ẏ = 16aγc − 3ac − 3αγ − 4αc2 + (16a2 γ − 3a2 3(αb − aβ) −3α2 + 8αac)x + (16abγ − 3ab − 3αβ + 16aβc − 8αbc)y  +12a2 αx2 + 8a(2aβ + αb)xy − 4b(αb − 4aβ)y 2 whose first integral is 1 H4 (x, y) = −24(ax + by + c)2 (γ + αx + βy) 16(γ + αx + βy) − 3  +9 ((ax + by + c)2 + (γ + αx + βy)2 ) + 16(ax + by + c)4 . Finally the differential system (V) after the change of variables (2) becomes 1 ẋ = 4βγ 2 + 8bγc + 3bc + 3βγ − 16βc2 + (8abγ + 8αβγ 3(αb − aβ) +3ab + 3αβ − 32aβc + 8αbc)x + (8b2 γ + 8β 2 γ + 3b2 + 3β 2 −24βbc)y + 4(α2 β − 4a2 β + 2αab)x2 + 8(αβ 2 − 3aβb + αb2 )xy  −4βy 2 (2b2 − β 2 )y 2 , 1 (6) ẏ = 16αc2 − 4αγ 2 − 8aγc − 3ac − 3αγ − (8a2 γ + 8α2 γ 3(αb − aβ) +3a2 + 3α2 − 24αac)x − (8abγ + 8αβγ + 3ab + 3αβ + 8aβc −32αbc)y + 4α(2a2 − α2 )x2 + 8(a2 (−β) − α2 β + 3αab)xy  −4(αβ 2 + 2aβb − 4αb2 )y 2 16 whose first integral is 1 H5 (x, y) = 9 ((ax + by + c)2 + (γ + αx + βy)2 ) (8(γ + αx + βy) + 3)4  +16(γ + αx + βy)4 + 24(γ + αx + βy)3 . III. PROOF OF THEOREM 1 A. Proof of Theorem 1 for system (I)–(II) We consider the planar linear differential system (I) with first integral H1 (x, y) in the half-plane x < 0 and the quadratic polynomial differential system (3) with first integral H2 (x, y) in the half-plane x > 0. If there exists a limit cycle of the discontinuous piecewise differential systems (I)-(3) it must intersect the discontinuity line x = 0 in two different points (0, y) and (0, Y ). Clearly these two points must satisfy the system H1 (0, y) − H1 (0, Y ) = (Y − y) (−4A2 y − 4A2 Y + 8B − yω 2 − ω 2 Y ) = (Y − y)P1 (y, Y ) = 0, (7) (Y − y)Q2 (y, Y ) H2 (0, y) − H2 (0, Y ) = = 0, [1 + 2(βy + γ)][1 + 2(βY + γ)] where P1 and Q2 are polynomials of degrees one and two, respectively. Since the points (0, y) and (0, Y ) are different, from P1 (y, Y ) = 0, we get Y as a function of y, that is, Y = f (y). Substituting this expression in equation Q2 (y, Y ) = 0 we obtain a quadratic equation in the variable y. Then the maximum number of solutions of (7) is two, namely (y1 , Y1 ) and (y2 , Y2 ), but in fact, these two solutions represent the same limit cycle because Y1 = y2 and Y2 = y1 . So for the discontinuous piecewise differential system (I)–(3), there exists at most one limit cycle. Now we give an example of a discontinuous piecewise differential system (I)–(3) having one limit cycle. On x > 0 we consider the linear differential system 5 ẋ = 1 − x − y, ẏ = x + y, (8) 4 17 whose first integral is H1 (x, y) = −8y + y 2 + 4(x + y)2 , and on x > 0 we consider the quadratic isochronous differential system of type (3) ẋ = −4 − 5x − 6y − x2 − 4xy − 2y 2 , ẏ = 1 + 3x + y + x2 + 2xy, (9) whose first integral is (x + y + 1)2 + (y + 1)2 H2 (x, y) = . 2(x + y + 1) + 1 We can take, without  loss of generality, the solution of (7) satisfying y < Y , and so the 1 √ 1 √ pair (y, Y ) = (4 − 3 14), (4 + 3 14) provides the limit cycle that exists for the 5 5 discontinuous differential piecewise system (8)-(9) shown in Figure 1. B. Proof of Theorem 1 for system (I)–(III) We consider the linear differential system (I) with first integral H1 (x, y) on the half- plane x < 0, and on the half-plane x > 0, we take the quadratic isochronous differential system (4) with its first integral H3 (x, y). Then if there exists some limit cycle for the discontinuous differential system (I)–(4), it must intersect the discontinuity line x = 0 at two different points (0, y) and (0, Y ), satisfying the equations H1 (0, y) − H1 (0, Y ) = (Y − y)P1 (y, Y ) = 0, (Y − y)Q3 (y, Y ) (10) H3 (0, y) − H3 (0, Y ) = = 0. (1 + βy + γ)2 (1 + βY + γ)2 In (10) P1 and Q3 are polynomials of degrees one and two, respectively. By following the same procedure as for the proof of system (I)–(II), we solve the equation P1 (y, Y ) = 0 obtaining the variable Y as a function of y, that is Y = f (y). By replacing Y in the equation Q3 (y, Y ) = 0, we obtain again a quadratic polynomial equation in the variable y, so that the equation has at most two different solutions. As in the proof for system (I)–(II), these two solutions represent, if they exist, the same limit cycle. Therefore system (10) has only one solution with y < Y , and then the discontinuous piecewise 18 differential system (I)–(4) has at most one limit cycle. Next we give a specific discontinuous piecewise differential system (I)–(4) having one limit cycle. On