An extension to the planar Markus-Yamabe Jacobian conjecture

We extend the planar Markus-Yamabe Jacobian Conjecture to differential systems having jacobian matrix with eigenvalues with negative or zero real parts.


Introduction
LetẊ = F (X), X ∈ IR n , F ∈ C 1 (IR n , IR n ) be a first order differential system. Let us denote by J F (X) the jacobian matrix of F (X). If O is a critical point of (1) and the eigenvalues of J F (O) have negative real parts, then O is asymptotically stable [2]. In particular, all orbits starting close enough to O tend asymptotically to O.
In [7] the question was raised, whether J F (X) having eigenvalues with negative real parts for every X ∈ IR n imply O to be globally asymptotically stable, i. e. whether all orbits in IR 2 tend asymptotically to O. Such a problem was named Markus-Yamabe Jacobian Conjecture and several 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 global asymptotic stability of O is equivalent to the injectivity of F (X). Such a result led to study the problem applying methods previously used to study injectivity. The Markus-Yamabe Jacobian Conjecture was solved in the positive in [4,5,6] for planar systems, and was proved to have negative answer in higher dimensions [1,3]. The three approaches proposed in in [4,5,6] 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 of negative real parts. In fact, it is sufficient to assume that the Jacobian matrix has nowhere real positive eigenvalues.
Such general results did not lead to similarly general results in the study of the systems dynamics. This is likely due to the fact that accepting the possibility of eigenvalues with different real parts (positive, zero or negative) at different points of the plane does not allow to apply the procedure developped in [8] to establish the equivalence of injectiviy and global asymptotic stability. On the other hand, eigenvalues with zero real parts are compatible with asymptotic stability, even if not sufficient to imply it.
In this paper we assume J F (X) to be non-singular and have eigenvalues with non-positive real parts for all X ∈ IR 2 . Differently from the classical case, in this case a system does not necessarily have a globally asymptotically stable critical point. If a critical point exists, we prove that either such a system has a global center, or there exists a globally asymptotically stable compact set. We show by an example that such a global attractor is not necessarily a critical point. If the system is analytic the conclusion can be sharpened, proving that either there exists a global center, or a globally asymptotically stable critical point. Our results follow from Olech approach to global attractivity [8] and Fessler theorem about global injectivity [4].

Results
We consider maps F ∈ C 1 (IR 2 , IR 2 ), F (x, y) = (P (x, y), Q(x, y)). We denote partial derivatives by subscripts. Let In what follows we consider the differential system associated to F : We denote by φ(t, x, y) the local flow defined by (2). We say that a critical point O of (2) (2) is asymptotically stable if it is stable and attractive [2]. In this case we denote by A O its attraction region. If A O = IR 2 then O is said to be globally asymptotically stable.
In the proof of theorem 2 we repeatedly use F injectivity. We report here the theorem applied, proved in [4].
2) there is a compact set K ⊂ IR 2 such that J F (x, y) has no real positive eigenvalues for any (x, y) ∈ K.
Then F is injective.
For 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 change after a translation. We set and denote by T − its closure. We denote by µ the 2-dimensional Lebesgue measure. iii) If T (x, y) does not vanish identically, then there exists a globally asymptotically stable compact set M. Proof.
i.2) Vice-versa, assume T (x, y) to vanish identically on a neighbourhood U O of O. Then the system is Hamiltonian on a simply connected neighbourhood V O ⊂ U O . Let H(x, y) be its Hamiltonian function. One has The Hessian matrix of H(x, y) is The Hessian determinant is H xx H yy −H xy H yx = P x Q y −P y Q x = 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 neighbourhood of O then it vanishes on all of IR 2 , hence the system is Hamiltonian on all of IR 2 . We claim that N O is unbounded. In fact, let us assume by absurd N O is bounded, hence also ∂N O is bounded. By F 's injectivity [4], Hence the Poincaré map is strictly monotone, which implies either attractivity or repulsivity of ∂N O . Repulsivity is not compatible with the sign of the divergence, hence ∂N O is attractive, and N O is asymptotically stable. Its global attractivity can be 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 in the following differential system, for which one has J F (x, y) = 0 1 −1 −3y 2 .
Its determinant is 1 + α 2 (r) + rα(r)α ′ (r) > 0 and its trace is −2α(r) − 2rα ′ (r) ≤ 0. For r ≤ 1 the trace is zero, for r > 1 the trace is negative. The system (5) 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 a global attractor, sincė r < 0 for r > 1.