This is a preprint of: “Crossing Periodic Orbits Via First Integrals”, Jaume Llibre, Durval J. Tonon, Mariana Q. Velter, Internat. J. Bifur. Chaos Appl. Sci. Engrg., vol. 30(11), 2050163 (9 pages), 2020. DOI: [10.1142/S0218127420501631] CROSSING PERIODIC ORBITS VIA FIRST INTEGRALS JAUME LLIBRE1 , DURVAL JOSÉ TONON2 AND MARIANA QUEIROZ VELTER3 Abstract. We characterize the families of periodic orbits of two discon- tinuous piecewise differential systems in R3 separated by a plane using their first integrals. One of these discontinuous piecewise differential systems is formed by linear differential systems, and the other by nonlinear differential systems. 1. Introduction and statement of the main results Discontinuous piecewise differential systems have been studied intensively in recent decades and this is mainly due to the fact that they provide in many applications more realistic models, as for instance in modeling electrical circuits [3, 10], in some mechanical problems [1], in control theory problems [2], etc. We define a discontinuous piecewise differential system following the rules of Filippov [6]. In the study of the dynamics of differential equations the periodic orbits play a main role when they exist, and the crossing periodic orbits play also a relevant role in the case of the discontinuous piecewise differential systems. A crossing periodic orbit of a discontinuous piecewise differential system in R3 is a periodic orbit which intersect the surface of discontinuity in a finite number n of points with n > 1. The objective of this paper is to study the crossing periodic orbits of dis- continuous piecewise differential systems of the form (1) X (x, y, z) = X1 (x, y, z) + sign (x) X2 (x, y, z) , where X1 and X2 are smooth vector fields in R3 using their first integrals. As usual the sign function is defined as   −1 if x < 0, sign (x) = 0 if x = 0,   1 if x > 0. 2010 Mathematics Subject Classification. 34C27, 34J45. Key words and phrases. Discontinuous piecewise differential systems, first integrals, pe- riodic orbits, homoclinic orbits. 1 2 J. LLIBRE, D.J. TONON AND M.Q. VELTER We note that the discontinuous piecewise differential system (1) is separated by the plane x = 0, and it is given by the system Y = X1 + X2 in x > 0, and by the system Z = X1 − X2 in x < 0. If H1 and H2 are first integrals of Y and Z respectively, then   H1 + H2 H1 − H2 H= + sign (x) , 2 2 is a first integral of the discontinuous piecewise differential system (1). More precisely, we shall characterize the crossing periodic orbits having two and four points in the plane x = 0 of the following two discontinuous piece- wise differential systems. First, the discontinuous piecewise linear differential system ẋ = z, (2) ẏ = z, ż = −sign(x) y, which has the two independent first integrals (3) H (x, y, z) = y 2 + sign (x) z 2 , F (x, y, z) = x − y, as it is easy to verify. Second, the discontinuous piecewise nonlinear differential system ẋ = (x − y) z, (4) ẏ = z, ż = −sign (x) y, which has the two independent first integrals (5) H (x, y, z) = y 2 + sign (x) z 2 , G (x, y, z) = e−y (1 − x + y) , which again are easy to check. This means that systems (2) and (4) are completely integrable, because they are defined in R3 and they have two independent first integrals. We recall that any n-dimensional linear differential system having n − 1 independent first integrals, is completely integrable, see for instance [5]. It is well known that, in general, the analytic study of the periodic orbits is not easy because they are non-local objects. Sometimes the crossing periodic orbits of the discontinuous piecewise differential systems are studied computing the fixed points of the Poincaré return map, see for instance [4, 8]. But the