the half-plane x < 0 we consider the linear differential system (8), and on the half-plane x > 0 we consider the quadratic isochronous differential system of type (4) ẋ = −2 − 2y + x2 + xy, ẏ = 2 + 2x + 3y + xy + y 2 , (11) with first integral (x + y + 1)2 + (y + 1)2 H3 (x, y) = . (y + 2)2 In this case, the unique solution for system (10) with y < Y is   1 √ 1 √ (y, Y ) = (4 − 3 14), (4 + 3 14) , 5 5 and the corresponding limit cycle of the discontinuous piecewise differential system (8)- (11) associated to this solution is shown in Figure 2. C. Proof of Theorem 1 for system (I)–(IV) We consider again on the half-plane x < 0 the linear differential system (I) with its first integral H1 (x, y), and on x > 0 we take the quadratic isochronous differential system (5) with its first integral H4 (x, y). Then if the discontinuous differential system (I)–(5) has a limit cycle, it must intersect the discontinuity line x = 0 at two different points (0, y) and (0, Y ). These points must satisfy the equations H1 (0, y) − H1 (0, Y ) = (Y − y)P1 (y, Y ) = 0, (Y − y)Q4 (y, Y ) (12) H4 (0, y) − H4 (0, Y ) = = 0, (−3 + 16yβ + 16γ)(−3 + 16Y β + 16γ) where P1 and Q4 are polynomials of degrees one and four, respectively. We solve the equation P1 (y, Y ) = 0 obtaining the variable Y as a function of y, that is Y = f (y). If we substitute Y = f (y) in the equation Q4 (y, Y ) = 0, we obtain a polynomial equation of degree four in the variable y, and so system (12) has at most four real solutions. Taking 19 into account the symmetry between these solutions, as in the previous statements there can be only two different solutions (y, Y ) of (12) satisfying y < Y . Now, we show a concrete discontinuous piecewise differential system (I)-(5) having a unique limit cycle. On the half-plane x < 0 we consider the linear differential system ẋ = 1 − x − 2y, ẏ = x + y, (13) with first integral H1 (x, y) = −2y + y 2 + (x + y)2 , and on the half-plane x > 0 we consider the quadratic isochronous differential system of type (4) given by ẋ = −8.98889 − 17.5383x − 20.7447y − 2.21606x2 − 9.76545xy − 7.54939y 2 , (14) ẏ = 14.6779 + 21.205x + 25.5383y + 3.54939x2 + 12.4321xy + 8.88273y 2 , with first integral 1 H4 (x, y) = (16(1 + x + y)4 16x + 0.0318943(1067y + 2091) −24(1 + x + y)2 (4.35569 + x + 2.12695y) + 9((1 + x + y)2 +(4.35569 + x + 2.12695y)2 )) . In this case the solution for system (12) with y < Y is (y1 , Y1 ) = (−1.898651543493539, 2.898651543493539), and the corresponding limit cycle of the discontinuous piecewise differential system (13)– (14) associated to these solutions are shown in Figure 3. Remark. For all these discontinuous piecewise differential systems it is possible that the upper bound found for the maximum number of limit cycles cannot be reached. This is due to the fact that the solutions (y, Y ) do not need to correspond necessarily to periodic solutions of the discontinuous piecewise differential systems. 20 D. Proof of Theorem 1 for system (I)–(V) We take again the linear differential system (I) with its first integral H1 (x, y) on the half-plane x < 0, and on x > 0 we consider the quadratic isochronous differential system (6) with its first integral H5 (x, y). Thus if the discontinuous differential system (I)–(6) has a limit cycle, it must intersect the discontinuity line x = 0 at two different points (0, y) and (0, Y ). These points must satisfy the equations H1 (0, y) − H1 (0, Y ) = (Y − y)P1 (y, Y ) = 0, (Y − y)Q5 (y, Y ) (15) H5 (0, y) − H5 (0, Y ) = = 0, (3 + 8yβ + 8γ)4 (3 + 8Y β + 8γ)4 where P1 and Q5 are polynomials of degrees one and five, respectively. We solve again the equation P1 (y, Y ) = 0 obtaining the variable Y as a function of y, that is Y = f (y). If we substitute Y = f (y) in the equation Q5 (y, Y ) = 0, we obtain a polynomial equation of degree 4 in the variable y, and so system (12) has at most four real solutions. Taking into account the symmetry between these solutions, as in the previous statements, there can be only two different solutions (y, Y ) of (15) satisfying y < Y . Finally we show a discontinuous piecewise differential system (I)–(6) having two limit cycles. On the half-plane x < 0 we consider the linear differential system ẋ = 1 + x − 2y, ẏ = x − y, (16) with first integral H1 (x, y) = −2y + x2 − 2xy + 2y 2 , and on the half-plane x > 0 we consider the quadratic isochronous differential system of type (6) ẋ = 0.00134583 + 7.88817x + 0.03269y − 5.33333x2 + 5.81077xy − 0.0735418y 2 , (17) ẏ = 3.82531 − 3.82531x + 5.44516y − 2.66667xy + 1.93692y 2 , 21 with first integral 1 H5 (x, y) = (9 ((−1x + 0.726346y + 1)2 + (−1y − 1.80949)2 ) (8(−1y − 1.80949) + 3)4 +16(−1y − 1.80949)4 + 24.