We extend the planar Markus–Yamabe Jacobian conjecture to differential systems having Jacobian matrix with eigenvalues with negative or zero real parts

Sabatini Marco

English

University of Szeged

Electronic Journal of Qualitative Theory of Differential Equations2021, No. 80, 1–6; https://doi.org/10.14232/ejqtde.2021.1.80 www.math.u-szeged.hu/ejqtde/An extension to the planarMarkus–Yamabe Jacobian conjectureMarco SabatiniBUniversità di Trento, Via Sommarive, 14, I-38123 Povo, Trento (TN), ItalyReceived 30 June 2021, appeared 13 October 2021Communicated by Armengol GasullAbstract. We extend the planar Markus–Yamabe Jacobian conjecture to differentialsystems having Jacobian matrix with eigenvalues with negative or zero real parts.Keywords: Markus–Yamabe, Jacobian conjecture, global asymptotic stability, globalcenter.2020 Mathematics Subject Classification: 34D23, 34D45.1 IntroductionLetẊ = F(X), X ∈ IRn, F ∈ C1(IRn, IRn) (1.1)be a first order differential system. Let us denote by JF(X) the Jacobian matrix of F(X). If O isa critical point of (1.1) and the eigenvalues of JF(O) have negative real parts, then O is asymp-totically stable [2]. In particular, all orbits starting close enough to O tend asymptotically toO.In [7] the question was raised, whether JF(X) having eigenvalues with negative real partsfor every X ∈ IRn imply O to be globally asymptotically stable, i. e. whether all orbits in IRntend asymptotically to O. Such a problem was named Markus–Yamabe Jacobian conjecture andseveral results were obtained under various additional hypotheses. A key step was made in[8], where it was proved that under Markus–Yamabe hypotheses, for planar systems the globalasymptotic stability of O is equivalent to the injectivity of F(X). Such a result led to studythe problem applying methods previously used to study injectivity. The Markus–YamabeJacobian conjecture was solved in the positive in [4–6] for planar systems, and was proved tohave negative answer in higher dimensions [1, 3]. The three approaches proposed in in [4–6]first prove the injectivity of F(X), then as a consequence get the global asymptotic stability.Actually, in all such papers injectivity is proved under much weaker hypotheses than that ofnegative real parts. In fact, it is sufficient to assume that the Jacobian matrix has nowhere realpositive eigenvalues.BEmail: marco.sabatini@unitn.it2 M. SabatiniSuch general results did not lead to similarly general results in the study of the systemsdynamics. This is likely due to the fact that accepting the possibility of eigenvalues withdifferent real parts (positive, zero or negative) at different points of the plane does not allowto apply the procedure developed in [8] to establish the equivalence of injectiviy and globalasymptotic stability. On the other hand, eigenvalues with zero real parts are compatible withasymptotic stability, even if not sufficient to imply it.In this paper we assume JF(X) to be non-singular and have eigenvalues with non-positivereal parts for all X ∈ IR2. Differently from the classical case, in this case a system does notnecessarily have a globally asymptotically stable critical point. If a critical point exists, weprove that either such a system has a global center, or there exists a globally asymptoticallystable compact set. We show by an example that such a global attractor is not necessarily acritical point. If the system is analytic the conclusion can be sharpened, proving that eitherthere exists a global center, or a globally asymptotically stable critical point. Our results followfrom Olech approach to global attractivity [8] and Fessler theorem about global injectivity [4].2 ResultsWe consider maps F ∈ C1(IR2, IR2), F(x, y) = (P(x, y), Q(x, y)). We denote partial derivativesby subscripts. LetJF(x, y) =(Px(x, y) Py(x, y)Qx(x, y) Qy(x, y)).be the Jacobian matrix of F at (x, y). We denote by D(x, y) = det JF(x, y) = Px(x, y)Qy(x, y)−Py(x, y)Qx(x, y) its determinant and by T(x, y) = Px(x, y) + Qy(x, y) its trace. T(x, y) is thedivergence of the vector field F(x, y).In what follows we consider the differential system associated to F:{ẋ = P(x, y),ẏ = Q(x, y).