The following problem of Markus and Yamabe is answered affirmatively: Let f be a local diffeomorphism of the euclidean plane whose jacobian matrix has negative trace everywhere. If f(0) = 0, is it true that 0 is a global attractor of the ODE dx/dt = f(x)? An old result of Olech states that this is equivalent to the question if such an f is injective. Here the problem is treated in the latter form by means of an investigation of the behaviour of f near infinity.
@article{bwmeta1.element.bwnjournal-article-apmv62z1p45bwm,
author = {Robert Fe\ss ler},
title = {A proof of the two-dimensional Markus-Yamabe Stability Conjecture and a generalization},
journal = {Annales Polonici Mathematici},
volume = {62},
year = {1995},
pages = {45-74},
zbl = {0835.34052},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-apmv62z1p45bwm}
}
Robert Feßler. A proof of the two-dimensional Markus-Yamabe Stability Conjecture and a generalization. Annales Polonici Mathematici, Tome 62 (1995) pp. 45-74. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv62z1p45bwm/
[000] [Ba] N. E. Barabanov, On a problem of Kalman, Siberian Math. J. 29 (3) (1988), 333-341.
[001] [Fe] R. Feßler, A solution of the Global Asymptotic Stability Jacobian Conjecture and a generalization, in: M. Sabatini (ed.), Recent Results on the Global Asymptotic Stability Jacobian Conjecture, Proc. Povo, 1993, Dipartimento di Matematica, Università di Trento.
[002] [GLS] A. Gasull, J. Llibre and J. Sotomayor, Global asymptotic stability of differential equations in the plane, J. Differential Equations 91 (1991), 327-335. | Zbl 0732.34045
[003] [Gu] C. Gutierrez, A solution of the bidimensional Global Asymptotic Stability Jacobian Conjecture, in: M. Sabatini (ed.), Recent Results on the Global Asymptotic Stability Jacobian Conjecture, Proc. Povo, 1993, Dipartimento di Matematica, Università di Trento.
[004] [Ha] P. Hartman, On stability in the large for systems of ordinary differential equations, Canad. J. Math. 13 (1961), 480-492. | Zbl 0103.05901
[005] [HO] P. Hartman and C. Olech, On global asymptotic stability of differential equations, Trans. Amer. Math. Soc. 104 (1962), 154-178.
[006] [Kr] N. N. Krasovskiĭ, Some Problems of the Stability Theory of Motion, Gos. Izdat. Fiz.-Mat. Literat., Moscow, 1959 (in Russian).
[007] [MY] L. Markus and H. Yamabe, Global stability criteria for differential systems, Osaka J. Math. 12 (1960), 305-317. | Zbl 0096.28802
[008] [MO] G. Meisters and C. Olech, Solution of the global asymptotic stability jacobian conjecture for the polynomial case, in: Analyse Mathématique et Applications, Gauthier-Villars, Paris, 1988, 373-381. | Zbl 0668.34048
[009] [Ol] C. Olech, On the global stability of an autonomous system on the plane, Contributions to Differential Equations 1 (1963), 389-400.