(−1y − 1.80949)3 ) . For this case, the two solutions for system (15) with y < Y are   1 4 (y1 , Y1 ) = (0, 1), (y2 , Y2 ) = , 5 5 and the corresponding limit cycles of the discontinuous piecewise differential system (8)–(17) associated to this solution is shown in Figure 4. IV. PROOF OF THEOREM 2 A. Proof of statement (a) of Theorem 2. We consider the quadratic polynomial differential system (3) with first integral H2 (x, y) in the half-plane x < 0. By changing the parameters (a, α, b, β, c, γ) to (a1 , α1 , b1 , β1 , c1 , γ1 ) in system (3) and in its first integral, we obtain a second isochronous quadratic differential system of type (3) with the first integral H e 2 (x, y), namely 1 ẋ = β1 γ12 + 2b1 γ1 c1 + b1 c1 + β1 γ1 − β1 c21 + (2a1 b1 γ1 + 2α1 β1 γ1 b 1 α 1 − a1 β 1 +a1 b1 + α1 β1 − 2a1 β1 c1 + 2α1 b1 c1 )x + (2γ1 + 1)(b21 + β12 )y + (−a21 β1 + α12 β1  +2α1 a1 b1 )x2 + 2α1 (b21 + β12 )xy + β1 (b21 + β12 )y 2 , 1 (18) ẏ = −α1 γ12 − 2a1 γ1 c1 − a1 c1 − α1 γ1 + α1 c21 − (2γ1 + 1)(a21 + α12 )x b1 α1 − a1 β1 +(−2a1 b1 γ1 − 2α1 β1 γ1 − a1 b1 − α1 β1 − 2a1 β1 c1 + 2α1 b1 c1 )y − α1 (a21 + α12 )x2  −2β1 (a21 + α12 )xy − (α1 β12 + 2a1 β1 b1 − α1 b21 )y 2 , whose first integral is 2 2 e 2 (x, y) = (c1 + a1 x + b1 y) + (xα1 + yβ1 + γ1 ) . H 1 + 2(xα1 + yβ1 + γ1 ) 22 If a limit cycle of the discontinuous piecewise differential system (3)-(18) has two different intersection points (0, y) and (0, Y ) with the line x = 0, then they must satisfy the system (Y − y)P2 (y, Y ) H2 (0, y) − H2 (0, Y ) = = 0, (2γ + 2βy + 1)(2γ + 2βY + 1) (19) e 2 (0, y) − H e 2 (0, Y ) = (Y − y)Q2 (y, Y ) H = 0, (2γ1 + 2β1 y + 1)(2γ1 + 2β1 Y + 1) where both polynomials P2 and Q2 are of degree two. From the equation Q2 (y, Y ) = 0, we get Y as a function of y, that is, Y = f (y), and substituting this expression in the equation P2 (y, Y ) = 0, we obtain a quadratic polynomial equation in the variable y. Then the maximum number of solutions of (19) is two, namely (y1 , Y1 ) and (y2 , Y2 ), but in fact, these two solutions represent the same limit cycle because they are symmetric in the sense of the proof of Theorem 1. Thus system (3)-(3) has at most one limit cycle. Now we give an example of a discontinuous piecewise differential system of type (3)- (3) having one limit cycle. On x < 0 we consider the quadratic isochronous differential system ẋ = −4 − x − 6y + x2 − 2y 2 , ẏ = 3 + 3x + 5y + 2xy + 2y 2 , (20) with a first integral (1 + y)2 + (1 + x + y)2 H2 (x, y) = , 1 + 2(1 + y) and on x > 0 we consider the quadratic isochronous differential system of type (3) ẋ = −2 + x − 6y + x2 − 4xy + 2y 2 , ẏ = 1 + 3x − 5y + x2 − 2xy, (21) whose first integral is 2 2 He 2 (x, y) = (1 + x − y) + (1 + y) . 1 + 2(1 + x − y)   1 √ 1 √ The solution of (19) satisfying y < Y is (y, Y ) = (−1 − 3), (−1 + 3) , which 2 2 provides the limit cycle for the discontinuous differential piecewise system (20)-(21) shown in Figure 5. 23 B. Proof of statement (b) of Theorem 2. We consider again the quadratic polynomial differential system (18) with first integral e 2 (x, y) in the half-plane x < 0 and for x > 0 we take the isochronous differential system H (4) whose first integral is H3 (x, y). If there exists a limit cycle of the discontinuous piecewise differential system (18)-(4), then it has two different intersection points (0, y) and (0, Y ) with the line x = 0, which satisfy the system e 2 (0, y) − H e 2 (0, Y ) = (Y − y)P2 (y, Y ) H = 0, (2γ1 + 2β1 y + 1)(2γ1 + 2β1 Y + 1) (Y − y)Q2 (y, Y ) (22) H3 (0, y) − H3 (0, Y ) = = 0, (γ2 + β2 y + 1)2 (γ2 + β2 Y + 1)2 where both polynomials P2 and Q2 have degree two. From the equation P2 (y, Y ) = 0, we get Y as a function of y, that is, Y = f (y), and substituting this expression in the equation Q2 (y, Y ) = 0, we obtain a polynomial equation of degree two in the variable y. Then the maximum number of solutions of (22) is two. But due to the symmetry of the solutions, the system (3)-(4) has at most one limit cycle. Now we write an example of a discontinuous piecewise differential system having a unique limit cycle. On x < 0 we consider the quadratic isochronous differential system (20), and on x > 0 we consider the quadratic isochronous differential system of type (4) ẋ = −2 − 4y − xy, ẏ = x − 3y − y 2 , (23) whose first integral is (x − y + 1)2 + (y + 1)2 H3 (x, y) = . (x − y + 2)2   4 Then, the obtained solution of system (22) satisfying y < Y is (y, Y ) = − , 0 . 