The Markus–Yamabe Conjecture Does not Hold for Discontinuous Piecewise Linear Differential Systems Separated by One Straight Line

The Markus–Yamabe conjecture is a conjecture on global asymptotic stability. The conjecture states that if a differentiable system x˙=f(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{x}=f(x)$$\end{document} has a singularity and the Jacobian matrix Df(x) has everywhere eigenvalues with negative real part, then the singularity is a global attractor. In this paper we consider discontinuous piecewise linear differential systems in R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^2$$\end{document} separated by one straight line Σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma $$\end{document} such that the unique singularity of the system is at Σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma $$\end{document} and the Jacobian matrix of the system has everywhere eigenvalues with negative real part. We prove that these discontinuous piecewise linear differential systems can have one crossing limit cycle and consequently these systems do not satisfy the Markus–Yamabe conjecture.


Introduction and Statement of the Main Results
Consider f (x) a C 1 map on an n-dimensional real vector space. Leṫ (1) be a differential system such that f (0) = 0. In 1960 Markus and Yamabe stated that if all eigenvalues of D f (x) have negative real part, then the origin of (1) is a global attractor. In their paper [11] 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 [12]. Considering vector fields of class C 1 defined in R 2 , Gutierrez [9], Fleber [4] and Glutsyuk [7] for this order provided different proofs of the Markus-Yamabe conjecture in the years 1994-1995. However counterexamples 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 [3] 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 R n with n ≥ 3. Other papers on the Markus-Yamabe conjecture are [2,6,8].
We define a discontinuous piecewise linear Markus-Yamabe differential system, or simply piecewise Markus-Yamabe system aṡ where z = (x, y) ∈ R 2 , X and Y are linear vector fields, the real part of the eigenvalues of D X(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 = {(x, y) ∈ R 2 ; x = 0} is called the discontinuity set. Hereż 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 (i N). We say that Z is a piecewise MYsystem of type L R, with L, R ∈ {F, N , N * , i N}, 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 [5] the discontinuity set 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 Fi N has at most one limit cycle, and there are systems of these types with exactly one limit cycle. The others piecewise MYsystems different from these three types have no limit cycles.
The rest of the paper is organized as follows. In Sect. 2 we provide some basic notions and results that we shall need for proving Theorem 1. In the short Sect. 3 we prove the last part of Theorem 1, i.e. that the piecewise MY-systems of type F N * , N N * , i N N, and i N N * have no limit cycles. Finally in the long Sect. 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 Fi N .
Recently the Markus-Yamabe conjecture has been considered for a class of differential systems in infinite dimension see [13].

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 anḋ where z = (x, y) ∈ R 2 , a piecewise linear differential system whose discontinuity is the We define the following singularities to system (3).
We say that a point (0, y) is an invisible fold point for the vector field X = (X 1 , X 2 ) when and an invisible fold point for the vector field A T -system is a linear differential system having a singularity of type T with T ∈ {F, N , N * , i N}. Next result provides a simpler way to rewrite a T -system in the plane and its proof can be found in Proposition 5 of [10].

Proposition 2
Let M = m i j be a 2 × 2 matrix. If the linear differential system 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 = M 1 (x, y) T ; (b) F-system then after a vertical lines-preserving linear change of variables and a timerescaling system (4) becomes (ẋ,ẏ) T = M 2 (x, y) T with a = 0; (c) i N -system then after a vertical lines-preserving linear change of variables and a time- The next result is the Lemma 6 of [10].

Lemma 3 We consider the functions
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 FN * , NN * , iNN and iNN * Observe that, if the piecewise MY-system has one virtual singularity of type N * , then the solution passing through the point (0, y) ∈ cannot return to 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 has one boundary singularity of type N * , N or i N, then the orbits of the vector field X passing through (0, y) ∈ coincides with or returns to at the singularity. Therefore the first return map is not defined. Consequently the piecewise MY-systems of type F N * , N N * , i N N and i N N * do not have limit cycles. This proves the last part of Theorem 1.