computation of the Poincaré return map is in general difficult because we need to know the times that orbits, starting at x = 0, need to return to CROSSING PERIODIC ORBITS VIA FIRST INTEGRALS 3 x = 0. Here using the first integrals of the completely integrable discontinuous piecewise differential systems we shall study their crossing periodic orbits, this tool already has been used previously in piecewise linear differential systems, see [7, 9], here we also use it for piecewise nonlinear differential systems. Figure 2. The homo- Figure 1. The crossing clinic orbit of Corollary 1 periodic orbit of Theorem 1 which intersects the plane which intersects the plane x = 0 in the points x = 0 in the points (0, 10, 8) (0, 10, 10) and (0, 10, −10). and (0, 10, −8). Figure 3. The crossing periodic orbits of The- Figure 4. The crossing orem 2 which intersects periodic orbits of The- the plane x = 0 the orem 2 which intersects first one at the points the plane x = 0 in the (0, −0.76803.., 0.5) and points (0, −0.82464.., 2.5), (0, −0.76803.., −0.5) and (0, −0.82464.., −2.5), the second one at the points (0, 2.02231.., 3.10800..) and (0, 1.67834.., 1.57384..) and (0, 2.02231.., −3.10800..). (0, 1.67834.., −1.57384..). Our main results are the following. Theorem 1. The discontinuous piecewise differential system (2) has a cross- ing periodic orbit which intersects the discontinuous plane x = 0 in two points 4 J. LLIBRE, D.J. TONON AND M.Q. VELTER if and only if the points are (0, y0 , z0 ) and (0, y0 , −z0 ) satisfying z0 6= 0, y0 > 0 and y0 2 − z0 2 > 0. One of these periodic orbits is shown in Figure 1. The proofs of Theorem 1 and its corollary are given in section 3. Corollary 1. For each pair of points (0, y0 , ±y0 ) such that y0 > 0, the system (2) has a homoclinic orbit to the equilibrium point (−y0 , 0, 0) that intersects the discontinuous plane x = 0 first in (0, y0 , y0 ), and after in (0, y0 , −y0 ) . One of these homoclinic orbits is shown in Figure 2. Figure 5. The homoclinic orbits of Corollary 2 which intersects the plane x = 0 in the points (0, y0 , −y0 ) and (0, y1 , −y1 ), where y0 = −0.76803.. and y1 = 1.67834... Theorem 2. The following statements hold. (i) The discontinuous piecewise differential system (4) has a periodic orbit which intersects the discontinuous plane x = 0 in two points if and only if the points are (0, y0 , z0 ) and (0, y0 , −z0 ) satisfying z0 6= 0, y0 2 − z0 2 > 0 and y0 is the solution of the equation 1 + y − g ey = 0 for g < 1. We show two of these periodic orbits in Figure 3. (ii) The discontinuous piecewise differential system (4) has a periodic orbit which intersects the discontinuous plane x = 0 in four points if and only if the points are (0, y0 , ±z0 ) and (0, y1 , ±z1 ) satisfying zi 6= 0, yi 2 − zi 2 < 0 for i = 0, 1, y0 is the negative solution of the equation 1 + y − g ey = 0 and y1 is the positive solution of the equation 1 + y − g ey = 0 for 0 < g < 1. One of these periodic orbits is shown in Figure 4. CROSSING PERIODIC ORBITS VIA FIRST INTEGRALS 5 The proofs of Theorem 2 and its corollary are given in section 3. Corollary 2. For each pair of points (0, y0 , ±y0 ) such that y0 is a solution of the equation 1 + y − g ey = 0 for g < 1, system (4) has two homoclinic orbits to the equilibrium point (1 − e−y0 (1 + y0 ) , 0, 0) that intersects the discontinuous plane x = 0 first in (0, y0 , y0 ), and after in (0, y0 , −y0 ) . Two of these homoclinic orbits are shown in Figure 5. 