3 This pair, provides the limit cycle that exists for the discontinuous differential piecewise system (20)-(23) shown in Figure 6. 24 C. Proof of statement (c) of Theorem 2. We take the quadratic polynomial differential system (18) with the first integral e 2 (x, y) in the half-plane x < 0 and for x > 0 we take the isochronous differential H system (5) with the first integral H4 (x, y). If there exists a limit cycle of the discontinuous piecewise differential system (18)-(5), then it has two different intersection points (0, y) and (0, Y ) with the separation line x = 0, satisfying the system e 2 (0, y) − H e 2 (0, Y ) = (Y − y)P2 (y, Y ) H = 0, (2γ1 + 2β1 y + 1)(2γ1 + 2β1 Y + 1) (Y − y)Q4 (y, Y ) (24) H4 (0, y) − H4 (0, Y ) = = 0, (−3 + 16yβ + 16γ)(−3 + 16Y β + 16γ) where both polynomials P2 and Q4 are of degree two and four respectively. From the equation P2 (y, Y ) = 0, we obtain Y as a function of y, that is, Y = f (y), and if we put this expression in the second equation Q4 (y, Y ) = 0, we obtain a polynomial equation of degree six in the variable y. Then, the maximum number of solutions of (22) is six, but due to the symmetry, there are at most three solutions of system (24) satisfying y < Y . Thus system (3)-(5) has at most three limit cycles. Next we give an example of discontinuous piecewise differential system of type (3)-(5) having two limit cycles. On x < 0 we consider the quadratic isochronous differential system ẋ = −0.427384 − 0.457523x − 4.27384y + 1.9317x2 + 6.83814xy − 0.427384y 2 , (25) ẏ = 0.116463 + 0.68296x + 1.87174y − 0.546368x2 + 0.136592xy + 3.652y 2 , with the first integral 2 √ √ e 2 (x, y) = 9 + 128x + 9y(18 + 89y) − 16x(1 − 2 2 + y − 20 2y) H , 16(5 − 8x + y) 25 and on x > 0 we consider the quadratic isochronous differential system of type (5) ẋ = −0.593985 − 6.89377x − 6.77032y + 3.03414x2 + 0.667964xy − 1.89916y 2 , (26) ẏ = 4.77286 + 7.53417x + 9.75481y − 1.87138x2 + 1.93171xy + 3.30143y 2 , whose first integral is 1 H4 (x, y) = (16(−0.35763 − x − 0.908852y)4 54.173 − 16x + 19.6577y −24(−0.35763 − x − 0.908852y)2 (3.57331 − x + 1.22861y) (27) +9((−0.35763 − x − 0.908852y)2 + (3.57331 − x + 1.22861y)2 )) . The solutions of (24) satisfying y < Y are (y1 , Y1 ) = (−1, 1), (y2 , Y2 ) = (−2, 3), which provide the two limit cycles for the discontinuous differential piecewise system (25)-(26) shown in Figure 7. D. Proof of statement (d) of Theorem 2. We consider the quadratic polynomial differential system (18) with the first integral e 2 (x, y) in the half-plane x < 0, and for x > 0 we take the isochronous differential H system (6) whose first integral is H5 (x, y). If a limit cycle exists for the discontinuous piecewise differential system (18)-(6), then it has two different intersection points (0, y) and (0, Y ) with the line x = 0, satisfying the closing equations e 2 (0, y) − H e 2 (0, Y ) = (Y − y)P2 (y, Y ) H = 0, (2γ1 + 2β1 y + 1)(2γ1 + 2β1 Y + 1) (28) (Y − y)Q5 (y, Y ) H5 (0, y) − H5 (0, Y ) = = 0, (8γ + 8βy + 3)(8γ + 8βY + 3) where both polynomials P2 and Q5 are of degree two and five respectively. From the equation P2 (y, Y ) = 0, we get Y as a function of y, that is, Y = f (y), and substituting 26 this expression in the equation Q5 (y, Y ) = 0, we obtain an equation of order six in the variable y. Then, the maximum number of solutions of (28) is six, but because of the symmetry property as in the previous statements, system (3)-(6) has three limit cycle at most. Next, we give an example of discontinuous piecewise differential system of type (3)-(6) having two limit cycles. On x < 0 we consider the quadratic isochronous differential system ẋ = −4.24371 + 0.170949x + 6.25388y + 3.65748x2 − 12.5078xy + 8.04071y 2 , (29) ẏ = −2.20072 − 1.28915x + 5.48591y + 1.28915x2 − 3.31496xy + 0.579633y 2 , with the first integral 2 e 2 (x, y) = 14(1.5 + x + x(0.414214 − 