Global attractor of a differentiable autonomous system on the plane

Nguyen Van Chau

Annales Polonici Mathematici (1995)

  • Volume: 62, Issue: 2, page 143-154
  • ISSN: 0066-2216

Abstract

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We study the structure of a differentiable autonomous system on the plane with non-positive divergence outside a bounded set. It is shown that under certain conditions such a system has a global attractor. The main result here can be seen as an improvement of the results of Olech and Meisters in [7,9] concerning the global asymptotic stability conjecture of Markus and Yamabe and the Jacobian Conjecture.

How to cite

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Nguyen Van Chau. "Global attractor of a differentiable autonomous system on the plane." Annales Polonici Mathematici 62.2 (1995): 143-154. <http://eudml.org/doc/262725>.

@article{NguyenVanChau1995,
abstract = {We study the structure of a differentiable autonomous system on the plane with non-positive divergence outside a bounded set. It is shown that under certain conditions such a system has a global attractor. The main result here can be seen as an improvement of the results of Olech and Meisters in [7,9] concerning the global asymptotic stability conjecture of Markus and Yamabe and the Jacobian Conjecture.},
author = {Nguyen Van Chau},
journal = {Annales Polonici Mathematici},
keywords = {Markus-Yamabe Conjecture; asymptotically stable; Jacobian Conjecture; global attractor; global asymptotic stability conjecture of Markus and Yamabe; Jacobian conjecture},
language = {eng},
number = {2},
pages = {143-154},
title = {Global attractor of a differentiable autonomous system on the plane},
url = {http://eudml.org/doc/262725},
volume = {62},
year = {1995},
}

TY - JOUR
AU - Nguyen Van Chau
TI - Global attractor of a differentiable autonomous system on the plane
JO - Annales Polonici Mathematici
PY - 1995
VL - 62
IS - 2
SP - 143
EP - 154
AB - We study the structure of a differentiable autonomous system on the plane with non-positive divergence outside a bounded set. It is shown that under certain conditions such a system has a global attractor. The main result here can be seen as an improvement of the results of Olech and Meisters in [7,9] concerning the global asymptotic stability conjecture of Markus and Yamabe and the Jacobian Conjecture.
LA - eng
KW - Markus-Yamabe Conjecture; asymptotically stable; Jacobian Conjecture; global attractor; global asymptotic stability conjecture of Markus and Yamabe; Jacobian conjecture
UR - http://eudml.org/doc/262725
ER -

References

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  1. [1] V. I. Arnold and Yu. S. Il'yashenko, Ordinary differential equations. I, in: Dynamical Systems I, Ordinary Differential Equations and Smooth Dynamical Systems, D. V. Anosov and V. I. Arnold (eds.), Springer, New York, 1988, 7-149. 
  2. [2] H. Bass, E. F. Connell and D. Wright, The Jacobian Conjecture, Bull. Amer. Math. Soc. 7 (1982), 287-330. Zbl0539.13012
  3. [3] A. Białynicki-Birula and M. Rosenlicht, Injective morphisms of algebraic varieties, Proc. Amer. Math. Soc. 13 (1962), 200-204. Zbl0107.14602
  4. [4] O. H. Keller, Ganze Cremona-Transformationen, Monatsh. Math. Phys. 47 (1939), 299-306. 
  5. [5] L. Markus and H. Yamabe, Global stability criteria for differential systems, Osaka Math. J. 12 (1960), 305-317. Zbl0096.28802
  6. [6] D. J. Newman, One-one polynomial maps, Proc. Amer. Math. Soc. 11 (1960), 867-870. Zbl0103.01102
  7. [7] C. Olech, On the global stability of an autonomous system on the plane, in: Contributions to Differential Equations 1 (1963), 389-400. 
  8. [8] C. Olech, Global phase-portrait of a plane autonomous system, Ann. Inst. Fourier (Grenoble) 14 (1964), 87-98. 
  9. [9] C. Olech and M. Meisters, 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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