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35 p, 627.1 KB

Criteria on the existence of limit cycles in planar polynomial differential systems / Giné, Jaume (Universitat de Lleida. Departament de Matemàtica) ; Grau, Maite (Universitat de Lleida. Departament de Matemàtica) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
We summarize known criteria for the non-existence, existence and on the number of limit cycles of autonomous real planar polynomial differential systems, and also provide new results. We give examples of systems which realize the maximum number of limit cycles provided by each criterion. [...]
2022 - DOI (Digital Object Identifier)

10.1016/j.exmath.2022.09.002
Expositiones Mathematicae, Vol. 40, Issue 4 (December 2022) , p. 1049-1083

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18 p, 333.3 KB

Solving polynomials with ordinary differential equations / Gasull, Armengol (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Giacomini, Hector (Université de Tours. Institut Denis Poisson)
In this work we consider a given root of a family of n-degree polynomials as a one-variable function that depends only on the independent term. Then we prove that this function satisfies several ordinary differential equations (ODE). [...]
2021 - DOI (Digital Object Identifier)

10.1016/j.exmath.2021.06.001
Expositiones Mathematicae, Vol. 39, Issue 4 (December 2021) , p. 624-643

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29 p, 1.1 MB

Phase portraits of separable quadratic systems and a bibliographical survey on quadratic systems / Li, Tao (Sichuan University. Department of Mathematics) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Although planar quadratic differential systems and their applications have been studied in more than one thousand papers, we still have no complete understanding of these systems. In this paper we have two objectives. [...]
2021 - DOI (Digital Object Identifier)

10.1016/j.exmath.2021.01.003
Expositiones Mathematicae, Vol. 39, Issue 4 (December 2021) , p. 540-565

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18 p, 798.2 KB

A survey on algebraic and explicit non-algebraic limit cycles in planar differential systems / Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Zhang, X. (Shanghai Jiao Tong University. School of Mathematical Sciences and MOE-LSC (People's Republic of China))
In the qualitative theory of differential equations in the plane one of the most difficult objects to study is the existence of limit cycles. There are many papers dedicated to this subject. Here we will present a survey mainly dedicated to the algebraic and explicit non-algebraic limit cycles of the polynomial differential systems in R and of the discontinuous piecewise differential systems in R formed by two linear differential systems separated by a straight line. [...]
2021 - DOI (Digital Object Identifier)

10.1016/j.exmath.2020.03.001
Expositiones Mathematicae, Vol. 39, Issue 1 (March 2021) , p. 48-61

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17 p, 402.2 KB

Limit cycles of the classical Liénard differential systems : A survey on the Lins Neto, de Melo and Pugh's conjecture / Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Zhang, Xiang (Shanghai Jiao Tong University. School of Mathematical Sciences)
In 1977 Lins Neto et al. (1977) conjectured that the classical Liénard system ẋ=y−F(x),ẏ=−x with F(x) a real polynomial of degree n, has at most [(n−1)/2] limit cycles, where [⋅] denotes the integer part function. [...]
2017 - DOI (Digital Object Identifier)

10.1016/j.exmath.2016.12.001
Expositiones Mathematicae, Vol. 35, Issue 3 (September 2017) , p. 286-299

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