2. Proof of Theorem 1 For studying the existence of crossing periodic orbits of the piecewise linear differential system (2) we must analyze the behavior of the two linear differ- ential systems (6) ẋ = z, ẏ = z, ż = y, for x < 0, and (7) ẋ = z, ẏ = z, ż = − y, for x > 0. From (3) two independent first integrals for system (6) are H1 (x, y, z) = y 2 − z 2 and F (x, y, z) = x − y, and two independent first integrals for system (7) are H2 (x, y, z) = y 2 + z 2 and F (x, y, z) = x − y. We want to study the intersection of the surfaces H1 (x, y, z) = h1 with F (x, y, z) = f in the half-space x < 0. The curve defined by this intersection is denoted by γh1 f , and clearly this curve is formed by orbits of system (6) when the intersection is non-empty. For h1 6= 0 the surfaces H1 (x, y, z) = h1 are hyperbolic cylinders, that is, their projections in the plane x = 0 describe a hyperbola  whosevertices are √  p the points 0, ± h1 , 0 if h1 > 0, or the points 0, 0, ± |h1 | if h1 < 0. The surface H1 (x, y, z) = 0 is formed by two planes whose intersection is the x-axis. Finally, the surfaces F (x, y, z) = f are planes. We observe that the curve γh1 f is well-defined, i.e. the intersection {H1 = h1 }∩ {F = f } ∩ {x < 0} is always non-empty, and γh1 f contains one √ orbit of system (6) contained in the half-space x < 0 if h1 > 0 and f < h1 . Otherwise, √ when h1 < 0 or when h1 > 0 and h1 ≤ f , the curve γh1 f contains two orbits of system (6) where each orbit is contained √ in distinct sheets of the hyperbolic cylinder if h1 < 0 or if h1 , f > 0 and h1 ≤ f (see Figure 6), or both orbits 6 J. LLIBRE, D.J. TONON AND M.Q. VELTER Figure 6. Intersection of Figure 7. Intersection of the surfaces H1 (x, y, z) = the surfaces H2 (x, y, z) = 0.1 and F (x, y, z) = −1 in 3.5 and F (x, y, z) = −0.3 the half-space x < 0. in the half-space x > 0. are contained √ in the same sheet of the hyperbolic cylinder if h1 > 0, f < 0 and f ≤ − h1 . Now, since γ0 f intersects the x-axis at the point (f, 0, 0), which is an equilibrium point of system (6), γ0 f is formed by two, three or five orbits of system (6) when f > 0, f = 0 and f < 0, respectively. The surface H2 = h2 > 0, is a cylinder whose projection√ in the plane x = 0 is a circle with center at the origin and with radius h2 . We want to study the intersection of H2 (x, y, z) = h2 with F (x, y, z) = f in the half-space √ x > 0, denoted √ by γh2 f . First, we note that γh2 f is empty if −f < − h2 . If −f ≥ − h2 , the curve γh2 f is formed by an orbit of system (7) if h2 > 0, or by the equilibrium point (f, 0, 0) of system (7) if h2 = 0 (see the Figure 7). A crossing periodic orbit of system (2) with two points in x = 0 occurs only if an orbit of system (6) and an orbit of system (7) meet in the same two crossing points of x = 0. So a periodic orbit of system (2) obtained from the connection of an orbit of γh1 f with an orbit of γh2 f is only possible if (8) h1 > 0 and f < 0. Let (0, y0 , z0 ) be a point such that h1 = H1 (0, y0 , z0 ), h2 = H2 (0, y0 , z0 ) and f = F (0, y0 , z0 ) satisfying conditions (8). Considering the orbits of the systems (9) H1 = h1 = y0 2 − z0 2 , F = f = −y0 , x = 0, and (10) H2 = h2 = y0 2 + z0 2 , F = f = −y0 , x = 0, CROSSING PERIODIC ORBITS VIA FIRST INTEGRALS 7 we obtain the points of the curves γh1 f and γh2 f that are in the plane x = 0. The orbits satisfying both systems intersect the plane x = 0 at the points (0, y0 , ±z0 ) such that y0 2 − z0 2 > 0 and y0 > 0. Finally, z0 6= 0 because the crossing periodic orbit must intersect the plane x = 0 in two distinct points. So the “if” part of Theorem 1 is proved. Now we shall prove the sufficient conditions of Theorem 1. Consider two points (0, y0 , ±z0 ) which satisfy y0 2 − z0 2 > 0, y0 > 0 and z0 6= 0. Our objective is to prove that the orbits of systems (6) and (7) that intersect these points form a crossing periodic orbit of system (2). Taking the variable x as a parameter the orbit of system (6) that intersects the plane x = 0 at the points (0, y0 , ±z0 ) is the curve {H1 = h1 } ∩ {F = f } ∩ {x ≤ 0}, i.e. (11)  q   2 p 2 2 2 2 x, x + y0 , ± (x + y0 ) − (y0 − z0 ) ; −y0 + y0 − z0 ≤ x ≤ 0 in x ≤ 0. In a similar way the orbit {H2 = h2 } ∩ {F = f } ∩ {x ≥ 0} of system (7) which intersect the plane x = 0 in those two points is (12)  q   2 p x, x + y0 , ± (y0 2 + z0 2 ) − (x + y0 ) ; 0 ≤ x ≤ −y0 + y0 2 + z0 2 in x ≥ 0. The connection of the curve (11) in x ≤ 0 with the curve (12) in x ≥ 0 provides a crossing periodic orbit. This completes the proof of Theorem 1. Proof of Corollary 1. Solving systems (9) end (10) with respect to the variables y0 and z0 under the conditions H1 (0, y0 , z0 ) = 0, H2 (0, y0 , z0 ) = h2 and F (0, y0 , z0 ) = f with f < 0, we obtain the points (0, y0 , ±y0 ) with y0 > 0. The solution of system (6) with initial condition (0, y0 , y0 ) with y0 > 0 is   (x (t) , y (t) , z (t)) = −1 + et y0 , et y0 , et y0 , for t ∈ (−∞, 0]. We note that this orbit begins at the point (−y0 , 0, 0) which is equilibrium point of system (2), because all points in the x-axis are equilibrium points of that system. The solution of system (7) with initial condition (0, y0 , y0 ) with y0 > 0 is (x (t) , y (t) , z (t)) = y0 (cos t + sin t − 1, cos t + sin t, cos t − sin t) , for t ∈ [0, π/2] and we have that (x (π/2) , y (π/2) , z (π/2)) = (0, y0 , −y0 ) . Finally, the solution of system (6) with initial condition (0, y0 , −y0 ) with y0 > 0 is   (x (t) , y (t) , z (t)) = −1 + e−t y0 , e−t y0 , −e−t y0 , for t ∈ [0, +∞), and it is ends at the equilibrium point (−y0 , 0, 0) when t = ∞. 8 J. LLIBRE, D.J. TONON AND M.Q. VELTER In summary, for each y0 > 0 system (2) has a homoclinic orbit to the equilibrium point (−y0 , 0, 0) obtained from the connection of the three pieces of orbits described above.  3. Proof of Theorem 2 Following the approach used to prove Theorem 1 we go to study the discon- tinuous piecewise differential system formed by the two differential systems (13) ẋ = (x − y) z, ẏ = z, ż = − y, for x > 0, and (14) ẋ = (x − y) z, ẏ = z, ż = y, for x < 0, by analyzing the intersections of the level surfaces of the two in- dependent first integrals of each system. From (5) the two independent first integrals of system (13) are H1 (x, y, z) = y 2 − z 2 and G (x, y, z) = e−y (1 − x + y) , and the two independent first integrals of system (14) are H2 (x, y, z) = y 2 + z 2 and G (x, y, z) = e−y (1 − x + y). Since G is a first integral for both systems, first we go to study the behavior of the surfaces G = g for all g ∈ R. As G does not depended on z, for studying these surfaces it is sufficient to analyze the curve G = g |z=0 , denoted by γ (g) . We have that the curve γ (g) is defined by the equation fg (y) = x with fg (y) = 1 + y − g ey . This curve coincides with the graphic (fg (y) , y) of the function fg : R → R. For g = 0 the curve γ (g) is a straight line which pass through the points (1, 0) and (0, −1) , and the surface G = 0 is the plane 1 − x + y = 0. When g > 0, we define y0 = log (1/g) , and y0 is a maximum of the function fg , i.e. fg (y0 ) = log (1/g) ≥ fg (y) for all y ∈ R. Moreover, we observe that for g = 1, fg (y0 ) = y0 = 0. So the graphic of fg intersects the y-axis exactly at the point (0, 0), that is γ (g) is contained in the half-space x ≤ 0. Additionally if g > 1 then fg (y0 ) = y0 < 0. In this case the graphic of the function fg does not intersect the y-axis. Although for 0 < g < 1, fg (y0 ) = y0 > 0, i.e. the graphic of the function fg intersects the y-axis at two points, denoted by α (g) and β (g), where −1 < α (g) < 0 and β (g) > 0. We observed that when y → −∞ the graphic of the function fg tends to the straight line x − y − 1 = 0 for any g ∈ R, and when y → +∞ we have that fg (y) → +∞ if g ≤ 0, else if g > 0 then fg (y) → −∞. So if g ≤ 0 and CROSSING