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ArticleOriginal scientific text

Title

A proof of the two-dimensional Markus-Yamabe Stability Conjecture and a generalization

Authors 1

Affiliations

  1. Mathematisches Institut der Universität Basel, Rheinsprung 21, CH-4051 Basel, Switzerland

Abstract

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.

Keywords

ENG
Markus-Yamabe conjecture, asymptotic behaviour of solutions of ODE's, immersions, embeddings, injectivity of mappings, curve lifting, foliations

Bibliography

  1. [Ba] N. E. Barabanov, On a problem of Kalman, Siberian Math. J. 29 (3) (1988), 333-341.
  2. [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.
  3. [GLS] A. Gasull, J. Llibre and J. Sotomayor, Global asymptotic stability of differential equations in the plane, J. Differential Equations 91 (1991), 327-335.
  4. [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.
  5. [Ha] P. Hartman, On stability in the large for systems of ordinary differential equations, Canad. J. Math. 13 (1961), 480-492.
  6. [HO] P. Hartman and C. Olech, On global asymptotic stability of differential equations, Trans. Amer. Math. Soc. 104 (1962), 154-178.
  7. [Kr] N. N. Krasovskiĭ, Some Problems of the Stability Theory of Motion, Gos. Izdat. Fiz.-Mat. Literat., Moscow, 1959 (in Russian).
  8. [MY] L. Markus and H. Yamabe, Global stability criteria for differential systems, Osaka J. Math. 12 (1960), 305-317.
  9. [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.
  10. [Ol] C. Olech, On the global stability of an autonomous system on the plane, Contributions to Differential Equations 1 (1963), 389-400.
Annales Polonici Mathematici
1995/62/1
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Pages:
45-74
Main language of publication
English
Received
1994-09-10
Accepted
1994-12-12
Published
1995
Exact and natural sciences