A weakened Markus-Yamabe condition for planar polynomial differential systems of degree

Jaume Llibre; Clàudia Valls

doi:10.1017/S0013091523000615

A weakened Markus-Yamabe condition for planar polynomial differential systems of degree

Jaume Llibre, Clàudia Valls

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Abstract

For a general autonomous planar polynomial differential system, it is difficult to find conditions that are easy to verify and which guarantee global asymptotic stability, weakening the Markus-Yamabe condition. In this paper, we provide three conditions that guarantee the global asymptotic stability for polynomial differential systems of the form, where f1 has degree one, f2 has degree and has degree one in the variable y. As a consequence, we provide sufficient conditions, weaker than the Markus-Yamabe conditions that guarantee the global asymptotic stability for any generalized Liénard polynomial differential system of the form, with g1 and g2 polynomials of degrees n and m, respectively.

Original language	English
Pages (from-to)	1110-1118
Number of pages	9
Journal	Proceedings of the Edinburgh Mathematical Society
Volume	66
Issue number	4
DOIs	https://doi.org/10.1017/S0013091523000615
Publication status	Published - 2023

Keywords

Global asymptotic stability
Markus-Yamabe conjecture
Planar polynomial vector fields

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10.1017/S0013091523000615

https://ddd.uab.cat/record/291528

STUDY OF CONTINUOUS AND DISCRETE DYNAMIC SYSTEMS WITH EMPHASIS ON THEIR BIFURCATIONS, PERIODIC ORBITS AND INTEGRABILITY
Llibre Salo, J. (Principal Investigator), Torregrosa i Arús, J. (Co-Investigador/a Principal), Cufi Cabre, C. (Collaborator), Carvalho, Y. R. (Collaborator), Cousillas, G. (Collaborator), Diz Pita, É. (Collaborator), Ferragut Amengual, A. M. (Collaborator), GOUVEIA, L. F. D. S. (Collaborator), Iván Sánchez Sánchez (Collaborator), Jiménez Ruíz, J. J. (Collaborator), RODERO, A. L. (Collaborator), Ramírez, V. (Collaborator), Artes Ferragud, J. C. (Investigator), Caubergh , M. M. (Investigator), Cima Mollet, A. M. (Investigator), Corbera Subirana, M. (Investigator), Cors Iglesias, J. M. (Investigator), Gasull Embid, A. (Investigator), Pantazi ., C. (Investigator), Soler Villanueva, J. (Investigator), Garrido Pelaez, J. M. (Collaborator) & Valls Anglés, C. (Collaborator)
1/06/20 → 29/02/24
Project: Research Projects and Other Grants

Cite this

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title = "A weakened Markus-Yamabe condition for planar polynomial differential systems of degree",

abstract = "For a general autonomous planar polynomial differential system, it is difficult to find conditions that are easy to verify and which guarantee global asymptotic stability, weakening the Markus-Yamabe condition. In this paper, we provide three conditions that guarantee the global asymptotic stability for polynomial differential systems of the form, where f1 has degree one, f2 has degree and has degree one in the variable y. As a consequence, we provide sufficient conditions, weaker than the Markus-Yamabe conditions that guarantee the global asymptotic stability for any generalized Li{\'e}nard polynomial differential system of the form, with g1 and g2 polynomials of degrees n and m, respectively.",

keywords = "Global asymptotic stability, Markus-Yamabe conjecture, Planar polynomial vector fields",

author = "Jaume Llibre and Cl{\`a}udia Valls",

year = "2023",

doi = "10.1017/S0013091523000615",

language = "English",

volume = "66",

pages = "1110--1118",

journal = "Proceedings of the Edinburgh Mathematical Society",

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T1 - A weakened Markus-Yamabe condition for planar polynomial differential systems of degree

AU - Llibre, Jaume

AU - Valls, Clàudia

PY - 2023

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N2 - For a general autonomous planar polynomial differential system, it is difficult to find conditions that are easy to verify and which guarantee global asymptotic stability, weakening the Markus-Yamabe condition. In this paper, we provide three conditions that guarantee the global asymptotic stability for polynomial differential systems of the form, where f1 has degree one, f2 has degree and has degree one in the variable y. As a consequence, we provide sufficient conditions, weaker than the Markus-Yamabe conditions that guarantee the global asymptotic stability for any generalized Liénard polynomial differential system of the form, with g1 and g2 polynomials of degrees n and m, respectively.

AB - For a general autonomous planar polynomial differential system, it is difficult to find conditions that are easy to verify and which guarantee global asymptotic stability, weakening the Markus-Yamabe condition. In this paper, we provide three conditions that guarantee the global asymptotic stability for polynomial differential systems of the form, where f1 has degree one, f2 has degree and has degree one in the variable y. As a consequence, we provide sufficient conditions, weaker than the Markus-Yamabe conditions that guarantee the global asymptotic stability for any generalized Liénard polynomial differential system of the form, with g1 and g2 polynomials of degrees n and m, respectively.

KW - Global asymptotic stability

KW - Markus-Yamabe conjecture

KW - Planar polynomial vector fields

U2 - 10.1017/S0013091523000615

DO - 10.1017/S0013091523000615

M3 - Article

SN - 0013-0915

VL - 66

SP - 1110

EP - 1118

JO - Proceedings of the Edinburgh Mathematical Society

JF - Proceedings of the Edinburgh Mathematical Society

IS - 4

ER -

A weakened Markus-Yamabe condition for planar polynomial differential systems of degree

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Keywords

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STUDY OF CONTINUOUS AND DISCRETE DYNAMIC SYSTEMS WITH EMPHASIS ON THEIR BIFURCATIONS, PERIODIC ORBITS AND INTEGRABILITY

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