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

Annales Polonici Mathematici (1995)

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

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topRobert 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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- [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.
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- [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.
- [Ha] P. Hartman, On stability in the large for systems of ordinary differential equations, Canad. J. Math. 13 (1961), 480-492. Zbl0103.05901
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- [Kr] N. N. Krasovskiĭ, Some Problems of the Stability Theory of Motion, Gos. Izdat. Fiz.-Mat. Literat., Moscow, 1959 (in Russian).
- [MY] L. Markus and H. Yamabe, Global stability criteria for differential systems, Osaka J. Math. 12 (1960), 305-317. Zbl0096.28802
- [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
- [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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