This is a preprint of: “The Markus-Yamabe conjecture for continuous and discontinuous piecewise linear differential systems”, Jaume Llibre, Xiang Zhang, Proc. Amer. Math. Soc., vol. 149(12), 5264–5274, 2021. DOI: [10.1090/proc/15601] THE MARKUS–YAMABE CONJECTURE FOR CONTINUOUS AND DISCONTINUOUS PIECEWISE LINEAR DIFFERENTIAL SYSTEMS JAUME LLIBRE1 AND XIANG ZHANG2 Abstract. In 1960 Markus and Yamabe made the following con- jecture: If a C 1 differential system ẋ = F (x) in Rn has a unique equilibrium point and the Jacobian matrix of F (x) for all x ∈ Rn has all its eigenvalues with negative real part, then the equilibrium point is a global attractor. Until 1997 we do not have the complete answer to this conjecture. It is true in R2 , but it is false in Rn for all n > 2. Here we extend the conjecture of Markus and Yamabe to con- tinuous and discontinuous piecewise linear differential systems in Rn separated by a hyperplane, and we prove that for the contin- uous piecewise linear differential systems it is true in R2 , but it is false in Rn for all n > 2. But for discontinuous piecewise linear differential systems it is false in Rn for all n ≥ 2. 1. Introduction and statement of the main results For the n–dimensional differential system (1) ẋ = Ax + ϕ(cT x)b, with ϕ ∈ C 0 (R) satisfying ϕ(0) = 0 and b, c ∈ Rn , we say that S(α, β) is a sector of linear stability (also called Hurwitz sector) if for all k ∈ (α, β) the matrix A + kbcT is Hurwitz (that is all its eigenvalues have negative real parts), and when this interval is bounded from below (respectively, bounded from above) then the matrix A+αbcT (respectively, A+βbcT ) has pure imaginary or zero eigenvalues. In 1957 Kalman formulated the following conjecture. 2010 Mathematics Subject Classification. 34D23, 34D45, 37C70, 34C05. Key words and phrases. Markus–Yamabe conjecture, Kalman conjecture, Hur- witz matrix, continuous piecewise linear differential system, discontinuous piecewise linear differential system. 1 2 JAUME LLIBRE AND XIANG ZHANG Kalman Conjecture. If S(α, β) is a sector of linear stability for the system (1) and ϕ : R → R is continuously differentiable satisfying ϕ0 (σ) ∈ (α, β) then the origin is a global attractor. We recall that an equilibrium point p is a global attractor for a differ- ential system defined in Rn if the ω–limit of any orbit of the differential system is p. See for instance [8] for the definition of ω–limit of an orbit. In 1988 Barabanov [2] proved the Kalman conjecture for n = 2, 3 and tried to provide a counterexample to Kalman conjecture in dimension 4 but his construction has some gaps. Bernat and Llibre [4] in 1996 pre- sented the first counterexample to that conjecture in dimension larger than 3. For a clear proof of Kalman conjecture for n = 3 see [7]. A C 1 differential system ẋ(t) = F (x(t)) defined in Rn is Hurwitz if the Jacobian matrix of F (x) is Hurwitz at every point x ∈ Rn . Consider a C 1 differential system ẋ(t) = F (x(t)) defined in Rn and having an equilibrium point at the origin of coordinates. If DF (0) is Hurwitz then by Hartman–Grobman Theorem [17] the origin is locally asymptotically stable. A natural question is: what hypotheses we have to add to the function F (x) in order to assure that the origin is a global attractor. Markus and Yamabe [23] in 1960 made the next conjecture. Markus–Yamabe conjecture. If we have a C 1 Hurwitz differential system x0 = F (x) defined in Rn and having a unique equilibrium point at the origin of coordinates, then the origin is a global attractor. The Markus-Yamabe conjecture for n = 1 follows easily. Some au- thors proved the Markus-Yamabe conjecture adding additional assump- tions. For instance, Markus and Yamabe in [23] proved the conjecture if F = (F1 , F2 ) and ∂Fi /∂xj = 0 for i 6= Rj, i, j ∈ {1, 2}. In 1963 the ∞ conjecture was proved by Olech in [25] if 0 min||x||=r ||F (x)||dr = ∞. Gasull, Llibre and Sotomayor in [14] provided a list with more than ten different additional sufficient conditions forcing the Markus–Yamabe conjecture for n = 2. Meistres and Olech [24] in 1988 proved the Markus-Yamabe conjecture for n = 2 when F1 and F2 are polynomials. Finally the conjecture was proved without additional hypotheses in independently by Gutierrez [18, 19] and by Fessler [9, 10]. Gutierrez had the proof in 1992 but it was published in 1993, and Fessler seems prove it in 1993, but it was published in 1995. In 1994 a simpler proof was provided by Glutsyuk [15, 16]. Also there are some additional assumptions forcing the Markus– Yamabe