The Markus–Yamabe Conjecture for Discontinuous Piecewise Linear Differential Systems in Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^n$$\end{document} Separated by a Conic×Rn-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times \mathbb {R}^{n-2}$$\end{document}

In 1960 Markus and Yamabe made the conjecture that if a C1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^1$$\end{document} differential 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} in Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^n$$\end{document} has a unique equilibrium point and DF(x) is Hurwitz for all x∈Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x \in \mathbb {R}^n$$\end{document}, then the equilibrium point is a global attractor. This conjecture was completely solved in 1997 and it turned out to be true 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} and false in Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^n$$\end{document} for all n≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \ge 3$$\end{document}. In (The Markus–Yamabe conjecture for continuous and discontinuous piecewise linear differential systems, 2020) the authors extended the Markus–Yamabe conjecture to continuous and discontinuous piecewise linear differential systems in Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^n$$\end{document} separated by a hyperplane, they proved for the continuous systems that the extended conjecture is true 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} and false in Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^n$$\end{document} for all n≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \ge 3$$\end{document}, but for discontinuous systems they proved that the conjecture is false in Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^n$$\end{document} for all n≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \ge 2$$\end{document}. In this paper first we show that there are no continuous piecewise linear differential systems separated by a conic×Rn-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times \mathbb {R}^{n-2}$$\end{document} except the linear differential systems in Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^n$$\end{document}. And after we prove that the extended Markus–Yamabe conjecture to discontinuous piecewise linear differential systems in Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^n$$\end{document} separated by a conic×Rn-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times \mathbb {R}^{n-2}$$\end{document} is false in Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^n$$\end{document} for all n≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \ge 2$$\end{document}.


Introduction and Statement of the Results
Consider a C 1 differential systemẋ = F(x) defined in R n and having an equilibrium point at the origin of coordinates. If D F(0) is Hurwitz (i.e. all eigenvalues of D F(0) have negative real B Clàudia Valls cvalls@math.tecnico.ulisboa.pt Jaume Llibre jllibre@mat.uab.cat 1 Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra, 08193 Barcelona, Catalonia, Spain parts), then by the Hartman-Grobman Theorem [11,14] the origin is locally asymptotically stable. A natural question arises: which are the additional hypotheses that one may add to the function F in order that the origin is a global attractor.
Markus and Yamabe in 1960 (see [18]) made the following conjecture: If we have a C 1 differential systemẋ = F(x) defined in R n such that D F(x) is Hurwitz for all x ∈ R n , and having a unique equilibrium point at the origin of coordinates, then the origin is a global attractor.
This conjecture follows easily when n = 1. This conjecture when n = 2 was proved independently by Gutierrez [12,13] in 1993 and by Fessler [6,7] in 1995. A simpler proof was then given by Glutsyuk [9,10]. The counterexample to Markus-Yamabe conjecture for n > 3 was given by Bernat and Llibre [2] and the counterexample for n ≥ 3 was given by Cima et al. [3]. In short, the Markus-Yamabe conjecture is true in R 2 and false in R n for n ≥ 3.
We recall that an equilibrium point p is a global attractor if it is globally asymptotically stable, that is, every solution tends to p as the time goes to infinity.
The natural step is to ask whether this conjecture is true for continuous systems and even more for discontinuous ones. Since the study of such systems is much more complicated, one natural thing to do is to start with the simpler ones, that is the continuous or discontinuous piecewise linear differential systems. The study of this class of continuous and discontinuous differential systems started with Andronov et al. [1]. Due to the fact that these systems model many real phenomena and different modern devices, they have became a topic of great interest these last twenty years. For more details see for instance [4,19] and the references therein.
In [15,17] the authors extended the Markus-Yamabe conjecture to continuous and discontinuous piecewise linear differential systems formed by two pieces of R n separated by a hyperplane. A Markus-Yannabe piecewise linear differential system is a piecewise linear differential system of the formẋ such that x = (x 1 , . . . , x n ), the matrices A + and A − are Hurwitz, and either only one of the systemsẋ = A + x + b + andẋ = A − x + b − has a real equilibrium point, or both systems have the same equilibrium point in . , x n ), then we say that system (1) is a continuous piecewise linear differential system, and otherwise we say that it is a discontinuous linear differential system. The dynamics of the discontinuous piecewise differential systems on the hyperplane x 1 = 0 is defined according with the definitions of the book of Filippov [8], and consequently the discontinuous piecewise differential systems define a flow in R n .
We recall that a linear differential systemẋ = A + x + b + has a real equilibrium point if the equilibrium point −(A + ) −1 b + exits, and it is in the closed half-space {x 1 ≥ 0}, otherwise we say that the equilibrium point is virtual. Similarly for the linear differential systemẋ = A − x + b − . More concretely in [17] it was proved the following result.