(2.1)We denote by φ(t, x, y) the local flow defined by (2.1). We say that a critical point O of (2.1) isa center if it has a punctured neighbourhood filled with non-trivial cycles surrounding O. Thelargest connected set NO filled with such cycles is called period annulus of O. If NO = IR2 \ {O},then O is said to be a global center. We say that a critical point O of (2.1) is asymptotically stableif it is stable and attractive [2]. In this case we denote by AO its attraction region. If AO = IR2then O is said to be globally asymptotically stable.In the proof of Theorem 2.2 we repeatedly use F injectivity. We report here the theoremapplied, proved in [4].Theorem 2.1. Let F ∈ C1(IR2, IR2) be such that:1) D(x, y) > 0 for all (x, y) ∈ IR2;2) there is a compact set K ⊂ IR2 such that JF(x, y) has no real positive eigenvalues for any(x, y) 6∈ K.Then F is injective.An extension to Markus–Yamabe Jacobian conjecture 3For the sake of simplicity, without loss of generality from now on we assume O = (0, 0).The hypotheses we consider rely only on derivatives properties, hence they do not changeafter a translation. We setT− = {(x, y) : T(x, y) < 0},and denote by T− its closure. We denote by µ the 2-dimensional Lebesgue measure.Theorem 2.2. Assume D(x, y) > 0 and T(x, y) ≤ 0 for all (x, y) ∈ IR2. Let O be a critical point of(2.1). Then:i) O is a center if and only if it has a neighbourhood UO such that T(x, y) vanishes identically onUO; in such a case (2.1) is Hamiltonian on all of NO; if, additionally, F is analytic, then thesystem is Hamiltonian and O is a global center.ii) O is asymptotically stable if and only if it belongs to T−; in such a case O is globally asymptoti-cally stable.iii) If T(x, y) does not vanish identically, then there exists a globally asymptotically stable compactset M.Proof. i.1) We claim that if O is a center, then T(x, y) vanishes identically on NO. By absurd,assume T(x∗, y∗) < 0 for some (x∗, y∗) ∈ NO. By continuity there exists a neighbourhoodU∗ of (x∗, y∗) such that T(x, y) < 0 for all (x, y) ∈ U∗. Let γ∗ be the cycle passing through(x∗, y∗) and ∆∗ the bounded planar region having γ∗ as boundary. ∆∗ is invariant, henceµ(∆∗) = µ(φ(t, ∆∗)) for all t ∈ IR. By Liouville’s theorem one has0 =ddtµ(φ(t, ∆∗)) =∫φ(t,∆∗)T(x, y) dx dy < 0,because T(x, y) < 0 on φ (t, ∆∗ ∩U∗), contradiction.i.2) Vice-versa, assume T(x, y) to vanish identically on a neighbourhood UO of O. Thenthe system is Hamiltonian on a simply connected neighbourhood VO ⊂ UO. Let H(x, y) be itsHamiltonian function. One hasThe Hessian matrix of H(x, y) isJF(x, y) =(Hxx HxyHyx Hyy)=(Qx Qy−Px − Py). (2.2)The Hessian determinant is Hxx Hyy−HxyHyx = PxQy− PyQx = D(x, y) > 0, hence H(x, y)has a minimum at O. As a consequence, O is a center.i.3) If additionally F is analytic, then also T(x, y) is analytic. If it vanishes in a neigh-bourhood of O then it vanishes on all of IR2, hence the system is Hamiltonian on all of IR2.We claim that NO is unbounded. In fact, let us assume by absurd NO is bounded, hencealso ∂NO is bounded. By F injectivity [4], ∂NO contains no critical points, hence by Poincaré–Bendixson theorem ∂NO is a non-trivial cycle. One can consider the Poincaré map defined ona section Σ of ∂NO. Such a map is analytic and coincides with the identity map on Σ ∩ NO,hence it coincides with the identity map on all of Σ. As a consequence every orbit meetingΣ ∩ ∂NO is a cycle, hence ∂NO is contained in the period annulus, contradicting the fact thatit is the boundary of NO. Moreover, every connected components of ∂NO is unbounded. Infact, if a connected components of ∂NO was bounded, then by its invariance and by Poincaré–Bendixson theorem either it would be a cycle or it would contain a critical point. The formercase has already been considered above, the latter one can be excluded by the injectivity of F.4 M. Sabatinii.4) In order to prove that O is a global center we use again the injectivity of F. For ε > 0let Bε be the open disk of radius ε > 0 centered at O. F is a diffeomorphism, hence the anti-image Dε = F−1 (Bε) is an open neighbourhood of O. By construction and by the injectivityof F, Dε contains all the points of (x, y) ∈ IR2 such that |F(x, y)| < ε, hence for all (x, y) 6∈ Dεone has |F(x, y)| ≥ ε. Let us choose ε small enough such that ∂NO ∩ Dε = ∅. Let ∂NuO be anunbounded component of ∂NO. Then working as in [8], since T(x, y) ≤ 0 and |F(x, y)| ≥ εoutside Dε, one proves that every orbit starting close enough to ∂NuO is unbounded too, henceit is not a cycle, contradicting the fact that ∂NuO is in the boundary of NO. As a consequence∂NO = ∅ and NO = IR2 \ {O}.ii) Assume O to be asymptotically stable and AO its region of attraction. By hypothesis,in every neighbourhood of O there are points such that T(x, y) < 0, and by continuity thisoccurs in an open subset of AO. If by absurd AO is bounded, then by its invariance, for all t0 =ddtµ(φ(t, AO)) =∫φ(t,AO)T(x, y) dx dy < 0,contradiction. Hence AO is unbounded. Assume by absurd there exists a bounded connectedcomponent ∂AbO of ∂AO. As above, by Poincaré–Bendixson theorem either it is a cycle orcontains a critical point. If it is a cycle, it cannot surround O, since in such a case AO would bebounded. Hence it surrounds another critical point, violating F injectivity. The same violationwould occur