conjecture when n > 2. Thus in 1961 the conjecture was MARKUS–YAMABE CONJECTURE 3 proved by Hartman in [20] if DF (x) is negative definite for all x ∈ Rn . Other additional sufficient conditions, under which the conjecture hold- s, were given in 1962 by Hartamn and Olech in [21]. The counterexample to the Kalman conjecture for n > 3 given in [4], also is a counterexample to Markus–Yamabe conjecture for n > 3. Cima, van den Essen, Gasull, Hubbers and Mañosas provided a counterexample to the Markus–Yamabe conjecture for n = 3 given by a polynomial differential system. In summary, now we know that the Markus–Yamabe conjecture hold- s in R2 , but does not hold in Rn for all n > 2. We can say that the study of the continuous or discontinuous piece- wise linear differential systems started with Andronov, Vitt and Khaikin in [1]. After these systems became a topic of great interest in the math- ematical community due to their applications in many areas, because they are used for modeling real phenomena and different modern de- vices, see for instance the books [3, 26] and references quoted therein. Here we extend the Markus–Yamabe conjecture to continuous and discontinuous piecewise linear differential systems formed by two pieces of Rn separated by a hyperplane. More precisely, without loss of gen- erality we consider the following class of piecewise linear differential systems in Rn  + A x + b+ if x1 ≥ 0, (2) ẋ = A− x + b− if x1 ≤ 0. If A+ x + b+ = A− x + b− in all points x = (0, x2 , . . . , xn ), then we say that the differential system (2) is a continuous piecewise linear d- ifferential system; otherwise we say that it is a discontinuous piecewise linear differential system. We note that the dynamics of this discontin- uous differential system on the straight line of discontinuity is defined according with the definitions of the book of Filippov [11]. Since we will study the global stability of system (2) at an equilibrium point, it forces us to consider the case that the systems ẋ = A+ x + b+ and ẋ = A− x + b− both have a unique equilibrium point. So we will consider without loss of generality that A+ and A− are both invertible. We say that the linear differential system ẋ = A+ x + b+ has a real equilibrium point if the equilibrium point −(A+ )−1 b+ exists, and it is in the closed half-space {x1 ≥ 0}, otherwise that equilibrium point is called virtual. Similarly the linear differential system ẋ = A− x + b− has a real equilibrium point if the equilibrium point −(A− )−1 b− exists, 4 JAUME LLIBRE AND XIANG ZHANG and it is in the closed half-space {x1 ≤ 0}, otherwise that equilibrium point is called virtual. A Markus–Yamabe piecewise linear differential system is a differential system (2) such that the matrices A+ and A− are Hurwitz, and either only one of the systems ẋ = A+ x + b+ and ẋ = A− x + b− has a real equilibrium point, or both systems have the same real equilibrium point in {x1 = 0}. The objective of this paper is to answer the following question: De- termine the values of n ≥ 2 for which all continuous (respectively discontinuous) Markus–Yamabe piecewise linear differential systems in Rn have a global attractor in their real equilibrium point. That is, our goal is to consider the conjecture of Markus–Yamabe made for C 1 d- ifferential systems to continuous and discontinuous Markus–Yamabe piecewise linear differential systems in Rn . Our main results are the following two theorems, which characterize when the extension of the Markus–Yamabe conjecture holds or not for continuous and discontinuous Markus–Yamabe piecewise linear differ- ential systems in Rn . Theorem 1. The following statements hold. (a) The equilibrium point of all the continuous Markus–Yamabe piecewise linear differential systems in R2 is a global attractor. (b) For all n > 2 there are continuous Markus–Yamabe piecewise linear differential systems in Rn for which their equilibrium point is not a global attractor. Theorem 1 is proved in section 2. A big part of this proof uses results of the articles [6] and [12]. We note that the answer to the extended Markus–Yamabe conjecture for continuous piecewise linear differential systems is the same than for the C 1 differential systems. But this is not the case for the discontinu- ous piecewise linear differential systems as it is shown in the following result. Theorem 2. For all n ≥ 2 there are discontinuous Markus–Yamabe piecewise linear differential systems in Rn for which their equilibrium point is not a global attractor. Theorem 2 is proved in section 2. We note that for a very special class of discontinuous piecewise linear differential systems in R2 , different from the classes here studied, the MARKUS–YAMABE CONJECTURE 5 extension of the Markus–Yamabe conjecture to them holds, see Llibre and Teixeira [22]. 