PERIODIC ORBITS VIA FIRST INTEGRALS 9 since for all y ∈ R we have that fg0 (y) > 0, the graphic of the fg intersects the y-axis in a unique point, denoted by ρ (g) ≤ −1. We note that ρ (0) = −1. In summary we have that the system (15) e−y (1 − x + y) = g, x = 0, has in R2 only the solution (0, ρ (g)) if g ≤ 0; two solutions (0, α (g)) and (0, β (g)) if 0 < g < 1; the trivial solution (0, 0) if g = 1; and it has no solutions if g > 1. Varying z we shall describe the behavior of the surface G = g for all g ∈ R. First we have that the solutions of system (15) considered in R3 are the straight line y = ρ (g) if g ≤ 0, the straight lines y = α (g) and y = β (g) if 0 < g < 1, and the straight line y = 0 if g = 1. Now we must analyze the intersection of the surfaces H1 (x, y, z) = h1 and G (x, y, z) = g in the half-space x > 0, denoted by γh1 g . Using the above analysis and taking the projections of these surfaces onto the plane√x = 0, it is easy to check√that γh1 g is empty when either g ≥ 1, or h1 > 0, or − h1 < α (g) and β (g) < h1 . When γh1 g is not empty it is formed by one or two orbits of system (13) if h1 6= 0. More precisely, γh1 g is one orbit of the system contained in the half-space x >√0, i.e. which does not intersect the plane x = 0 if h1 > 0 and ρ (g) √= − h1 ; and γh√ 1 g is also one orbit of the system when h1 > 0, α (g) = − h1and β (g) > h1 , but in this case it intersects the plane x = 0 q at the points 0, β (g) , ± β (g)2 − h1 . Otherwise γh1 g contains two orbits of system (13). However, when √ (a) h1 > 0 and ρ (g) < − h1 one of the orbits of the system is contained in the half-space  x > 0 and q the other orbit  intersects the plane x = 0 at the points 0, ρ (g) , ± ρ (g)2 − h1 ; √ √ (b) h1 > 0, α (g) < − h1 and β (g) > h1 both  orbits intersect the plane  q 2 x = 0 in two points; one orbit at the points 0, α (g) , ± α (g) − h1 ,  q  2 and the other orbit at the points 0, β (g) , ± β (g) − h1 (see Fig- ure 8); 10 J. LLIBRE, D.J. TONON AND M.Q. VELTER Figure 8. Intersection of Figure 9. Intersection of the surfaces H1 = 0.05 and the surfaces H1 = −0.05 and G = 0.8 in the half-space G = 0.7 in the half-space x > 0. x > 0. Figure 10. Intersection of the surfaces H2 = 2.5 and G = 0.8 in the half-space x < 0. (c) h1 < 0 and g ≤ 0 both orbits  intersect the planein only one point: an q orbit does it at the point 0, ρ (g) , ρ (g)2 − h1 , and the other orbit  q  2 does it at the point 0, ρ (g) , − ρ (g) − h1 ; (d) h1 < 0 and 0 < g < 1 one of the orbits is contained in the positive sheet of the hyperbolic cylinder and it intersects the plane x = 0 at the points CROSSING PERIODIC ORBITS VIA FIRST INTEGRALS 11  q   q  2 2 0, α (g) , α (g) − h1 and 0, β (g) , β (g) − h1 . The other or- bit is contained in the negative sheet of the hyperbolic  cylinder q and this orbit intersects the plane x = 0 at the points 0, α (g) , − α (g)2 − h1  q  2 and 0, β (g) , − β (g) − h1 , as shown in Figure 9. When h1 = 0 the curve γh1 g is formed by five orbits of system (13) for any g because the surface G = g intersects the x-axis at the point (1 − g, 0, 0) which is an equilibrium point of system (4). Moreover, when g ≤ 0 two of these orbits intersect the plane x = 0: one orbit at the point (0, ρ (g) , ρ (g)) and the other at the point (0, ρ (g) , −ρ (g)); when 0 < g < 1 each orbit intersects the plane x = 0 at one point, namely: (0, α (g) , −α (g)), (0, α (g) , α (g)), (0, β (g) , −β (g)) and (0, β (g) , −β (g)) . On the other hand using again the analysis of the function fg and looking at the projections of the surfaces H2 (x, y, z) = h2 and G (x, y, z) = g onto the plane x = 0 we have √ that γh2 g = √ {H2 = h2 } ∩ {G = g} ∩ {x < 0} is not empty when |ρ (g)| ≤ h2 or |α (g)| ≤ h2 . So under these assumptions γh2 