This is a preprint of: “The Markus–Yamabe conjecture does not hold for piecewise discontinuous linear differential systems separated by one straight line”, Jaume Llibre, Lucyjane de A. S. Menezes, J. Dyn. Diff. Equat., 2020. DOI: [10.1007/s10884-020-09825-8] THE MARKUS–YAMABE CONJECTURE DOES NOT HOLD FOR PIECEWISE DISCONTINUOUS LINEAR DIFFERENTIAL SYSTEMS SEPARATED BY ONE STRAIGHT LINE JAUME LLIBRE1 , LUCYJANE DE A. S. MENEZES2 Abstract. The Markus–Yamabe conjecture is a conjecture on global as- ymptotic stability. The conjecture states that if a differentiable system ẋ = f (x) has a singularity and the Jacobian matrix Df (x) has everywhere eigenvalues with negative real part, then the singularity is a global attrac- tor. In this paper we consider piecewise discontinuous linear differential systems in R2 separated by one straight line Σ such that the unique sin- gularity of the system is at Σ and the Jacobian matrix of the system has everywhere eigenvalues with negative real part. We prove that these piece- wise discontinuous linear differential systems can have one crossing limit cycle and consequently these systems do not satisfy the Markus–Yamabe conjecture. 1. Introduction and statement of the main results Consider f (x) a C 1 map on an n-dimensional real vector space. Let (1) ẋ = f (x) be a differential system such that f (0) = 0. In 1960 Markus and Yamabe stated that if all eigenvalues of Df (x) have negative real part, then the origin of (1) is a global attractor. In their paper [8] the conjecture has been proved under some strong additional hypotheses. This statement became known as the Markus–Yamabe conjecture. Many authors have dedicated in proving the Markus–Yamabe conjecture. The conjecture is true for C 1 two-dimensional systems. In 1988 Meisters and Olech proved the Markus–Yamabe conjecture for polynomial vector fields in the plane, see [9]. Considering vector fields of class C 1 defined in R2 , Gutierrez [6], Fleber [3] and Glutsyuk [5] for this order provided different proofs of the Markus–Yamabe conjecture in the years 1994–5. However counterexamples 2010 Mathematics Subject Classification. Primary 34C05, 34C07, 37G15. Key words and phrases. discontinuous differential system, limit cycle, Markus–Yamabe conjecture. 1 http://www.gsd.uab.cat 2 J. LLIBRE AND L.A.S. MENEZES have been constructed in higher dimension. Bernat and Llibre [1] in 1996 presented a counterexample to the conjecture in dimension larger than 3. In the paper [2] of 1997, Cima et al. provided a counterexample for the case n = 3. More precisely, they proved that the Markus–Yamabe conjecture is false for polynomial vector fields in Rn with n ≥ 3. We define a piecewise discontinuous linear Markus–Yamabe differential sys- tem, or simply piecewise Markus–Yamabe system as   X(z) if x > 0, (2) ż =  Y (z) if x < 0, where z = (x, y) ∈ R2 , X and Y are linear vector fields, the real part of the eigenvalues of DX(z) and DY (z) are negative, Y (0) = 0, and the singularity of X is virtual, i.e. it leaves in the half–plane x < 0. The straight line Σ = {(x, y) ∈ R2 ; x = 0} is called the discontinuity set. Here ż denotes the derivative of z with respect to the independent variable t, usually called the time. In order to simplify the notation we denote the piecewise Markus–Yamabe system as Z = (X, Y ) and call it simply as piecewise MY–system. Observe that X and Y are linear differential systems whose singularities are of type foci (F) or nodes (N). Furthermore, the nodes can diagonalize with distinct eigenvalues (N ), or with equal eigenvalues (N ∗ ), or do not diagonalize (iN ). We say that Z is a piecewise MY–system of type LR, with L, R ∈ {F, N, N ∗ , iN }, when Y has a singularity of type L and X has a singularity of type R. The extension of the conjecture of Markus–Yamabe to piecewise MY–system claims: The origin of any piecewise MY–system is a global attractor. Our main goal is to prove that this conjecture does not hold for piecewise MY–systems. According to Filippov conventions [4] the discontinuity set Σ of the piecewise MY–system is decomposed in escape, sliding and sewing regions. We define a crossing limit cycle as a limit cycle that is concatenation of two orbits one of the vector field X and the other of the vector field Y which connect in two sewing points. In what follows a crossing limit cycle will be called simply limit cycle. In order to see that the Markus–Yamabe