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

• Autorzy: Robert Feßler
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

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

Kategorie tematyczne

• 34D45: Attractors
• 57R30: Foliations; geometric theory
• 34D05: Asymptotic properties
• 57R40: Embeddings
• 57R42: Immersions

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 1995
• Tom: 62
• Numer: 1
• Strony: 45-74

Daty

wydano

1995

otrzymano

1994-09-10

poprawiono

1994-12-12

Twórcy

autor

Robert Feßler

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

Bibliografia

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• [GLS] A. Gasull, J. Llibre and J. Sotomayor, Global asymptotic stability of differential equations in the plane, J. Differential Equations 91 (1991), 327-335.
• [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.
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• [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.
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