2. Proof of the results In order to prove Theorem 1 we shall use the canonical forms of the piecewise linear differential systems (2) provided in [13]. We consider the piecewise linear differential systems        ẋ 2` −1 x 0 (3) = 2 2 + , ẏ ` −α 0 y a defined in {x ≤ 0}, and        ẋ 2r −1 x b (4) = + , ẏ r2 − β 2 0 y c defined in {x ≥ 0}, where α, β ∈ {i, 0, 1}, being i is the imaginary unit. If α = i then the equilibrium point of system (3) has eigenvalues ` ± i, so it is a focus if ` 6= 0, and a center if ` = 0. If α = 0 then system (3) is a node with eigenvalue ` 6= 0 of multiplicity 2 whose linear part does not diagonalize. If α = 1 then system (3) is a saddle with eigenvalues ` − 1 and ` + 1 when |`| < 1, and a node with eigenvalues ` − 1 and ` + 1 whose linear part diagonalize when |`| > 1. Let U be an open subset of R2 . We say that the homeomorphism h between U and its image by h is a topological equivalence between the piecewise linear differential system (2) and the piecewise linear differential system (3)+(4) if h applies orbits of system (2) contained in U into orbits of system (3)+(4) contained in h(U ). From Propositions 1 and 2 of [13] it follows that there exists a topo- logical equivalence between the phase portrait of the piecewise linear differential system (2) and the phase portrait of the piecewise linear differential system (3)+(4) restricted to the orbits that do not have points in common with the sliding set of these systems. Therefore, since we are interested in studying when the unique equilibrium point of a Markus–Yamabe piecewise linear differential system (2) is a global attractor and the sliding set for our piecewise linear differential sys- tems is at most formed by one orbit we do not need to take care of the sliding set. For a definition of the sliding set see [11]. Since we shall use the canonical forms for studying the continuous piecewise linear differential systems, we can restricted the canonical 6 JAUME LLIBRE AND XIANG ZHANG forms (3)+(4) to the continuous canonical forms on x = 0, i.e. to the continuous piecewise linear differential systems        ẋ 2` −1 x 0 (5) = + , ẏ `2 − α 2 0 y a defined in {x ≤ 0}, and        ẋ 2r −1 x 0 (6) = + , ẏ r2 − β 2 0 y a defined in {x ≥ 0}. Proof of Theorem 1. Statement (a) of Theorem 1 when both linear dif- ferential systems of (2) have the same real equilibrium point in {x = 0} is proved in [12]. Now we shall prove statement (a) when only one of the two linear differential systems of (2) has a real equilibrium point, without loss of generality we can assume that this real equilibrium point is −(A+ )−1 b+ . So we can assume that the continuous piecewise linear differential systems (5)+(6) has a unique equilibrium point in the open half–plane {x > 0}, and we must prove that this equilibrium point is a global attractor. We claim that the straight line x = 0 is tranversal under the flow of the continuous piecewise linear differential systems (5)+(6) except at the origin. Recall that a line segment is transversal under the flow of systems (5)+(6) if it has no contact points with the flow of systems (5)+(6), and that a contact point p of x = 0 is a point where the vector field (2`x − y, (`2 − α2 )x + a) associated to systems (5)+(6) at p is parallel to the straight line x = 0. That is, if p = (0, y0 ) is a contact point of a system (5)+(6) with the straight line x = 0, then the inner product (2`x − y, (`2 − α2 )x + a) · (1, 0)|(x,y)=(0,y0 ) = −y0 . This means that system (5)+(6) has a unique contact point with the straight line x = 0, which is at the origin (0, 0). Since by assumptions the matrices of the continuous piecewise linear differential systems (5)+(6) are Hurwitz, the equilibrium point con- tained in the open half–plane {x > 0} is a local attractor. Therefore on the straight line x = 0 the flow of systems (5)+(6) enters in forward time at the half–plane {x > 0}. Hence, since systems (5)+(6) restrict- ed to {x > 0} is a linear differential system, all the orbits contained in {x ≥ 0} have ω–limit the equilibrium point contained in {x > 0}. MARKUS–YAMABE CONJECTURE 7 On the other hand, since systems (5)+(6) are Markus-Yamabe in the half–plane {x < 0} they do not have equilibria, the equilibrium point of a linear system (5) is virtual, i.e. it is in {x > 0} which is also a global attractor for the orbits of the linear system (5) in R2 , so all the orbits of a system (5)+(6) starting at a point of {x < 0} cross the straight line x = 0 and enter in the open half–plane {x > 0}, and consequently the equilibrium point contained in {x > 0} is a global attractor. This concludes the proof of statement (a). Statement (b) of Theorem 1 follows directly from Theorem 1 of [6]. More precisely, in [6] the authors proved the existence of continuous Markus–Yamabe piecewise linear differential systems in R3 having a unique unstable equilibrium point at the origin of coordinates. In fact, the origin has an one-dimensional stable manifold and a two- dimensional invariant manifold, which is an attractive cone, on which the dynamics can be of stable or unstable focus type, or of center type. Of course, these examples of continuous Markus–Yamabe piecewise linear differential systems in R3 for which the origin is unstable, can be extended to continuous Markus–Yamabe piecewise linear differential systems in Rn with n > 3 adding to the 3–dimensional differential system the equations ẋk = −xk for k = 4, . . . , n.  Proof of Theorem 2. It is sufficient to prove the theorem for n = 2, be- cause we can extend a discontinuous Markus–Yamabe piecewise linear differential system in R2 for which the unique equilibrium point of the system is not a global attractor, to a discontinuous Markus–Yamabe piecewise linear differential system in Rn with n > 2 for which its u- nique equilibrium point will not be a global attractor, adding to the 2–dimensional system the equations ẋk = −xk for k = 2, . . . , n. First we consider the discontinuous piecewise linear differential sys- tem in R2 defined by ẋ = y − 1, ẏ = 1 − x, in x ≤ 0, (7) ẋ = y, ẏ = 1 − x, in x ≥ 0. Note that this discontinuous piecewise linear differential system is formed by two linear centers. Moreover the orbits in the half-plane x ≤ 0 are contained in the arcs of the circles {(x − 1)2 + (y − 1)2 = r2 } ∩ {x ≤ 0} with r ≥ 1, while the orbits in the half-plane x ≥ 0 are contained in the arcs of the circles {(x − 1)2 + y 2 = r2 } ∩ {x ≥ 0} with r ≥ 0. Then the orbit of the discontinuous piecewise linear differential sys- tem (7) starting at the point (0, 2) crosses in forward time the straight 8 JAUME LLIBRE AND XIANG ZHANG line x = 0 by the first time at the point (0, −2), by the second time at the point (0, 4), by the third time at the point (0, −4), by the fourth time at the point (0, 6), ... So this orbit in forward time escapes to infin- ity spiraling intersecting the straight line x = 0 at the points (0, ±2k) for all k = 1, 2, . . . Now we perturb slightly the discontinuous piecewise linear differen- tial system (7) as follows ẋ = y − 1 − εx, ẏ = 1 − x, in x ≤ 0, (8) ẋ = y − εx, ẏ = 1 − x, in x ≥ 0, with ε > 0 sufficiently small. Note that the two matrices A+ and A− of system (8) are Hurwitz. So system (8) is a discontinuous Markus–Yamabe piecewise linear dif- ferential system having the unique real equilibrium point (1, ε), which is a stable focus of the right subsystem. Since ε > 0 is sufficiently small the orbit starting at the point (0, 2) of system (7) which escapes spiraling to infinity, now for system (8) continues escaping to infini- ty. Consequently the equilibrium point (1, ε) is not a global attractor. This completes the proof of the theorem.  Acknowledgements The first author is partially supported by the Ministerio de Ciencia, Innovación y Universidades, Agencia Estatal de Investigación grants MTM2016-77278-P (FEDER) and MDM-2014-0445, the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911. The second author is partially supported by National Natural Science Foundation of China grants 11671254 and 11871334. References [1] A. Andronov, A. Vitt and S. Khaikin, Theory of Oscillations, Pergamon Press, Oxford, 1966. [2] N.E. Barabanov, On the Kalman Problem, Sibirskii Matematicheskii Zhur- nal 29 (1988), 3–11. [3] M. Di Bernardo, C.J. Budd, A.R. 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[26] D.J.W. Simpson, Bifurcations in Piecewise–Smooth Continuous Systems, World Scientific Series on Nonlinear Science A, vol 69, World scientific, Sin- gapure, 2010. 1 Departament de Matemàtiques, Universitat Autònoma de Barcelon- a, 08193 Bellaterra, Barcelona, Catalonia, Spain E-mail address: jllibre@mat.uab.cat 2 School of Mathematical Sciences and MOE–LSC, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China E-mail address: xzhang@sjtu.edu.cn