conjecture does not hold for piecewise MY–system, we should characterize what piecewise MY–systems can have limit cycles. This characterization is done in the next result. Theorem 1. A piecewise MY–system of types F F , F N and F iN have at most one limit cycle, and there are systems of these types with exactly one THE MARKUS–YAMABE CONJECTURE FOR DISCONTINUOUS SYSTEMS 3 limit cycle. The others piecewise MY–systems different from these three types have no limit cycles. The rest of the paper is organized as follows. In Section 2 we provide some basic notions and results that we shall need for proving Theorem 1. In the short section 3 we prove the last part of Theorem 1, i.e. that the piecewise MY– systems of type F N ∗ , N N ∗ , iN N , and iN N ∗ have no limit cycles. Finally in the long section 4 we prove the first part of Theorem 1, which we have divided it in three subsections one for each piecewise MY–system of type F F , F N and F iN . 2. Preliminary results In this section we present the essential definitions and results that we need in this paper. Consider X and Y linear vector fields and   X(z) if x > 0, (3) ż =  Y (z) if x < 0, where z = (x, y) ∈ R2 , a piecewise linear differential system whose discontinu- ity is the straight line Σ = {(x, y) ∈ R2 ; x = 0}. We define the following singularities to system (3). • Real singularity: p = (x, y) such that x > 0 and X(p) = 0, or x < 0 and Y (p) = 0. • Virtual singularity: p = (x, y) such that x > 0 and Y (p) = 0, or x < 0 and X(p) = 0. • Boundary singularity: p ∈ Σ such that X(p) = 0 or Y (p) = 0. We say that a point (0, y) is an invisible fold point for the vector field X = (X1 , X2 ) when ∂X1 X1 (0, y) = 0 and (0, y)X2 (0, y) < 0; ∂y and an invisible fold point for the vector field Y = (Y1 , Y2 ) when ∂Y1 Y1 (0, y) = 0 and (0, y)Y2 (0, y) > 0. ∂y A T –system is a linear differential system having a singularity of type T with T ∈ {F, N, N ∗ , iN }. Next result provides a simpler way to rewrite a T –system in the plane and its proof can be found in Proposition 5 of [7]. 4 J. LLIBRE AND L.A.S. MENEZES Proposition 2. Let M = (mij ) be a 2 × 2 matrix. If the linear differential system (4) (ẋ, ẏ)T = M (x, y)T is a (a) N or N ∗ –system then after a vertical lines–preserving linear change of variables and a time–rescaling system (4) becomes (ẋ, ẏ)T = M1 (x, y)T ; (b) F –system then after a vertical lines–preserving linear change of vari- ables and a time–rescaling system (4) becomes (ẋ, ẏ)T = M2 (x, y)T with a 6= 0; (c) iN –system then after a vertical lines–preserving linear change of vari- ables and a time–rescaling system (4) becomes (ẋ, ẏ)T = M3 (x, y)T , where     a 1 a 1 M1 =   with |a| > 1; M2 =   with a ∈ R; 1 a −1 a   λ λ and M3 =   with λ = ±1. 0 λ The next result is the Lemma 6 of [7]. Lemma 3. We consider the functions F (t) = e−at csc t − cot t, G(t) = e−at cscht − coth t, (5) e−t − 1 H(t) = . t The following statements hold. (a) For every a < 0, the function F (t) is monotonic increasing in the interval (0, π) and F (t) > −a for t ∈ (0, π). (b) For a < −1, the function G(t) is monotonic increasing on R and G(t) > −a for t > 0. (c) The function H(t) is monotonic increasing on R and H(t) < −1 for t < 0. 3. The MY–systems of type F N ∗ , N N ∗ , iN N and iN N ∗ Observe that, if the piecewise MY–system have one virtual singularity of type N ∗ , then the solution passing through the point (0, y) ∈ Σ cannot return THE MARKUS–YAMABE CONJECTURE FOR DISCONTINUOUS SYSTEMS 5 to Σ because the eigenvalues of the Jacobian matrix are equal, that is the orbits of the vector field Y leave in straight lines. If the piecewise MY–system have one boundary singularity of type N ∗ , N or iN , then the orbits of the vector field X passing through (0, y) ∈ Σ coincides with Σ or returns to Σ at the singularity. Therefore the first return map is not defined. Consequently the piecewise MY–system of type F N ∗ , N N ∗ , iN N and iN N ∗ do not have limit cycles. This proves the last part of Theorem 1. 