if ∂AbO contained a critical point. Then the argument proceeds as in point i.4),showing that ∂AO = ∅ and AO = IR2.Vice-versa, assume O ∈ T−. Then T(x, y) does not vanish identically on any neighbour-hood UO of O, hence by point i) it is not a center. By the hypotheses on D(0, 0) and T(0, 0), Ois a non degenerate elementary critical point of center-focus type, according to the real part ofits eigenvalues. If such real parts are negative O is a focus, hence asymptotically stable. If suchreal parts are zero, one proves, as at the beginning of point i), that O cannot be accumulationpoint of cycles, hence it is asymptotically stable. Working as in point i.4) one proves that it isglobally asymptotically stable.iii) If O ∈ T−, then point ii) applies and one can take M = {O}.If O 6∈ T−, it has a neighbourhood UO where T(x, y) vanishes identically, hence it is acenter. We claim that NO is bounded. In fact, if NO is unbounded one can proceed as inpoint i.4), in order to prove that every orbit starting close enough to ∂NuO is unbounded,contradicting the fact that ∂NuO is part of the boundary. The boundedness of NO implies theboundedness of ∂NO, which is a cycle, by the absence of critical points on ∂NuO. Let us considera section Σ of ∂NO and its Poincaré map. Such a map is the identity on Σ ∩ NO, and has nofixed points on Σ \ NO, otherwise there would be a cycle γ containing ∂NO, T(x, y) wouldvanish identically inside γ and every orbit inside γ would be a cycle, contradicting the factthat ∂NO is the boundary of NO. Hence the Poincaré map is strictly monotone, which implieseither attractivity or repulsivity of ∂NO. Repulsivity is not compatible with the sign of thedivergence, hence ∂NO is attractive, and NO is asymptotically stable. Its global attractivity canbe proved as in i.4) and ii), proving that the boundary of its region of attraction is empty.An example of globally asymptotically stable critical point belonging to T− is the origin inthe following differential system, {ẋ = y,ẏ = −x− y3,(2.3)An extension to Markus–Yamabe Jacobian conjecture 5for which one hasJF(x, y) =(0 1−1 −3y2).One has D(x, y) = 1, T(x, y) = −3y2 ≤ 0, hence T− is x-axis.If (2.1) is not analytic, then a center need not be global. We construct a system satisfying thehypotheses of Theorem 2.2, having a non-global center and a globally asymptotically stablecompact set. Let α ∈ C∞(IR, IR) be such thatα(r) = 0, r ≤ 1,α(r) > 0, r > 1,α′(r) > 0, r > 1.Let us set r =√x2 + y2. The vector field defined by the system{ẋ = y− x α(r),ẏ = −x− y α(r).(2.4)Setting cr = x, sr = y, the Jacobian matrix of the vector field isJF(x, y) =(−α(r)− xcα′(r) 1− xsα′(r)−1− ycα′(r) − α(r)− ysα′(r)).Its determinant is 1+ α2(r) + rα(r)α′(r) > 0 and its trace is −2α(r)− 2rα′(r) ≤ 0. For r ≤ 1the trace is zero, for r > 1 the trace is negative. The system (2.4) is Hamiltonian for r ≤ 1,with a center at O whose central region is the disk of radius 1 centered at O. Such a disk is aglobal attractor, since ṙ < 0 for r > 1.References[1] J. Bernat, J. Llibre, Counterexample to Kalman and Markus–Yamabe conjectures in di-mension larger than 3, Dynam. Contin. Discrete Impuls. Systems 2(1996), No. 3, 337–379.MR1411771; Zbl 0889.34047[2] N. P. Bhatia, G. P. Szegö, Stability theory of dynamical systems, Die Grundlehren dermathematischen Wissenschaften, Band 161, Springer-Verlag, New York–Berlin, 1970.MR0289890; Zbl 0213.10904[3] A. Cima, A. van den Essen, A. Gasull , E. Hubbers, F. Mañosas, A polynomial coun-terexample to the Markus–Yamabe conjecture, Adv. Math. 131(1997), No. 2, 453–457.https://doi.org/10.1006/aima.1997.1673; MR1483974; Zbl 0896.34042[4] R. Fessler, A proof of the two-dimensional Markus–Yamabe stability conjecture anda generalization, Ann. Pol. Math. 62(1995), No. 1, 45–74. https://doi.org/10.4064/ap-62-1-45-74; MR1348217; Zbl 0835.34052[5] A. A. Glutsyuk, Complete solution of the Jacobian problem for planar vector fields(in Russian), Uspekhi Mat. Nauk 49(1994), No. 3, 179–180, translation in Russian Math.Surveys 49(1994), No. 3, 185–186. https://doi.org/10.1070/RM1994v049n03ABEH002262;MR1289394; Zbl 0828.340416 M. Sabatini[6] C. Gutiérrez, A solution to the bidimensional global asymptotic stability conjecture,Ann. Inst. H. Poincaré Anal. Non Linéaire 12(1995), No. 6, 627–671. https://doi.org/10.1016/S0294-1449(16)30147-0; MR1360540; Zbl 0837.34057[7] L. Markus, H. Yamabe, Global stability criteria for differential systems, Osaka Math. 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An extension to the planar Markus-Yamabe Jacobian conjecture

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An extension to the planar Markus-Yamabe Jacobian conjecture

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