Limit Cycles for Piecewise MY-Systems
Consider a discontinuous piecewise linear differential system defined in R 2 where z = (x, y). We denote the solution of (6) by x,ȳ) are the solutions of vector fields X and Y , respectively.
Assume that t + (ȳ) > 0 and t − (ȳ) < 0 are defined. There exists a limit cycle passing through the point Thus, in this case the limit cycles are in correspondence with the zeros y * of the function 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 Thus, in this case we must study the zeros y * of the function on the domain J * .
Since the vector fields X and Y are linear, then a limit cycle passing through the point (x 0 , y 0 ) must contain points of the form (0, y * ) and (0, y * ) such that y * ∈ J * and y * ∈ J * . Therefore all the zeros of (7) or (8) determine all the limit cycles.
In [10] the authors proved that the discontinuous piecewise linear differential systems of type F F, F N and Fi N have at most two limit cycles. Here assuming that these systems are piecewise MY-systems we shall prove that they have at most one limit cycle.

Piecewise MY-Systems of Type FF
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 has at most one limit cycle.
Assume that system (6) has 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 M 2 and the associated matrix of the linear vector field Y is transformed is a general matrix B = (b i j ). Then we can rewrite system (6) as where 4b 12 b 21 +(b 11 −b 22 ) 2 < 0 because we have a focus for the vector field Y . Furthermore u 1 > 0 because the focus of the vector field X is virtual. Since (9) is a discontinuous piecewise linear differential system, we can compute its solution. Thus the solution ϕ We have the following result. The first part of its proof is in the proof of Proposition 11 of [10].

Theorem 4
Assume that the eigenvalues of the vector fields X and Y have negative real parts, then the discontinuous piecewise 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 Observe that the orbits of vector field Y expends a time Doing a convenient translation in the plane so that it preserves the straight line and the half-plane x > 0, we can take u 2 = 0. This implies that the singularity of X is (− u 1 , 0). Furthermore the point (0, −a u 1 ) ∈ is an invisible fold point for the vector field X . It follows that the function t + (ȳ) > 0 is defined for everyȳ > −a u 1 (see Fig. 1).
In what follows we assume A < 0. Consider the functions and note that equation (14) holds if and only if h 1 (t) = h 2 (t). We will prove that there exists at most onet ∈ (0, π) such that h 1 (t) = h 2 (t).

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.
In this case we have that y M = max{1/10, 1} = 1, y + (t) = (e t/2 csc t − cot t)/5, and the function defined in (13) is Thus in the interval (0, π) we have that g 1 (t) = 0 if and only if t ≈ 2.588 . . . and, therefore system (16) has one limit cycle passing through y + (2.588 . . .) ≈ 1.714 . . . > y M . Changing the coordinates to u = x and v = y − 1, we rewrite the system (16) aṡ whose boundary focus is at (0, 0) and the virtual focus is at (− 1/5, − 1). Observe that system (17) is a piecewise MY-system and it has one limit cycle passing through point

Piecewise MY-Systems of Type FN
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 has at most one limit cycle. Suppose that the discontinuous piecewise 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 M 1 and the matrix of the vector field Y is a general matrix B = (b i j ). Therefore system (6) after this transformation becomes where 4b 12 b 21 + (b 11 − b 22 ) 2 < 0. Note that u 1 > 0 because the vector field X has a virtual node. The solution ϕ + (t,x,ȳ) of (18) withx > 0, such that ϕ x,ȳ)), where ϕ − 1 and ϕ − 2 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 b 11 + b 22 < 0, we will prove that system (18) has at most one limit cycle. In order to fix the clockwise orientation of the orbits of system (18) we assume that Y 1 (0, 1−v 2 ) = b 12 > 0.
The first part of the proof of the next result is in the proof of Proposition 14 of [10].