g is formed by one orbitof system (14) if g ≤ 0 and  it intersects the discontinuous q plane at the points 0, ρ (g) , ± h2 − ρ (g)2 , or the curve γh2 g is formed by two orbits of system (14) if 0 < g < 1 and each orbit has two points in the q 2 plane x = 0: one orbit at the points 0, α (g) , ± h2 − α (g) and the other  q  2 orbit at the point 0, β (g) , ± h2 − β (g) . For g > 1, the curve γh2 g is a closed orbit contained in a half-space x < 0 (see Figure 10). In order that system (4) has a crossing periodic orbit it is necessary that the orbits of system (13) connect with the orbits of system (14), i.e. the orbits for both systems must intersect the discontinuous plane x = 0 at the same points. This is only possible if either (16) h1 > 0, h2 > 0 and g < 1, or (17) h1 < 0, h2 > 0, and 0 < g < 1. This means that taking a point (0, y0 , z0 ) in the plane x = 0 such that H1 (0, y0 , z0 ) = h1 , H2 (0, y0 , z0 ) = h2 and G (0, y0 , z0 ) = g satisfying condi- tions (16) and considering the solutions of the systems (18) H1 = h1 = y0 2 − z0 2 , G = g = e−y0 (1 + y0 ) , x = 0, 12 J. LLIBRE, D.J. TONON AND M.Q. VELTER and (19) H2 = h2 = y0 2 + z0 2 , G = g = e−y0 (1 + y0 ) , x = 0, we obtain the connection points of the curves γh1 g and γh2 g , that they will be two points (0, y0 , ±z0 ) such that y0 2 − z0 2 > 0 and where y0 is a solution of the equation 1 + y − g ey = 0 if g ≤ 1, or they will be four points (0, y0 , ±z0 ) and (0, y1 , ±z1 ) satisfying the same conditions. That is in this second case the crossing periodic orbits appear in pairs. Let (0, y0 , z0 ) be a point of the plane x = 0 such that H1 (0, y0 , z0 ) = h1 , H2 (0, y0 , z0 ) = h2 and G (0, y0 , z0 ) = g satisfying conditions (17), then the solutions of systems (18) and (19) are of the form (0, y0 , ±z0 ) and (0, y1 , ±z1 ) with yi 2 − zi 2 < 0 and being yi a solution of the equation 1 + y − g ey = 0, i = 0, 1. Again z0 6= 0 because the crossing periodic orbit intersects the discontinu- ous plane at least in two distinct points and this completes the proof of the necessary conditions of statements (i) and (ii) of Theorem 2. Now we present the proof of the sufficient conditions of these statements. Consider the two points (0, y0 , ±z0 ) such that z0 6= 0, y0 2 − z0 2 > 0 and where y0 is a solution of the equation 1 + y − g ey = 0 for some g < 1. We will show that the orbits of systems (13) and (14) that intersect these points form a crossing periodic orbit of system (4). Taking the variable y as a parameter we have that the orbits of system (13) and (14) that intersect the plane x = 0 at the points (0, y0 , ±z0 ) are respectively, the curves n p  p o y 2 2 2 2 2 (20) 1 + y − g e , y, ± (y0 + z0 ) − y ; − y0 + z0 ≤ y ≤ y0 in x ≤ 0 and n p  p o (21) 1 + y − g ey , y, ± y 2 − (y0 2 − z0 2 ) ; y0 ≤ y ≤ − y0 2 − z0 2 in x ≥ 0, if y0 is the negative solution of the equation 1 + y − g ey = 0, or are the curves n p  p o (22) 1 + y − g ey , y, ± y 2 − (y0 2 − z0 2 ) ; y0 2 − z0 2 ≤ y ≤ y0 in x ≥ 0 and n p  p o (23) 1 + y − g ey , y, ± (y0 2 + z0 2 ) − y 2 ; y0 ≤ y ≤ y0 2 + z0 2 in x ≤ 0, if y0 is the positive solution of the equation 1 + y − g ey = 0. The connection of the orbit (20) in x ≤ 0 with the orbit (21) in x ≥ 0 pro- vides a crossing periodic orbit of system (4) which intersects the discontinuous plane in two points. The same occurs for the connection of the orbit (22) in CROSSING PERIODIC ORBITS VIA FIRST INTEGRALS 13 x ≥ 0 with the orbit (23) in x ≤ 0 and this completes the proof of statement (i) of Theorem 2. Finally consider the points (0, y0 , ±z0 ) and (0, y1 , ±z1 ) satisfying the as- sumptions of statement (ii) of Theorem 2. Now the orbits of system (14) that intersect the plane x = 0 in these points are two disjoint curves of the