Theorem 1
The following statements hold. (b) For all n ≥ 2 there are discontinuous Markus-Yamabe piecewise linear differential systems in R n for which their equilibrium point is not a global attractor.
Using an affine change of coordinates, any conic in R 2 can be written in one of the following nine canonical forms: (p) x 2 1 + x 2 2 = 0 two complex straight lines intersecting at a real point; We do not consider conics of type (p), (CL) or (CE) because they do not separate the plane in connected regions. Moreover the case (DL) is completely analogous to the one proved in [17] and so we do not consider it here. In short we are left with cases (PL), (LV), (E), (H) and (P).
In this paper we will focus on continuous and discontinuous piecewise linear differential systems formed by the pieces of R n separated by a hyperconic C(x) = 0. Following the definition in [17] a Markus-Yamabe piecewise linear differential system separated by a hyperconic C(x) = 0 is a piecewise linear differential system in R n of the forṁ such that the matrices A + and A − are Hurwitz and either only one of the systemsẋ = , then we say that system (2) is a continuous piecewise linear differential system. Otherwise we say that it is a discontinuous linear differential system separated by the hyperconic C(x) = 0. The dynamics of the discontinuous piecewise differential systems on the hyperconic of discontinuity is defined according with the definitions of the book of Filippov [8]. Moreover, a linear differential systemẋ = A + x + b + has a real equilibrium point if the equilibrium point −(A + ) −1 b + exists, and it is contained in {C(x) ≥ 0}, otherwise we say that the equilibrium point is virtual. Similarly for the linear differential systemẋ = A − x + b − .

Proposition 1
There are no continuous Markus-Yamabe piecewise linear differential systems in R n separated by a hyperconic of the form (PL), (LV), (H), (E) or (P), other than the linear differential systems.
The proof of Proposition 1 is given in Sect. 2. Our main result is the following.

Theorem 2 For all n ≥ 2 there are discontinuous Markov-Yamabe piecewise linear differential systems separated by a hyperconic (PL), (LV), (E), (H) and (P) in R n for which their equilibrium point is not a global attractor.
The proof of Theorem 2 is given in Sect. 2.

Proof of the Results
Proof of Proposition 1 We will do it only for (P) since for the others the proof is completely analogous.
Note that setting , j≤n we get fromẋ 1 the equality Doing a similar process withẋ k for k = 2, . . . , n we get that A + = A − and b + = b − . This proves the proposition for the hyperconic (P). The other cases are analogous.
We recall that a crossing limit cycle is a periodic solution isolated in the set of all periodic solutions of the discontinuous piecewise linear differential system, which only have two points of intersection with the discontinuity set C(x) = 0.

Proof of Theorem 2 (PL)
It is sufficient to prove the theorem for n = 2, because then we can extend a discontinuous Markus-Yamabe piecewise linear differential system in R 2 separated by a conic (PL) for which the unique equilibrium point of the system is not a global attractor, to a discontinuous Markus-Yamabe piecewise linear differential system in R n with n ≥ 3 separated by a hyperconic (PL) for which its unique equilibrium point will not be a global attractor by adding to the 2-dimensional system the equationṡ We consider the discontinuous piecewise linear differential system in R 2 with coordinates (x 1 , x 2 ) = (x, y) separated by two real parallel straight lines, a conic (PL), defined bẏ x = 2 − x,ẏ = −y, in the region |x| ≥ 1, x = −2 − x,ẏ = −y, in the region |x| ≤ 1. Note that this system is formed by two stable star nodes, i.e. there solutions leave on invariant straight lines. Clearly this is a discontinuous Markus-Yamabe piecewise linear differential system.
The star node at (2, 0) of the system in the region |x| ≥ 1 is real, and the start node at (−2, 0) of the system in the region |x| ≤ 1 is virtual. Since all the orbits of the system in the region |x| ≤ 1 runs from the right to the left, these orbits cannot go to the stable start node at (2, 0), so this node is not a global attractor. This completes the proof of Theorem 2 for the discontinuous piecewise linear differential systems separated by a two real parallel straight lines.

Proof of Theorem 2 (LV)
As in the proof for the hyperconic (LP) it is sufficient to prove the theorem for n = 2. Consider the discontinuous piecewise linear differential system in R 2 separated by two real straight lines intersecting in a point, a conic (LV), defined bẏ in the region x y ≥ 0, x = −1 − x,ẏ = −1 − y, in the region x y ≤ 0. Note that this system is formed by two stable star nodes. Hence it is a discontinuous Markus-Yamabe piecewise linear differential system.
The star node at (1, 1) of the system in the region x y ≥ 0 is real, and the start node at (−1, −1) of the system in the region x y ≤ 0 is virtual. Since all the orbits in the quadrant {(x, y) : x < 0, y > 0} of the system in the region x y ≤ 0 runs from top to bottom, these orbits cannot go to the stable start node at (1, 1), so this node is not a global attractor. This completes the proof of Theorem 2 for the discontinuous piecewise linear differential systems separated by two real straight lines intersecting in a point.