4. Limit cycles for Piecewise MY–systems Consider a piecewise discontinuous linear differential system defined in R2   X(z) if x > 0, (6) ż =  Y (z) if x < 0, where z = (x, y). We denote the solution of (6) by   ϕ+ (t, x̄, ȳ) if x̄ > 0, ϕ(t, x̄, ȳ) =  ϕ− (t, x̄, ȳ) if x̄ < 0,  satisfying ϕ(0, x̄, ȳ) = (x̄, ȳ), where ϕ+ (t, x̄, ȳ) = ϕ+ + 1 (t, x̄, ȳ), ϕ2 (t, x̄, ȳ) and ϕ− (t, x̄, ȳ) = ϕ− − 1 (t, x̄, ȳ), ϕ2 (t, x̄, ȳ) are the solutions of vector fields X and Y , respectively. Set t+ (ȳ) > 0 the smallest positive time such that ϕ+ (t+ (ȳ), 0, ȳ) ∈ Σ, and t+ (ȳ) < 0 be the biggest negative time such that ϕ+ (t+ (ȳ), 0, ȳ) ∈ Σ. Analo- gously, let t− (ȳ) < 0 be the biggest negative time for which ϕ− (t− (ȳ), 0, ȳ) ∈ Σ, and t− (ȳ) > 0 be the smallest positive time such that ϕ− (t− (ȳ), 0, ȳ) ∈ Σ. Observe that the functions t+ (ȳ), t+ (ȳ), t− (ȳ), and t− (ȳ) are not necessarily always defined. Assume that t+ (ȳ) > 0 and t− (ȳ) < 0 are defined. There exists a limit cycle passing through the point (0, ȳ) with ȳ ∈ J ∗ = Dom(t+ ) ∩ Dom(t− ) if and only − − if ϕ+ + 2 (t (ȳ), 0, ȳ) = ϕ2 (t (ȳ), 0, ȳ). Thus, in this case the limit cycles are in correspondence with the zeros y ∗ of the function − − (7) f (ȳ) = ϕ+ + 2 (t (ȳ), 0, ȳ) − ϕ2 (t (ȳ), 0, ȳ), on the domain J ∗ . Equivalently, if t+ (ȳ) < 0 and t− (ȳ) > 0 are defined then there exists a limit cycle passing through the point (0, ȳ) with ȳ ∈ J∗ = Dom(t+ ) ∩ Dom(t− ) if 6 J. LLIBRE AND L.A.S. MENEZES − and only if ϕ+ 2 (t+ (ȳ), 0, ȳ) = ϕ2 (t− (ȳ), 0, ȳ). Thus, in this case we must study the zeros y∗ of the function − (8) f (ȳ) = ϕ+ 2 (t+ (ȳ), 0, ȳ) − ϕ2 (t− (ȳ), 0, ȳ), on the domain J∗ . Since the vector fields X and Y are linear, then a limit cycle passing through the point (x0 , y0 ) must contain points of the form (0, y ∗ ) and (0, y∗ ) such that y ∗ ∈ J ∗ and y∗ ∈ J∗ . Therefore detecting all the zeros of (7) or (8) we must detect all the limit cycles. In [7] the authors proved that the discontinuous piecewise linear differential systems of type F F , F N and F iN have at most two limit cycles. Here as- suming that these systems are piecewise MY–systems we shall prove that they have at most one limit cycle. 4.1. Piecewise MY–systems of type F F . In this subsection we prove that a piecewise MY–system having at the origin a boundary stable focus of Y and having X a virtual stable focus have at most one limit cycle. Assume that system (6) have a singularity of type boundary focus for x < 0 and a virtual focus for x > 0. By Proposition 2 the matrix corresponding to the linear vector field X of (6) can be transformed into M2 and the associated matrix of the linear vector field Y is transformed is a general matrix B = (bij ). Then we can rewrite system (6) as         a 1   x + u1       if x > 0,   −1 a y + u ẋ 2 (9)  =    ẏ     b b x    11 12    if x < 0,   b21 b22 y + v2 where 4b12 b21 + (b11 − b22 )2 < 0 because we have a focus for the vector field Y . Furthermore u1 > 0 because the focus of the vector field X is virtual. Since (9) is a piecewise discontinuous linear differential system, we can com- pute its solution. Thus the solution ϕ+ (t, x̄, ȳ) of (9) for  x̄ > 0 such that ϕ+ (0, x̄, ȳ) = (x̄, ȳ) is ϕ+ (t, x̄, ȳ) = ϕ+ 1 (t, x̄, ȳ), ϕ+ 2 (t, x̄, ȳ) where ϕ+ at 1 (t, x̄, ȳ) = e ((u1 + x̄) cos t + (u2 + ȳ) sin t) − u1 , ϕ+ at 2 (t, x̄, ȳ) = e ((u2 + ȳ) cos t − (u1 + x̄) sin t) − u2 , and the solution ϕ− (t, x̄, ȳ) of (9), for x̄ < 0, such that ϕ− (0, x̄, ȳ) = (x̄, ȳ) is given by ϕ− (t, x̄, ȳ) = ϕ− − 1 (t, x̄, ȳ), ϕ2 (t, x̄, ȳ) , where THE MARKUS–YAMABE CONJECTURE FOR DISCONTINUOUS SYSTEMS 7    Γt ((b11 − b22 )x̄ + 2b12 (ȳ + v2 )) ϕ− 1 (t, x̄, ȳ) = sin 2 Γ (10)   (b11 + b22 )t Γt +x̄ cos e 2 , 2    ((b22 − b11 )(v2 + ȳ) + 2b21 x̄) Γt ϕ− 2 (t, x̄, ȳ) = sin 2 Γ (11)   (b11 + b22 )t Γt +(v2 + ȳ) cos e 2 − v2 , 2 p with Γ = −(b11 − b22 )2 − 4b12 b21 . We have the following result. The first part of its proof is in the proof of Proposition 11 of [7]. Theorem 4. Assume that the eigenvalues of the vector fields X and Y have negative real parts, then the piecewise discontinuous linear differential system (9) has at most one limit cycle. Proof. In order to fix the clockwise orientation of the flow of system (9) we assume that Y1 (0, 1 − v2 ) = b12 > 0. Observe that the orbits of vector field Y expends a time t− (ȳ) = −2π/Γ in the half–plane Σ− if ȳ > −v2 or t− (ȳ) = 2π/Γ if ȳ < −v2 , where the point (0, −v2 ) is the singularity. Therefore the functions t− (ȳ) < 0 and t− (ȳ) > 0 are define for ȳ > −v2 and ȳ < −v2 , respectively. Doing a convenient translation in the plane so that it preserves the straight line Σ and the half–plane x > 0, we can take u2 = 0. This implies that the singularity of X is (−u1 , 0). Furthermore the point (0, −a u1 ) ∈ Σ is an invisible fold point for the vector field X. It follows that the function t+ (ȳ) > 0 is defined for every ȳ > −a u1 (see Figure 1). ȳ Consider ȳ > −a u1 and y = x + ȳ the straight line L passing through the u1 points (−u1 , 0) and (0, ȳ). Then (−u1 (eaπ + 1), −ȳeaπ ) = ϕ+ (π, 0, ȳ) ∈ L, and this implies that ϕ+ (t, 0, ȳ) ∈ Σ for some t < π. Thus t+ (ȳ) ∈ (0, π) for every ȳ > −a u1 . 