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

Robert Feßler

Annales Polonici Mathematici (1995)

  • Volume: 62, Issue: 1, page 45-74
  • ISSN: 0066-2216

Abstract

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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.

How to cite

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Robert Feßler. "A proof of the two-dimensional Markus-Yamabe Stability Conjecture and a generalization." Annales Polonici Mathematici 62.1 (1995): 45-74. <http://eudml.org/doc/262348>.

@article{RobertFeßler1995,
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.},
author = {Robert Feßler},
journal = {Annales Polonici Mathematici},
keywords = {Markus-Yamabe conjecture; asymptotic behaviour of solutions of ODE's; immersions; embeddings; injectivity of mappings; curve lifting; foliations; two-dimensional Markus-Yamabe stability conjecture; global attractor},
language = {eng},
number = {1},
pages = {45-74},
title = {A proof of the two-dimensional Markus-Yamabe Stability Conjecture and a generalization},
url = {http://eudml.org/doc/262348},
volume = {62},
year = {1995},
}

TY - JOUR
AU - Robert Feßler
TI - A proof of the two-dimensional Markus-Yamabe Stability Conjecture and a generalization
JO - Annales Polonici Mathematici
PY - 1995
VL - 62
IS - 1
SP - 45
EP - 74
AB - 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.
LA - eng
KW - Markus-Yamabe conjecture; asymptotic behaviour of solutions of ODE's; immersions; embeddings; injectivity of mappings; curve lifting; foliations; two-dimensional Markus-Yamabe stability conjecture; global attractor
UR - http://eudml.org/doc/262348
ER -

References

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  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. Zbl0732.34045
  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. Zbl0103.05901
  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. Zbl0096.28802
  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. Zbl0668.34048
  10. [Ol] C. Olech, On the global stability of an autonomous system on the plane, Contributions to Differential Equations 1 (1963), 389-400. 

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