4.12284y) + y(−2.72659 + 4.85117y)) , H −7 + 14x − 18y and on x > 0 we consider the quadratic isochronous differential system of type (6) ẋ = 0.550125 + 9.00984x − 1.24812y + 6.05959x2 − 0.600689xy − 0.03414y 2 , (30) ẏ = 3.90588 + 49.7765x − 6.63534y + 17.3476x2 + 1.21415xy − 0.491153y 2 , whose first integral is 0.000244141 H5 (x, y) = (9((−0.825206 + x − 0.195584y)2 (−0.450206 + x − 0.195584y)4 ) +(0.178088 + x − 0.118725y)2 ) + 24(−0.825206 + x − 0.195584y)3 (31) +16(−0.825206 + x − 0.195584y)4 ) . The solutions of (28) satisfying y < Y which provide the two limit cycles for the discon- tinuous differential piecewise system (29)-(30) are (y1 , Y1 ) = (0.1., 1.), (y2 , Y2 ) = (0.166666, 0.833335), shown in Figure 8. 27 V. PROOF OF THEOREM 3 A. Proof of statement (a) of Theorem 3. We consider the quadratic polynomial differential system (4) with first integral H3 (x, y) in the half-plane x < 0. By changing the parameters (a, α, b, β, c, γ) to (a1 , α1 , b1 , β1 , c1 , γ1 ) in system (4) and in its first integral, we obtain a second isochronous quadratic differential system of type (4) with the first integral H e 3 (x, y), namely 1 ẋ = (b2 y(1 + xα1 + γ1 ) + b1 (c1 + a1 x)(1 + xα1 − yβ1 + γ1 ) b1 α1 − a1 β1 1 +β1 (−c21 − 2a1 c1 x − a21 x2 + xα1 + yβ1 + γ1 )) 1 (32) ẏ = (α1 (c21 + 2b1 c1 y + b21 y 2 − xα1 − yβ1 − γ1 ) − a21 x(1 + yβ1 + γ1 ) b 1 α 1 − a1 β 1 −a1 (c1 + b1 y)(1 − xα1 + yβ1 + γ1 )) , whose first integral is 2 2 e 3 (x, y) = (c1 + a1 x + b1 y) + (xα1 + yβ1 + γ1 ) . H (1 + xα1 + yβ1 + γ1 )2 If a limit cycle of the discontinuous piecewise differential system (4)-(32) has two different intersection points (0, y) and (0, Y ) with the line x = 0, then they must satisfy the system (Y − y)P2 (y, Y ) H3 (0, y) − H3 (0, Y ) = = 0, (1 + yβ + γ)2 (1 + Y β + γ)2 (33) e 3 (0, y) − H e 3 (0, Y ) = (Y − y)Q2 (y, Y ) H = 0, (1 + yβ1 + γ1 )2 (1 + Y β1 + γ1 )2 where both polynomials P2 and Q2 are of degree two. From the equation Q2 (y, Y ) = 0, we get Y as a function of y, that is, Y = f (y), and after substituting this expression in the equation P2 (y, Y ) = 0, we obtain a quadratic polynomial equation in the variable y. Then the maximum number of solutions of (33) is two, namely (y1 , Y1 ) and (y2 , Y2 ), but in fact, these two solutions represent the same limit cycle because they are symmetric as usual. Thus system (4)-(4) has at most one limit cycle. 28 Now we give an example of a discontinuous piecewise differential system of type (4)- (4) having one limit cycle. On x < 0 we consider the quadratic isochronous differential system (23) and on x > 0 we take the quadratic isochronous differential system of type (4) ẋ = −2 − 2y + x2 + xy, ẏ = 2 + 2x + 3y + xy + y 2 , (34) with a first integral (1 + y)2 + (1 + x + y)2 H3 (x, y) = . (2 + y)2   4 The solution of (33) satisfying y < Y is (y, Y ) = − , 0 , which provides the limit 3 cycle for the discontinuous differential piecewise system (23)-(34) shown in Figure 9. B. Proof of statement (b) of Theorem 3. We consider again the quadratic polynomial differential system (32) with first integral e 3 (x, y) in the half-plane x < 0 and for x > 0 we take the isochronous differential system H (5) whose first integral is H4 (x, y). If there exists a limit cycle of the discontinuous piecewise differential system (32)-(5), then it has two different intersection points (0, y) and (0, Y ) with the line x = 0, which satisfy the system e 3 (0, y) − H e 3 (0, Y ) = (Y − y)P2 (y, Y ) H = 0, (1 + yβ1 + γ1 )2 (1 + Y β1 + γ1 )2 ) (Y − y)Q4 (y, Y ) (35) H4 (0, y) − H4 (0, Y ) = = 0, (−3 + 16yβ + 16γ)(−3 + 16Y β + 16γ) where the polynomials P2 and Q4 have degree two and four, respectively. From the equation P2 (y, Y ) = 0, we get Y as a function of y, that is, Y = f (y), and substituting this expression in the equation Q4 (y, Y ) = 0, we obtain a polynomial equation of degree six in the variable y. Then, the maximum number of solutions of (35) is six. But due to the symmetry of the solutions, the system (4)-(5) has at most three limit cycles. Now, we show an example of discontinuous piecewise differential system of type (4)- (5) having two limit cycles. On x < 0 we consider the quadratic isochronous differential 29 system ẋ = −5.92592 − 47.9127x − 58.6606y + x2 − 0.101841xy, (36) ẏ = −246.99 + 9.5224x − 26.911y + xy − 0.101841y 2 , with