8 J. LLIBRE AND L.A.S. MENEZES y L ȳ −a u1 −u1 x Figure 1. Virtual focus for the vector field X. The shaded half-line is the domain of definition of the function t+ (ȳ) > 0. By definition of t+ (ȳ), we have that ϕ+ + 1 (t (ȳ), 0, ȳ) = 0 for every ȳ > −a u1 , that is +  eat (ȳ) u1 cos t+ (ȳ) + ȳ sin t+ (ȳ) − u1 = 0. Then taking y + (t) = u1 F (t), with F defined in (5), for t ∈ (0, π) we have that y + (t+ (ȳ)) = ȳ for every ȳ > −a u1 . Furthermore, by Lemma 3 we have that y + (t) > −au1 is an injective function on the interval (0, π). It follows that given t0 ∈ (0, π) and y0 = y + (t0 ) we obtain y + (t+ (y0 )) = y0 . Therefore y + (t+ (y0 )) = y + (t0 ) and this implies that t+ (y + (t0 )) = t+ (y0 ) = t0 . Since t0 was arbitrarily chosen, we conclude that t+ (y + (t)) = t for every t ∈ (0, π). Therefore the function t+ : (−a u1 , ∞) → (0, π) is invertible with inverse equal to y + : (0, π) → (−a u1 , ∞). In this case we have that the function (7) is given by (b11 + b22 )π − +  (12) f (ȳ) = v2 + e Γ (v2 + ȳ) + ea t (ȳ) ȳ cos t+ (ȳ) − u1 sin t+ (ȳ) . THE MARKUS–YAMABE CONJECTURE FOR DISCONTINUOUS SYSTEMS 9 Thus, defining yM = max{−a u1 , −v2 } we obtain that the functions t+ (ȳ) and t− (ȳ) are defined for every ȳ > yM . We conclude that computing the zeros of the function (12) for ȳ > yM is equivalent to compute the zeros of the function g1 (t) = f (y + (t)) given by (13) g1 (t) = v2 (1 + δ) + u1 (cot t − eat csc t) − δu1 (cot t − e−at csc t) , for t ∈ I = t+ ((YM , ∞)) ⊂ (0, π) and δ = e−(b11 +b22 )π/Γ . Since the eigenvalues of Y are negative we have that δ > 1. Moreover I = (0, π) provided v2 ≥ au1 . We will prove that the function (13) has at most one zero for t ∈ (0, π). Notice that, for t ∈ (0, π), g1 (t) = 0 is equivalent to (14) A sin t + (1 − δ) cos t − eat + δe−at = 0, v2 (1 + δ) where A = . Defining h(t) = (1 − δ) cos t − eat + δe−at , we get u1 h0 (t) = −(1 − δ) sin t − aeat − aδe−at > 0, for t ∈ (0, π), because δ > 1 and a < 0. This implies that h(t) is increasing in (0, π). Since h(0) = 0, we have that h(t) > 0 in (0, π). Therefore if A > 0 equation (14) does not have solution in (0, π) and, consequently the function (13) does not have zeros. In what follows we assume A < 0. Consider the functions h1 (t) = −A sin t − (1 − δ) cos t, (15) h2 (t) = −eat + δe−at , and note that equation (14) holds if and only if h1 (t) = h2 (t). We will prove that there exists at most one t̄ ∈ (0, π) such that h1 (t̄) = h2 (t̄). Observe that h1(0) = δ− 1, h1 (π/2) = −A, h1 (π) = 1 − δ, and h1 (t0 ) = 0, δ−1 where t0 = arctan . Furthermore, A h01 (t) = −A cos t + (1 − δ) sin t, h001 (t) = A sin t + (1 − δ) cos t, where h01 (t) > 0 (respectively <) for every t < t1 = arctan (A/(1 − δ)) (re- spectively >) andh001 (t)< 0 ( respectively  π  >) for every t < t0 (respectively >). π Notice that t0 ∈ , π and t1 ∈ 0, . It follows that the graph of h1 (t) is 2 2 given in Figure 2–(a). 10 J. LLIBRE AND L.A.S. MENEZES (a) (b) δ−1 t1 t0 π δ−1 π 1−δ Figure 2. (a) Graph of the function h1 (t); (b) Graph of the function h2 (t). Since h02 (t) > 0 for t ∈ (0, π) we have h2 (t) is increasing in t ∈ (0, π), and h2 (0) = δ − 1 > 0. Furthermore, h002 (t) = −a2 eat + a2 δe−at = a2 eat (−1 + δe−2at ) > 0 for t ∈ (0, π). The graph of the function h2 (t) is given in Figure 2–(b). Analyzing the graphs of the functions h1 (t) and h2 (t) and taking into account their convexities, we conclude that if h02 (0) < h01 (0), then there exists a unique t̄ ∈ (0, t0 ) ⊂ (0, π) such that h1 (t̄) = h2 (t̄). If h02 (0) ≥ h01 (0) then h2 (t) > h1 (t) for every t ∈ (0, π). Therefore equation (14) has at most one zero in (0, π).  Corollary 5. A piecewise MY–system of type F F has at most one limit cycle. Proof. The result follows applying Theorem 4 to piecewise MY–systems whose singularities for the vector fields X and Y are a virtual stable focus and a boundary stable focus at the origin, respectively.  