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On the global attractivity and asymptotic stability for autonomous systems of differential equations on the plane


Author: László Hatvani
Journal: Proc. Amer. Math. Soc. 145 (2017), 1121-1129
MSC (2010): Primary 34D23; Secondary 34D20
DOI: https://doi.org/10.1090/proc/13213
Published electronically: November 29, 2016
MathSciNet review: 3589312
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Abstract: The autonomous system of differential equations

$\displaystyle x'=f(x),\qquad (x=(x_1,x_2)^T\in \mathbb{R}^2,\ f(x)=(f_1(x),f_2(x))^T),$

is considered, and sufficient conditions are given for the global attractivity of the unique equilibrium $ x=0$. This property means that all solutions tend to the origin as $ t\to \infty $. The two cases (a) $ \operatorname {div} f(x)\le 0$ ( $ x\in \mathbb{R}^2$) and (b) $ \operatorname {div} f(x)\ge 0$ ( $ x\in \mathbb{R}^2$) are treated, where $ \operatorname {div} f(x):=\partial f_1(x)/\partial x_1+\partial f_2(x)/\partial x_2$. Earlier results of N. N. Krasovskiĭ and C. Olech about case (a) are improved and generalized to case (b). Three types of assumptions are required: certain stability properties of the origin (local attractivity, stability), boundedness above in some sense for $ \operatorname {div} f(x)$, and assumptions that $ \vert f(x)\vert$ is not as small as $ \vert x\vert\to \infty $. The conditions of the second and third types are connected with each other.

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Additional Information

László Hatvani
Affiliation: Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, H-6720 Szeged, Hungary
Email: hatvani@math.u-szeged.hu

DOI: https://doi.org/10.1090/proc/13213
Keywords: Green Formula, limit sets, Poincar\'e-Bendixson Theorem, phase volume method
Received by editor(s): January 19, 2016
Received by editor(s) in revised form: March 2, 2016
Published electronically: November 29, 2016
Additional Notes: This work was supported by the Hungarian National Foundation for Scientific Research (OTKA 109782) and Analysis and Stochastics Research Group of the Hungarian Academy of Sciences.
Dedicated: Dedicated to the memory of Czesłav Olech
Communicated by: Yingfei Yi
Article copyright: © Copyright 2016 American Mathematical Society