Theorem 7
Assume that the eigenvalues of vector fields X and Y have negative real parts, then the discontinuous piecewise 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 , we can assume that u 2 = 0. Clearly the point (0, −a u 1 ) ∈ is an invisible fold point, and (− u 1 , 0) is the singularity of X . Furthermore the invariant straight lines of the node intersect the line at the points (0, y s ) and (0, y ss ), respectively, where y s = u 1 < −a u 1 and y ss = − u 1 < u 1 . It follows that the function t + (y) > 0 is defined for every y > −au 1 , see Fig. 4.
Computing the zeros of the function (7) for y > Y M = max{−au 1 , −v 2 } is equivalent to compute the zeros of the function We will prove that the function g 2 (t) has at most one zero for t > 0. Notice that δ > 1 and g 2 (t) = 0 is equivalent to where

Consider the functions
with t > 0, and note that equation (21) holds if and only if h 2 (t) = h 3 (t). We will prove that there exists at most onet ∈ R + such that h 2 (t) = h 3 (t).
By the proof of Theorem 4 we have that graph of h 2 (t) is given in Fig. 2. Observe that from which it follows that, if A < 0 then h 3 (t) > 0 and h 3 (t) > 0. Furthermore we have that h 3 (0) = δ − 1 > 0, so h 3 (t) is a positive strictly increasing function and its graph is given in Fig. 5a. On the other hand if A > 0 we have three cases to consider Consequently the graph of the function h 3 (t) is given in Fig. 5b.
Analyzing the graph of the functions h 2 (t) and h 3 (t) we obtain that, for A < δ − 1 there exists at most onet ∈ R + such that h 2 (t) = h 3 (t), and for for every t ∈ R + . Therefore, equation (21) holds for at most one t ∈ R + and, consequently the function g 2 (t) has at most one zero t ∈ R + .

Corollary 8 A piecewise MY-system of type F N has 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.

The Piecewise MY-System of Type FiN
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 i N has at most one limit cycle.
Suppose that X has a virtual improper node i N, and Y has a boundary focus. By Proposition 2 the matrix of the linear vector field X is transformed into M 3 and the matrix of the linear vector field Y is a general matrix B = (b i j ). Therefore we rewrite system (6) as where λ = ±1 and 4b 12 b 21 + (b 11 − b 22 ) 2 < 0. Furthermore u 1 > 0 because the vector field X has a virtual improper node for system (24). In order to fix the clockwise orientation of the orbits of vector field Y we assume that The solution of (24 x,ȳ)) where ϕ − 1 and ϕ − 2 are given by equations (10) and (11), respectively. Applying a translation that preserves the line and the half-planex > 0, we can assume that u 2 = 0. The solution ϕ + (t,x,ȳ) of (24) forx < 0, such that ϕ We have the following result.
Theorem 10 Assume that the vector fields X and Y have only eigenvalues with negative real part, then the discontinuous piecewise linear differential system (24) has 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 (− u 1 , 0) is the improper node and (0, −u 1 ) is an invisible fold point of X . The invariant straight line of the node intersects at the origin, see Fig. 7. Thus the function t + (y) < 0 is defined for every y < −u 1 .
Computing the derivative of the function h 1 (t) we obtain h 1 (t) = − A. Furthermore we have that h 1 (0) = δ − 1 > 0 and h 1 (t 1 ) = 0, where t 1 = (δ − 1)/A. Therefore if A < 0 On the other hand, if A > 0 then t 1 > 0 and h 1 (t) is a strictly decreasing function and its graph is given by Fig. 8c. Now analyzing the graphs of the functions h 1 (t) and h 2 (t) we conclude that to A > 0 if h 1 (0) < h 2 (0) there exists a uniquet ∈ R − such that h 1 (t) = h 2 (t), and if h 1 (0) > h 2 (0) then h 1 (t) < h 2 (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 Fi N 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 Fi N having one limit cycle.
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 Fig. 9.