The following is an example of a piecewise MY–system of type F F having one limit cycle. With this example Theorem 1 is proved for a piecewise MY– systems of type F F . THE MARKUS–YAMABE CONJECTURE FOR DISCONTINUOUS SYSTEMS 11 Example 6. Consider the following piecewise discontinuous linear differential system      x 1 y 1  X(z) = y − − , −x − − for x > 0,  2 10 2 5  (16) ż =   π2x  Y (z) = y − x − 1, − −y+1 for x < 0. (ln 2)2 Observe that the point (−1/5, 0) is a virtual focus of X, and that the point (0, 1) 1 is a boundary focus of Y . Furthermore the eigenvalues of (16) are λ+ 1,2 = − ±i 2 − π for the point (−1/5, 0), and λ1,2 = −1 ± i, for the point (0, 1). Thus, by ln 2 Theorem 4 system (16) has at most one limit cycle. In this case we have that yM = max{1/10, 1} = 1, y + (t) = (et/2 csc t − cot t)/5, and the function defined in (13) is 1 2et/2 e−t/2 g1 (t) = −3 − cot t + csc t − csc t. 5 5 5 Thus in the interval (0, π) we have that g1 (t) = 0 if and only if t ≈ 2.588... and, therefore system (16) has one limit cycle passing thorough y + (2.588...) ≈ 1.714... > yM . u v Figure 3. The phase portrait of the piecewise MY–system (17). 12 J. LLIBRE AND L.A.S. MENEZES Changing the coordinates to u = x and v = y − 1, we rewrite the system (16) as      u 9 v 7  v − + , −u − − for u > 0,  2 10 2 10 (17) ż =   π2u  v − u, − −v for u < 0, (ln 2)2 whose boundary focus is at (0, 0) and the virtual focus is at (−1/5, −1). Ob- serve that system (17) is a piecewise MY–system and it has one limit cycle passing through point (u, v) = (0, 0.714...), see Figure 3. 4.2. Piecewise MY–systems of type F N . In this subsection we prove that a piecewise MY–system having at the origin a boundary stable focus of Y and having X a virtual stable node N have at most one limit cycle. Suppose that the piecewise discontinuous linear differential system (6) has a virtual node N for x > 0 and a boundary focus for x < 0. By Proposition 2 the matrix of the linear vector field X can be transformed into M1 and the matrix of the vector field Y is a general matrix B = (bij ). Therefore system (6) after this transformation becomes         a 1   x + u1      if x > 0,   1 a y + u ẋ 2 (18)  =    ẏ     b b x    11 12    if x < 0,   b21 b22 y + v2 where 4b12 b21 + (b11 − b22 )2 < 0. Note that u1 > 0 because the vector field X has a virtual node. The solution ϕ+ (t, x̄, ȳ) of (18) with  x̄ > 0, such that ϕ+ (0, x̄, ȳ) = (x̄, ȳ) is + + ϕ+ (t, x̄, ȳ) = ϕ1 (t, x̄, ȳ), ϕ2 (t, x̄, ȳ) where ϕ+ at 1 (t, x̄, ȳ) = −u1 + e [(u1 + x̄) cosh t + (u2 + ȳ) sinh t] (19) ϕ+ at 2 (t, x̄, ȳ) = −u2 + e [(u2 + ȳ) cosh t + (u1 + x̄) sinh t] . The solution of (18) with x̄ < 0 is ϕ− (t, x̄, ȳ) = (ϕ− − 1 (t, x̄, ȳ), ϕ2 (t, x̄, ȳ)), where ϕ− − 1 and ϕ2 are given in (10) and (11), respectively. Assuming that the real parts of the eigenvalues of system (18) are negative, that is a < −1 and b11 + b22 < 0, we will prove that system (18) has at most THE MARKUS–YAMABE CONJECTURE FOR DISCONTINUOUS SYSTEMS 13 one limit cycle. In order to fix the clockwise orientation of the orbits of system (18) we assume that Y1 (0, 1 − v2 ) = b12 > 0. The first part of the proof of the next result is in the proof of Proposition 14 of [7]. Theorem 7. Assume that the eigenvalues of vector fields X and Y have nega- tive real parts, then the piecewise discontinuous linear differential system (18) has at most one limit cycle. Proof. Doing a translation to system (18) that preserves the half–plane x > 0 and the discontinuity line Σ, we can assume that u2 = 0. Clearly the point (0, −a u1 ) ∈ Σ is an invisible fold point, and (−u1 , 0) is the singularity of X. Furthermore the invariant straight lines of the node intersect the line Σ at the points (0, y s ) and (0, y ss ), respectively, where y s = u1 < −a u1 and y ss = −u1 < u1 . It follows that the function t+ (y) > 0 is defined for every y > −au1 , see Figure 4. y −a u1 ys −u1 x Figure 4. Virtual diagonalizable node of the vector field X. The shaded line is the domain of definition of the function t+ (ȳ) > 0. By definition of t+ (ȳ), from (19) we get that + (ȳ)  ϕ+ + 1 (t (ȳ), 0, ȳ) = −u1 + e at u1 cosh t+ (ȳ) + ȳ sinh t+ (ȳ) = 0, for every ȳ > −au1 . Thus defining y + (t) = u1 G(t) for t > 0, with G(t) given in (5), we have that y + (t+ (ȳ)) = ȳ for every ȳ > −au1 . 