form n p  o (24) 1 + y − g ey , y, y 2 − (y0 2 − z0 2 ) ; y0 ≤ y ≤ y1 and n p  o y 2 2 2 (25) 1 + y − g e , y, − y − (y0 − z0 ) ; y0 ≤ y ≤ y1 in x ≥ 0. We observed that the orbit (24) stars at the point (0, y0 , z0 ) and ends at point (0, y1 , z1 ), and the orbit (25) stars at the point (0, y0 , −z0 ) and ends at point (0, y1 , −z1 ) . Therefore, connecting the orbits (20), (24), (23) and (25) we obtained a crossing periodic orbit for system (4), see figure 5. This completes the proof of Theorem 2. Proof of Corollary 2. First we observed that all points in the x-axis are equi- librium points of system (4). According with the description of possible inter- sections of the surfaces H1 (x, y, z) = h1 and G (x, y, z) = g made previously, the existence of homoclinic orbits for system (4) is only possible if h1 = 0. Let (0, y0 , y0 ) be a point such that y0 is a negative solution of the g = e−y (1 + y) for some g < 1. Again, taking the variable y as a parameter, we have that the orbits of system (14) that intersect the plane x = 0 at the points (0, y0 , y0 ) are respectively, the curves   (26) 1 + y − (1 + y0 ) e(y−y0 ) , y, ± y ; y0 ≤ y ≤ 0 . We observe that when y = 0, the two curves meet at the point (1−e−y0 (1 + y0 ), 0, 0) which is an equilibrium point of system (4). The same occurs when y0 is the positive solution of the equation g = e−y (1 + y) for some g < 1, but the curves which describes the orbits of system (14) in this case are   (27) 1 + y − (1 + y0 ) e(y−y0 ) , y, ± y ; 0 ≤ y ≤ y0 . Connecting the orbit (20) in x ≤ 0 with the orbits (26) in x ≥ 0 and connecting the orbit (22) in x ≤ 0 with the orbits (27) in x ≥ 0 we have two homoclinic orbits to the equilibrium point (1 − e−y0 (1 + y0 ) , 0, 0) .  Acknowledgements The first author is partially supported by the Ministerio de Ciencia, Inno- vación y Universidades, Agencia Estatal de Investigación grants MTM2016- 77278-P (FEDER), the Agència de Gestió d’Ajuts Universitaris i de Recerca 14 J. LLIBRE, D.J. TONON AND M.Q. VELTER grant 2017SGR1617, and the H2020 European Research Council grant MSCA- RISE-2017-777911. The second author is partially supported by CNPq-Brazil grant 443302/2014-6, PROCAD/CAPES grant 88881.0 68462/2014-01 and PRONEX/FAPEG grant 2017/10267000-508. The third author is partially supported by the Coordenação Aperfeiçoamento de Pessoal de Nı́vel Superior - Brasil (CAPES) - Finance Code 001. References [1] A. Andronov, A. Vitt and S. Khaikin, Theory of Oscillations, Pergamon Press, Oxford, 1966. 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Teruel, Limit cycles for planar piecewise linear differential systems via first integrals, Qual. Theory Dyn. Syst. 3 (2002), 29–50. [8] J. Llibre and E. Ponce, Three nested limit cycles in discontinuous piecewise linear differential systems with two zones, Dyn. Contin. Discrete Impuls Syst. Ser. B 19 (2012), 325–335. [9] J. Llibre and M.A. Teixeira, Periodic orbits of continuous and discontinuous piece- wise linear differential systems via first integrals, São Paulo J. Math. Sci. 12 (2018), 121–135. [10] D.J.W. Simpson, Bifurcations in Piecewise-Smooth Continuous Systems, World Sci- entific Series on Nonlinear Science A, vol 69, World Scientific, Singapore, 2010. 1 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain. Email address: jllibre@mat.uab.cat 2 Institute of Mathematics and Statistics of Federal University of Goiás, Avenida Esperança s/n, Campus Samambaia, CEP 74690-900, Goiânia, Goiás, Brazil. Email address: djtonon@ufg.br CROSSING PERIODIC ORBITS VIA FIRST INTEGRALS 15 3 Institute of Mathematics and Statistics of Federal University of Goiás, Avenida Esperança s/n, Campus Samambaia, CEP 74690-900, Goiânia, Goiás, Brazil. Email address: marianaqueirozvelter@gmail.com