the first integral 2 2 e 3 (x, y) = (24.9412 − x + 0.101841y) + (602.838 − x + 61y) , H (603.838 − x + 61y)2 and on x > 0 we consider the quadratic isochronous differential system (26) of type (5) whose first integral is 27. The solutions of (35) satisfying y < Y which provide the two limit cycles for the discontinuous differential piecewise system (36)-(26) are (y1 , Y1 ) = (−1, 1), (y2 , Y2 ) = (−2, 3), shown in Figure 10. C. Proof of statement (c) of Theorem 3. We take the quadratic polynomial differential system (32) with the first integral e 3 (x, y) in the half-plane x < 0 and for x > 0 we take the isochronous differential H system (6) with the first integral H5 (x, y). If there exists a limit cycle of the discontinuous piecewise differential system (32)-(6), then it has two different intersection points (0, y) and (0, Y ) with the separation line x = 0, satisfying the system e 3 (0, y) − H e 3 (0, Y ) = (Y − y)P2 (y, Y ) H = 0, (1 + yβ1 + γ1 )2 (1 + Y β1 + γ1 )2 ) (37) (Y − y)Q5 (y, Y ) H5 (0, y) − H5 (0, Y ) = = 0, (3 + 8yβ + 8γ)4 (3 + 8Y β + 8γ)4 where the polynomials P2 and Q5 are of degree two and five, respectively. From the equation P2 (y, Y ) = 0, we obtain Y as a function of y, that is, Y = f (y), and putting this expression in the second equation Q5 (y, Y ) = 0, we obtain a polynomial equation of degree six in the variable y. Then, the maximum number of solutions of (37) is six, but 30 due to the symmetry, there are at most three solutions of system (37) satisfying y < Y . Thus, system (4)-(6) has at most three limit cycles. Next, we give an example of discontinuous piecewise differential system of type (4)- (6) having two limit cycles. On x < 0 we consider the quadratic isochronous differential system ẋ = 6781.71 + 363.957x − 15585.4y + x2 − 45.2674xy, (38) ẏ = 140.782 + 7.24945x − 308.767y + xy − 45.2674y 2 , with the first integral 2 2 e 3 (x, y) = (−70.4328 + x − 57.2448y) + (18.3967 + x − 45.2674y) , H (−69.4328 + x − 57.2448y)2 and on x > 0 we consider the quadratic isochronous differential system (30) of type (6) whose first integral is 31. The solutions of (37) satisfying y < Y which provide the two limit cycles for the discontinuous differential piecewise system (38)-(30) are (y1 , Y1 ) = (0.1, 1), (y2 , Y2 ) = (0.166666, 0.833335), shown in Figure 11. VI. PROOF OF THEOREM 4 A. Proof of Theorem 4 for system (IV)-(IV) We consider the quadratic polynomial differential system (5) with the first inte- gral H4 (x, y) in the half-plane x < 0. By changing the parameters (a, α, b, β, c, γ) to (a1 , α1 , b1 , β1 , c1 , γ1 ) in system (5) and its first integral, we obtain a second isochronous 31 e 4 (x, y), namely quadratic differential system of type (5) with the first integral H 1 ẋ = (b1 (c1 + a1 x)(−3 + 16xα1 + 8yβ1 + 16γ1 ) + b21 y(−3 + 16xα1 + 12yβ1 + 16γ1 ) 3a1 β1 − 3b1 α1 −β1 (4c21 + 8a1 c1 x + 4a21 x2 + 3(xα1 + yβ1 + γ1 )) , 1 ẏ = a1 (c1 + b1 y) (−3 + 8xα1 + 16yβ1 + 16γ1 ) + a21 x(−3 + 12xα1 + 16yβ1 + 16γ1 ) 3b1 α1 − 3a1 β1 −α1 (4c21 + 8b1 c1 y + 4b21 y 2 + 3(xα1 + yβ1 + γ1 )) , (39) whose first integral is e 4 (x, y) = 1 H −24(a1 x + b1 y + c1 )2 (γ1 + α1 x + β1 y) 16(γ1 + α1 x + β1 y) − 3  +9 ((a1 x + b1 y + c1 )2 + (γ1 + α1 x + β1 y)2 ) + 16(a1 x + b1 y + c1 )4 . If there exists a limit cycle of the discontinuous piecewise differential system (5)-(39), then it has two different intersection points (0, y) and (0, Y ) with the separation line x = 0, satisfying the system e 4 (0, y) − H e 4 (0, Y ) = (Y − y)P4 (y, Y ) H = 0, (−3 + 16yβ1 + 16γ1 )(−3 + 16Y β1 + 16γ1 ) (Y − y)Q4 (y, Y ) (40) H4 (0, y) − H4 (0, Y ) = = 0, (−3 + 16yβ + 16γ)(−3 + 16Y β + 16γ) where both polynomials P4 and Q4 are of degree four. From the equation P4 (y, Y ) = 0, we obtain Y as a function of y, that is, Y = f (y), and substituting it in the second equation Q4 (y, Y ) = 0, we obtain a polynomial equation of degree six in the variable y. Then, the maximum number of solutions of (40) is six, but due to the symmetry, there are at most three solutions of system (40) satisfying y < Y . Thus, system (5)-(5) has at most three crossing limit cycles. Next, we give an example of discontinuous piecewise differential system of type (5)- (5) having two limit cycles. On x < 0 we consider the quadratic isochronous differential 32 system ẋ = −0.593985 + 6.89377x − 