14 J. LLIBRE AND L.A.S. MENEZES Computing implicitly the derivative in the variable ȳ of the identity y + (t+ (ȳ)) = ȳ we obtain dt+ (ȳ) sinh t+ (ȳ) = dȳ u1 cscht+ (ȳ) − u1 e−at+ (ȳ) (a + coth t+ (ȳ)) sinh2 t+ (ȳ) = > 0, u1 [1 − (a sinh t+ (ȳ) + cosh t+ (ȳ))e−at+ (ȳ) ] for t+ (ȳ) > 0, because a < −1 and u1 > 0. We claim that t+ (y + (t)) = t for every t > 0. Consider y0 = y + (t0 ) for some t0 > 0. From Lemma 3 we have that the function y + (t) is injective on R+ and y + (t) > −au1 . Therefore y0 > −au1 and from the previous results y0 = y + (t+ (y0 )). Thus y + (t0 ) = y + (t+ (y0 )) from which it follows that t0 = t+ (y0 ) = t+ (y + (t0 )). Since t0 > 0 was arbitrarily chosen we conclude that t+ (y + (t)) = t for every t > 0. Therefore the function t+ : (−au1 , ∞) → R+ is invertible with inverse equal to y + : R+ → (−au1 , ∞). Computing the zeros of the function (7) for y > YM = max{−au1 , −v2 } is equivalent to compute the zeros of the function (20) g2 (t) = f (y + (t)) = v2 (1 + δ) + u1 (1 − δ) coth t − u1 (eat − δe−at )cscht for t ∈ I = t+ ((YM , ∞)) ⊂ R+ and δ = e−(b11 +b22 )π/Γ . Moreover I = R+ provided v2 ≥ a u1 . We will prove that the function g2 (t) has at most one zero for t > 0. Notice that δ > 1 and g2 (t) = 0 is equivalent to (21) A sinh t + (1 − δ) cosh t − eat + δe−at = 0, v2 (1 + δ) where A = . u1 Consider the functions h2 (t) = −eat + δe−at , h3 (t) = −A sinh t − (1 − δ) cosh t, with t > 0, and note that equation (21) holds if and only if h2 (t) = h3 (t). We will prove that there exists at most one t̄ ∈ R+ such that h2 (t̄) = h3 (t̄). By the proof of Theorem 4 we have that graph of h2 (t) is given in Figure 2. Observe that h03 (t) = −A cosh t + (δ − 1) sinh t, h003 (t) = −A sinh t + (δ − 1) cosh t, THE MARKUS–YAMABE CONJECTURE FOR DISCONTINUOUS SYSTEMS 15 from which it follows that, if A < 0 then h03 (t) > 0 and h003 (t) > 0. Furthermore we have that h3 (0) = δ−1 > 0, so h3 (t) is a positive strictly increasing function and its graph is given in Figure 5-(a). On the other hand if A > 0 we have three cases to consider (1) A < δ − 1. In  this case h3 (t) 6= 0 and h03 (t1 ) = 0, where t1 = 1 A+δ−1 ln − , and h003 (t) > 0. Consequently the graph of the 2 A+1−δ function h3 (t) is given in Figure 5–(b). (2) A > δ − 1. In this case we obtain that h3 (t0 ) = 0, where t0 = 1 A+δ−1 ln , h03 (t) < 0 and h003 (t0 ) > 0 (respectively <) for every 2 A+1−δ t < t0 (respectively >). It follows that the graph of the function h3 (t) is given in Figure 5–(c). (3) A = δ − 1. We have that h3 (t) > 0, h03 (t) < 0 and h003 (t) > 0, and this implies that the function h3 (t) is strictly decreasing for t > 0. The graph of the function h3 (t) is given in Figure 5–(d). (a) (b) δ−1 δ−1 t1 (c) (d) δ−1 δ−1 t0 Figure 5. Graph of h3 (t): (a) A < 0, (b) 0 < A < δ − 1, (c) A > δ − 1, and (d) A = δ − 1. 16 J. LLIBRE AND L.A.S. MENEZES Analyzing the graph of the functions h2 (t) and h3 (t) we obtain that, for A < δ − 1 there exists at most one t̄ ∈ R+ such that h2 (t̄) = h3 (t̄), and for A ≥ δ − 1 we have h2 (t) 6= h3 (t) for every t ∈ R+ . Therefore, equation (21) holds for at most one t ∈ R+ and, consequently the function g2 (t) have at most one zero t ∈ R+ .  Corollary 8. A piecewise MY–system of type F N have at most one limit cycle. Proof. The result follows applying Theorem 7 to piecewise MY–systems whose singularities for the vector fields X and Y are a virtual diagonalizable stable node and a boundary stable focus at the origin, respectively.  The following is an example of piecewise a MY–system of type F N having one limit cycle. Example 9. Let (22)      2 1  X(z) = y − 2x − , x − 2y + for x > 0,  5  5     ż =   x 3 3 y 1  Y (z) = y − ln − 1, −x − ln − for x < 0, π 2 2 π π be a piecewise discontinuous linear differential system. The singularities of (22) are (−1/5, 0) which is a virtual node of X, and (0, 1) which is a boundary + + focus of Y . Observe that, the eigenvalues  of (22) are λ1 = −1 and λ2 = −3, 1 3 for the point (−1/5, 0), and λ− 1,2 = ln ± i, for the point (0, 1). Thus, by π 2 Theorem 7 system (22) has at most one limit cycle. e2t coth t In this case we have that yM = max{1, 2/5} = 1, y + (t) = cscht − , 5 5 and the function defined in (20) is 5 coth t e−2t 3e2t g2 (t) = − − − cscht + cscht. 2 10 5 10 Then for t > 0, g2 (t) = 0 if and only if t ≈ 1.412... and, therefore system (22) has one limit cycle passing thorough y + (1.412...) ≈ 1.520... > yM . Changing the coordinates to u = x and v = y − 1, we rewrite system (22) as      3 9  v − 2u + , u − 2v − for u > 0,  5 5 (23) ż =   u 3 v 3  v − ln , −u − ln for u < 0, π 2 π 2 THE MARKUS–YAMABE CONJECTURE FOR DISCONTINUOUS SYSTEMS 17 u v Figure 6. The phase portrait of the piecewise MY–system (23). . whose boundary focus is at (0, 0) and the virtual focus is at (−1/5, −1). We conclude that system (23) is a piecewise MY–system and it has one limit cycle passing through point (u, v) = (0, 0.520...), see Figure 6. 