Proof of Theorem 2 (E)
As in the proof for the hyperconic (LP) it is sufficient to prove the theorem for n = 2. Consider a discontinuous piecewise linear differential system in R 2 separated by an ellipse (E) defined bẏ Note that this discontinuous piecewise linear differential system is formed by two linear centers, one of them being virtual. Moreover, the orbits in the region x 2 + y 2 ≥ 1 are the level curves of the first integral while in the region x 2 + y 2 ≤ 1 the orbits are in the level curves of the first integral It was proved in [16] that this piecewise linear differential system has two crossing limit cycles. One of the crossing limit cycles has the points (1, 0) and (0, 1) of intersection with the ellipse (see Fig. 1). Fig. 1 The crossing limit cycle of the discontinuous piecewise linear differential system (3) Now we shall see that this crossing limit cycle is unstable. The orbit of the system in the region x 2 + y 2 ≤ 1 through the point (cos(−1/100), sin(−1/100)) = (0.9999500004166.., −0.00999983333416..), intersects the circle x 2 + y 2 = 1 at the point (−0.0129688565617.., 0.999915900843..). Then the orbit through this last point of the system in the region x 2 + y 2 ≥ 1 intersects the circle x 2 + y 2 = 1 in the point (0.999927123997.., −0.01207255955763..). So the crossing limit cycle in its inner part is unstable. Consider now the orbit of the system in the region x 2 + y 2 ≤ 1 through the point  (0.01288913205622.., 0.999916931687..). Then the orbit through this last point of the system in the region x 2 + y 2 ≥ 1 intersects the circle x 2 + y 2 = 1 in the point (0.999927745390.., 0.0120209815765..). So the crossing limit cycle in its outer part is also unstable.
We perturb the discontinuous piecewise linear differential system (3) as followṡ with ε > 0 sufficiently small. Note that the two matrices A + and A − of system (4) are Hurwitz. So system (4) is a discontinuous Markus-Yamabe piecewise linear differential system having the unique real equilibrium point which is a stable focus of the region {x 2 + y 2 ≥ 1}. Since the crossing limit cycle of system (3) is unstable, it persists for system (4) for ε sufficiently small and so the equilibrium point P is not a global attractor. This completes the proof of Theorem 2 for the discontinuous piecewise linear differential systems separated by the ellipse (E).

First proof of Theorem 2 (H)
As in the proof for the hyperconic (LP) it is sufficient to prove the theorem for n = 2. Consider a discontinuous piecewise linear differential system in R 2 separated by a hyperbola (H) defined bẏ Note that this system is formed by two stable star nodes. Therefore it is a discontinuous Markus-Yamabe piecewise linear differential system. The star node at (2, 0) of the system iin the region x 2 − y 2 ≥ 1 is real, and the start node at (−2, 0) of the system in the region x 2 − y 2 ≤ 1 is virtual. Since all the orbits of the system in the region x 2 − y 2 ≤ 1 runs from the right to the left, these orbits cannot go to the stable start node at (2, 0), hence this node is not a global attractor. This completes the proof of Theorem 2 for the discontinuous piecewise linear differential systems separated by a hyperbola (H).

Second proof of Theorem 2 (H)
As in the proof for the hyperconic (LP) it is sufficient to prove the theorem for n = 2. Consider a discontinuous piecewise linear differential system in R 2 separated by a hyperbola (H) defined bẏ Note that this system is formed by two linear centers. The linear differential system in {x 2 − y 2 ≤ 1} has the real center at the point (0, 4), so it is a real equilibrium point. The linear differential system in {x 2 − y 2 ≥ 1} has a virtual equilibrium point. The first integrals of these systems are respectively. Proceeding as in the proof of the ellipse (E) we can prove that this piecewise linear differential system has one crossing limit cycle with the points (1, 0) and ( √ 89/5, 8/5) of intersection with the hyperbola (see Fig. 2). Moreover, proceeding as in the proofs of the conics (LP) and (LV) one can show that is a stable limit cycle. Fig. 2 The crossing limit cycle of the discontinuous piecewise linear differential system (5) Now we perturb the discontinuous piecewise linear differential system (5) as followṡ in the region x 2 − y 2 ≥ 1, x = 1 40 (89 − 5 √ 89) − εx − y,ẏ = −1 + x − εy, in the region x 2 − y 2 ≤ 1.
Note that for ε > 0 sufficiently small the two matrices of this piecewise system A + and A − are Hurwitz, and that this piecewise system has a unique real equilibrium p = 4ε 4ε 2 + 1 , 4 4ε 2 + 1 , Fig. 3 The crossing limit cycle of the discontinuous piecewise linear differential system (7)- (8) Now we perturb the discontinuous piecewise linear differential system (5)  in the region y ≤ x 2 . Note that for ε > 0 sufficiently small the two matrices of this piecewise system A + and A − are Hurwitz, and that this piecewise system has a unique real equilibrium near P 1 (still in the region y ≤ x 2 ) which is a stable focus, and it has a stable limit cycle near 2 . Hence this local stable focus is not a global attractor. This completes the proof of Theorem 2 for the discontinuous piecewise linear differential systems separated by the parabola (P).
Again since the focus p and the limit cycle 1 are stable, implies by the Poincaré-Bendixson Theorem that an unstable limit cycle must exist between them.