6.77032y + 3.03414x2 − 0.667964xy − 1.89916y 2 , (41) ẏ = −4.77286 + 7.53417x − 9.75481y + 1.87138x2 + 1.93171xy − 3.30143y 2 , with the first integral e 4 (x, y) = 1 H (16(−0.35763 + x − 0.908852y)4 −3 + 16(3.57331 + x + 1.22861y) −24(−0.35763 + x − 0.908852y)2 (3.57331 + x + 1.22861y) (42) +9((−0.35763 + x − 0.908852y)2 + (3.57331 + x + 1.22861y)2 )) , and on x > 0 we consider the quadratic isochronous differential system ẋ = −4.35274 + 17.666x − 44.1167y + 1.65212x2 − 2.11558xy − 4.00174y 2 , (43) ẏ = 279.554 + 16.3982x + 89.6236y − 0.305758x2 + 4.69575xy + 5.2283y 2 , with the first integral 1 H4 (x, y) = (16(−13.4112 − x − 1.04263y)4 −3 + 16(47.304 − x + 12.0396y) −24(−13.4112 − x − 1.04263y)2 (47.304 − x + 12.0396y) +9((−13.4112 − x − 1.04263y)2 + (47.304 − x + 12.0396y)2 )) . The solutions of (40) satisfying y < Y which provide the two limit cycles for the discon- tinuous differential piecewise system (41)-(43) are (y1 , Y1 ) = (−1, 1), (y2 , Y2 ) = (−2, 3), shown in Figure 12. B. Proof Theorem 4 for system (IV)-(V) We take the quadratic polynomial differential system (39) with the first integral e 4 (x, y) in the half-plane x < 0 and for x > 0 we take the isochronous differential H 33 system (6) with the first integral H5 (x, y). If there exists a limit cycle of the discontinuous piecewise differential system (39)-(6), then there exist two different intersection points (0, y) and (0, Y ) with the separation line x = 0, satisfying the system e 4 (0, y) − H e 4 (0, Y ) = (Y − y)P4 (y, Y ) H = 0, (−3 + 16yβ1 + 16γ1 )(−3 + 16Y β1 + 16γ1 ) (44) (Y − y)Q5 (y, Y ) H5 (0, y) − H5 (0, Y ) = = 0, (3 + 8yβ + 8γ)4 (3 + 8Y β + 8γ)4 where the polynomials P4 and Q5 are of degree four and five, respectively. Again, we obtain Y as a function of y, that is, Y = f (y), from the equation P4 (y, Y ) = 0, and putting this expression in the second equation Q5 (y, Y ) = 0, we obtain a polynomial equation of degree seven in the variable y. Then, the maximum number of solutions of (44) is seven, but due to the symmetry, there are at most three solutions of system (44) which satisfy the condition y < Y . Then, system (5)-(6) has at most three limit cycles. Now, we show an example of discontinuous piecewise differential system of type (5)-(6) having two limit cycles. On x < 0 we consider again the quadratic isochronous differ- ential system (41) whose first integral is (42), and on x > 0 we consider the quadratic isochronous differential system ẋ = −0.0135199 + 6.44812x − 0.134872y + 4.00057x2 + 0.155285xy − 0.00453093y 2 , ẏ = 15.1375 + 132.575x − 1.28187y + 22.8872x2 + 5.3322xy − 0.0778083y 2 , (45) with the first integral 1 H5 (x, y) = (9((−1.24193 + x − 0.0582816y)2 (3 + 8(−1.24193 + x − 0.0582816y))4 +(0.387469 + x − 0.0000248632y)2 ) + 24(−1.24193 + x − 0.0582816y)3 +16(−1.24193 + x − 0.0582816y)4 ) . The solutions of (44) satisfying y < Y which provide the two limit cycles for the discon- 34 tinuous differential piecewise system (41)-(45) are (y1 , Y1 ) = (−1, 1), (y2 , Y2 ) = (−2, 3), shown in Figure 13. VII. PROOF OF THEOREM 5 We consider the quadratic polynomial differential system (6) with the first inte- gral H5 (x, y) in the half-plane x < 0. If we change the parameters (a, α, b, β, c, γ) to (a1 , α1 , b1 , β1 , c1 , γ1 ) in system (6) and its first integral, we obtain a second isochronous quadratic differential system of type (6) with the first integral H e 5 (x, y), namely 1 ẋ = 4β1 γ12 + 8b1 γ1 c1 + 3b1 c1 + 3β1 γ1 − 16β1 c21 + (8a1 b1 γ1 + 8α1 β1 γ1 3(α1 b1 − a1 β1 ) +3a1 b1 + 3α1 β1 − 32a1 β1 c1 + 8α1 b1 c1 )x + (8b21 γ1 + 8β12 γ1 + 3b21 + 3β12 −24β1 b1 c1 )y + 4(α12 β1 − 4a21 β1 + 2α1 a1 b1 )x2 + 8(α1 β12 − 3a1 β1 b1 + α1 b21 )xy  −4β1 y 2 (2b21 − β12 )y 2 , 1 ẏ = 16α1 c21 − 4α1 γ12 − 8a1 γ1 c1 − 3a1 c1 − 3α1 γ1 − (8a21 γ1 + 8α12 γ1 3(α1 b1 − a1 β1 ) +3a21 + 3α12 − 24α1 a1 c1 )x − (8a1 b1 γ1 + 8α1 β1 γ1 + 3a1 b1 + 3α1 β1 + 8a1 β1 c1 −32α1 b1 c1 )y + 4α1 (2a21 − α12 )x2 + 8(a21 (−β1 ) − α12 β1 + 3α1 a1 b1 )xy  −4(α1 β12 + 2a1 β1 b1 − 4α1 b21 )y 2 (46) whose first integral is 1 H5 (x, y) = 9 ((a1 x + b1 y + c1 )2 + (γ1 + α1 x + β1 y)2 ) (8(γ1 + α1 x + β1 y) + 3)4  +16(γ1 + α1 x + β1 y)4 + 24(γ1 + α1 x + β1 y)3 . If there exists a limit cycle of the discontinuous piecewise differential system (6)-(46), then it has two different intersection points (0, y) and (0, Y ) with the separation line 35 x = 0, satisfying the system e 5 (0, y) − H e 5 (0, Y ) = (Y − y)P5 (y, Y ) H = 0, (3 + 8yβ1 + 8γ1 )4 (3 + 8Y β1 + 8γ1 )4 (Y − y)Q5 (y, Y ) (47) H5 (0, y) − H5 (0, Y ) = = 0, (3 + 8yβ + 8γ)4 (3 + 8Y β + 8γ)4 where both polynomials P5 and Q5 are of degree five. Then, by Bézout Theorem, the maximum number of solutions (y, Y ) ∈ C of equation (47) is 25. Since our solutions appear in pairs because of the symmetry, the maximum number of solutions satisfying y < Y is twelve. Next, we give an example of discontinuous piecewise differential system of type (6)- (46) having two limit cycles. On x < 0 we consider the quadratic isochronous differential system (48) ẋ = 0.103814 + 0.0255717x − 0.0978649y − 0.0368897x2 + 0.218646xy − 0.322863y 2 , ẏ = 0.03051 + 0.0170376x − 0.0573298y − 0.0153809x2 + 0.0905173xy − 0.132811y 2 , (48) with the first integral e 5 (x, y) = 1 H (9((−3.79477 + 2x − 5.613y)2 (3 + 8(−4.15332 + x − 2.61606y))4 +(−4.15332 + x − 2.61606y)2 ) + 24(−4.15332 + x − 2.61606y)3 +16(−4.15332 + x − 2.61606y)4 ) , and on x > 0 we consider the quadratic isochronous differential system ẋ = 0.0000696792 + 0.00114119x − 0.000158087y + 0.000767513x2 − 0.0000760837xy −0.0000043242y 2 , ẏ = 0.000494721 + 0.00630472x − 0.000840437y + 0.00219727x2 + 0.000153785xy −0.0000622098y 2 , (49) 36 with the first integral 1 H5 (x, y) = (9((−0.825206 + x − 0.195584y)2 (3 + 8(−0.825206 + x − 0.195584y))4 +(0.178088 + x − 0.118725y)2 ) + 24(−0.825206 + x − 0.195584y)3 +16(−0.825206 + x − 0.195584y)4 ) . The solutions of (47) satisfying y < Y which provide the two limit cycles for the discon- tinuous differential piecewise system (48)-(49) are (y1 , Y1 ) = (0.1, 1), (y2 , Y2 ) = (0.166666, 0.833335), shown in Figure 14. Remark. Equations (47) can have at most 12 solutions using Bézout Theorem, but numerical evidences obtained given arbitrary values to the parameters of the class of discontinuous piecewise differential systems of type (V)-(V), we could only find at most 8 real solutions, but these solutions in general do not provide crossing limit cycles. In all the particular piecewise differential systems studied numerically we could only find at most 2 crossing limit cycles. VIII. CONCLUSIONS There is a unique family of linear isochronous centers and four families of quadratic isochronous centers. Considering all the possibilities of choosing two pairs of these isochronous centers, eventually repeated, we obtain fifteen pairs. Therefore we have fifteen classes of discontinuous piecewise differential systems formed by two differen- tial systems separated by a straight line when these two differential systems are linear isochronous centers or quadratic isochronous centers. For these fifteen classes of discontinuous piecewise differential systems we provide an upper bound for the maximum number of limit cycles that they can exhibit, i.e. for these classes of differential systems we have solved the 16th Hilbert problem. Moreover for seven of these fifteen classes of discontinuous piecewise differential systems the upper 37 bound on the maximum number obtained is reached. More precisely, it was known that discontinuous piecewise differential systems formed by two linear isochronous centers separated by a straight line cannot have limit cycles, see27 . If one of the systems is a linear isochronous center and the other is a quadratic isochronous center then their maximum number of limit cycles is studied in Theorem 1. While if the two systems are quadratic isochronous centers then their maximum number of limit cycles are studied in Theorems 2, 3, 4 and 5. ACKNOWLEDGMENTS The first author is partially supported by the Ministerio de Economía y Competitivi- dad in the frame of project PGC2018-096265-B-I00, and by the Consejería de Economía y Conocimiento de la Junta de Andalucía, under grant P12-FQM-1658. The second author is supported by the Ministerio de Ciencia, Innovación y Universidades, Agencia Estatal de Investigación grant PID2019-104658GB-I00 (FEDER), the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Re- search Council grant MSCA-RISE-2017-777911. The third author is partially supported by by FCT/Portugal through UID/MAT/04459/2019. REFERENCES 1 J.C. Artés, J. Llibre, J.C. 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