4.3. The piecewise MY–system of type F iN . In this subsection we prove that the piecewise MY–systems Z = (X, Y ) having at the origin a boundary stable focus of Y and having X a stable virtual node iN has at most one limit cycle. Suppose that X has a virtual improper node iN , and Y has a boundary focus. By Proposition 2 the matrix of the linear vector field X is transformed into M3 and the matrix of the linear vector field Y is a general matrix B = (bij ). Therefore we rewrite system (6) as         λ λ   x + u1       if x > 0,   0 λ y + u ẋ 2 (24)  =    ẏ     b b x    11 12    if x < 0,   b21 b22 y + v2 where λ = ±1 and 4b12 b21 + (b11 − b22 )2 < 0. Furthermore u1 > 0 because the vector field X has a virtual improper node for system (24). 18 J. LLIBRE AND L.A.S. MENEZES In order to fix the clockwise orientation of the orbits of vector field Y we assume that Y1 (0, 1 − v2 ) = b12 > 0. The solution of (24) for x̄ < 0 is ϕ− (t, x̄, ȳ) = (ϕ− − 1 (t, x̄, ȳ), ϕ2 (t, x̄, ȳ)) where ϕ− 1 − and ϕ2 are given by equations (10) and (11), respectively. Applying a translation that preserves the line Σ and the half–plane x̄ > 0, we can assume that u2 = 0. The solution ϕ+ (t, x̄, ȳ) of (24) for x̄ < 0,  such that ϕ+ (0, x̄, ȳ) = (x̄, ȳ) is given by ϕ+ (t, x̄, ȳ) = ϕ+ 1 (t, x̄, ȳ), ϕ+ 2 (t, x̄, ȳ) where ϕ+ λt 1 (t, x̄, ȳ) = −u1 + e (λtȳ + u1 + x̄), ϕ+ λt 2 (t, x̄, ȳ) = ȳe . We have the following result. Theorem 10. Assume that the vector fields X and Y have only eigenvalues with negative real part, then the piecewise discontinuous differential system (24) have at most one limit cycle. Proof. Observe that, if λ = −1 the first return map would not be defined, because we fixed the clockwise orientation of the orbits of vector field Y , and there would not exist limit cycles. Therefore we assume that λ = 1, and this implies that the linear vector field X has an unstable singularity. Thus, we assume that the singularity of Y also is unstable. We shall prove that the system (24) has at most one cycle limit and our result will follow by rescaling time τ = −t. In what follows we assume that the real parts of the eigenvalues of system (24) are positive and we prove that system (24) has at most one limit cycle. Observe that (−u1 , 0) is the improper node and (0, −u1 ) is an invisible fold point of X. The invariant straight line of the node intersects Σ at the origin, see Figure 7. Thus the function t+ (y) < 0 is defined for every y < −u1 . Defining y+ (t) = u1 H(t) for t < 0, we have that y+ (t) < −u1 , because by Lemma 3 we have H(t) < −1. Furthermore since ϕ+ 1 (t+ (ȳ), 0, ȳ) = −u1 + e t+ (ȳ) (u1 + ȳt+ (ȳ)) = 0, for every ȳ < −u1 , we have that u1 (e−t+ (ȳ) − 1) y+ (t+ (ȳ)) = = ȳ t+ (ȳ) for every ȳ < −u1 . THE MARKUS–YAMABE CONJECTURE FOR DISCONTINUOUS SYSTEMS 19 y −u1 −u1 x Figure 7. Virtual node iN for the vector field X when λ = 1. In this case the shaded line represents the domain of the function t+ (ȳ) < 0. We claim that the image of the function t+ (ȳ) is R− = (−∞, 0) and t+ (y+ (t)) = t. Indeed, computing the derivative of t+ (ȳ) we obtain dt+ (ȳ) et+ (ȳ) t+ (ȳ)2 = > 0, dȳ u1 (et+ (ȳ) − t+ (ȳ) − 1) for t+ (ȳ) < 0. So t+ (ȳ) is strictly increasing with t+ (−u1 ) = 0 and this implies that t+ (ȳ) < 0 for every ȳ < −u1 . Furthermore, by previous comments, taking t0 < 0 and y0 = y+ (t0 ), we have that y0 < 0 and y0 = y+ (t+ (y0 )). By Lemma 3 we have that y+ (t) is injective and this implies that t0 = t+ (y0 ) and, therefore t+ (y+ (t)) = t because we chose t0 arbitrarily. Consequently t+ : (−∞, −u1 ) −→ R− is an invertible function whose inverse function is y+ : R− −→ (−∞, −u1 ). Define ym = min{−u1 , −v2 }, then computing the zeros of the function (8) for y < ym is equivalent to compute the zeros of the u1 (1 − et ) δu1 (1 − e−t ) (25) g3 (t) = v2 (1 + δ) + − , t t for t ∈ I = y+ (−∞, ym ) ⊂ R− , where δ = e(b11 +b22 )π/Γ . Note that if v2 ≥ u1 then I = R− . In what follows we prove that the function (25) has at most one zero in R− . 20 J. LLIBRE AND L.A.S. MENEZES Since the singularity of Y is unstable we have that δ > 1. Moreover g3 (t) = 0 is equivalent to (26) At + 1 − δ − et + δe−t = 0, v2 (1 + δ) where A = . We will prove that there exists at most one t ∈ R− such u1 that equation (26) holds. (a) (b) (c) δ−1 δ−1 t1 δ−1 Figure 8. (a) Graph of the function h2 (t), (b) Graph of the function h1 (t) for A < 0, and (c) Graph of the function h1 (t) for A > 0. Consider the following functions h1 (t) = −At + δ − 1, (27) h2 (t) = −et + δe−t , defined for every t ∈ R− and note that equation (26) holds if and only if h1 (t) = h2 (t). We will show that there exists at most one t̄ ∈ R− such that h1 (t̄) = h2 (t̄). Observe that h2 (t) > 0 for every t < 0, h2 (0) = δ − 1 > 0 and h02 (t) = −et − δe−t < 0. Then h2 (t) is strictly decreasing positive function in R− . Furthermore h002 (t) = −et + δe−t > 0 implies that h2 (t) does not have inflexion points and its graph is given in Figure 8–(a). Computing the derivative of the function h1 (t) we obtain h01 (t) = −A. Fur- thermore we have that h1 (0) = δ −1 > 0 and h1 (t1 ) = 0, where t1 = (δ − 1)/A. Therefore if A < 0 then h1 (t) is a strictly increasing function in R− and the graph of h1 is given in Figure 8–(b). So there is not solution for equation h1 (t) = h2 (t). On the other hand, if A > 0 then t1 > 0 and h1 (t) is a strictly decreasing function and its graph is given by Figure 8–(c). THE MARKUS–YAMABE CONJECTURE FOR DISCONTINUOUS SYSTEMS 21 Now analyzing the graphs of the functions h1 (t) and h2 (t) we conclude that to A > 0 if h01 (0) < h02 (0) there exists a unique t̄ ∈ R− such that h1 (t̄) = h2 (t̄), and if h01 (0) > h02 (0) then h1 (t) < h2 (t) . Hence there exists at most one zero of the function (25) in R− . Therefore system (24) has at most one limit cycle  Corollary 11. The piecewise MY–system of type F iN has at most one limit cycle. Proof. The result follows applying Theorem 10 to piecewise MY–systems whose singularities for the vector fields X and Y are a virtual improper stable node and a boundary stable focus at the origin, respectively.  In what follows we present an example of a piecewise MY–system of type F iN having one limit cycle. Example 12. Consider the piecewise discontinuous linear differential system      1  X(z) = x + y + , y for x > 0,  2  (28) ż =   π2  Y (z) = x + y + 5, y − x+5 for x < 0. ln(2)2 The singularities of (28) are (−1/2, 0), which is a virtual improper node of X, and (0, −5), which is a boundary focus of Y . Observe that the eigenvalues of π (28) are λ+ = 1 for x > 0, and λ−1,2 = 1 ± i for x < 0. ln 2 In this case we have that ym = min{−1/2, −5} = −5, e−t − 1 y + (t) = , 2t and the function defined in (25) is 1 − et 1 − e−t g3 (t) = 15 + − . 2t t Then g3 (t) = 0 for t ∈ R− if and only if t ≈ −4.136... and, therefore the system (28) has one limit cycle passing thorough y + (−4.136...) ≈ −7.441... < ym . Now applying the rescaling of time τ = −t and the change of coordinates u = x and v = y − 5, we rewrite the system (22) as     9  −u − v + , −v + 5  for u > 0,  2  (29) ż =   π2  −u − v, −v + u for u < 0, ln(2)2 whose boundary stable focus is (0, 0) and its virtual improper stable node is (−1/2, 5). 22 J. LLIBRE AND L.A.S. MENEZES u v Figure 9. The phase portrait of the piecewise MY–system (29). Notice that the eigenvalues of (29) have negative real parts. We conclude that system (29) is a piecewise MY–system and it has one limit cycle passing through point (u, v) = (0, −2.441...), see Figure 9. Acknowledgements The first author is partially supported by a MINECO-FEDER grant MTM2016- 77278-P, a MINECO grant MTM2013-40998-P, and an AGAUR grant number 2014SGR-568. The second author is partially supported by a CAPES PDSE- 88881.133794/2016-01 and a Ph.D. CAPES grant. References [1] J. Bernat, J. 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Non Linéaire 12 (1995), 627–671. [7] J. Llibre, D. D. Novaes, M. A. Teixeira, Maximum number of limit cycles for certain piecewise linear dynamical systems, Nonlinear Dyn. 82 (2015), 1159–1175. [8] L. Markus, H. Yamabe, Global Stability Criteria for Differential Systems, Osaka Math J. 12 (1960), 305–317. [9] G. H. Meisters, Cz. Olech, Solution of the Global Asymptotic Stability Jacobian Conjecture for the Polynomial Case, Analyse mathèmatiques et applications. Contri- butions a l’honneur de Jacques–Louis Lions, Gauthier–Villars, Paris 1988, 373–381. 1 Departament de Matematiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain E-mail address: jllibre@mat.uab.cat 2 Instituto de Matemática e Estatı́stica, Universidade Federal de Goiás, 74690-900, Goiânia, GO, Brazil E-mail address: lucyjanedealmeida@gmail.com