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1.
Number of limit cycles for planar systems with invariant algebraic curves / Gasull, Armengol (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Giacomini, Hector (Centre National de la Recherche Scientifique. Université de Tours. Institut Denis Poisson)
For planar polynomials systems the existence of an invariant algebraic curve limits the number of limit cycles not contained in this curve. We present a general approach to prove non-existence of periodic orbits not contained in this given algebraic curve. [...]
2023 - 10.1007/s12346-023-00746-7
Qualitative theory of dynamical systems, Vol. 22, Issue 2 (June 2023) , art. 44  
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2.
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 - 10.1016/j.exmath.2021.06.001
Expositiones Mathematicae, Vol. 39, Issue 4 (December 2021) , p. 624-643  
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3.
23 p, 359.5 KB Effectiveness of the Bendixson-Dulac theorem / Gasull, Armengol (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Giacomini, Hector (Centre National de la Recherche Scientifique. Université de Tours. Institut Denis Poisson)
We illustrate with several new applications the power and elegance of the Bendixson-Dulac theorem to obtain upper bounds of the number of limit cycles for several families of planar vector fields. In some cases we propose to use a function related with the curvature of the orbits of the vector field as a Dulac function. [...]
2021 - 10.1016/j.jde.2021.10.011
Journal of differential equations, Vol. 305 (December 2021) , p. 347-367  
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4.
21 p, 392.8 KB A new approach for the study of limit cycles / García-Saldaña, Johanna Denise (Universidad Católica de la Santísima Concepción. Departamento de Matemática y Física Aplicadas (Chile)) ; Gasull, Armengol (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Giacomini, Hector (Centre national de la recherche scientifique (França). Institut Denis Poisson)
We prove that star-like limit cycles of any planar polynomial system can also be seen either as solutions defined on a given interval of a new associated planar non-autonomous polynomial system or as heteroclinic solutions of a 3-dimensional polynomial system. [...]
2020 - 10.1016/j.jde.2020.04.038
Journal of differential equations, Vol. 269, Issue 7 (September 2020) , p. 6269-6292  
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5.
27 p, 521.7 KB Some results on homoclinic and heteroclinic connections in planar systems / Gasull, Armengol (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Giacomini, Hector (Université de Tours. Laboratoire de Mathématiques et Physique Théorique (France)) ; Torregrosa, Joan (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Consider a family of planar systems depending on two parameters (n, b) and having at most one limit cycle. Assume that the limit cycle disappears at some homoclinic (or heteroclinic) connection when Φ(n, b) = 0. [...]
2010 - 10.1088/0951-7715/23/12/001
Nonlinearity, Vol. 23, Issue 12 (December 2010) , p. 2977-3001  
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6.
4 p, 235.4 KB Localizing limit cycles : from numeric to analytical results / Gasull, Armengol (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Giacomini, Hector (Université de Tours (França). Laboratoire de Mathemátiques et Physique Théorique) ; Grau, Maite (Universitat de Lleida. Departament de Matemàtica)
This note presents the results of [4]. It deals with the problem of location and existence of limit cycles for real planar polynomial differential systems. We provide a method to construct Poincaré-Bendixson regions by using transversal curves, that enables us to prove the existence of a limit cycle that has been numerically detected. [...]
Cham, etc : Springer Nature Switzerland AG, cop. 2018 (Trends in mathematics. Research perspectives CRM Barcelona ; 10) - 10.1007/978-3-030-01153-6_2
Extended abstracts Summer 2016, slow-fast systems and hysteresis, theory and applications, 2015, p. 17-21  
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7.
23 p, 585.1 KB Effective construction of Poincaré-Bendixson regions / Gasull, Armengol (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Giacomini, Hector (Université de Tours (França). Laboratoire de Mathemátiques et Physique Théorique) ; Grau, Maite (Universitat de Lleida. Departament de Matemàtica)
This paper deals with the problem of location and existence of limit cycles for real planar polynomial differential systems. We provide a method to construct Poincaré--Bendixson regions by using transversal curves, that enables us to prove the existence of a limit cycle that has been numerically detected. [...]
2017 - 10.11948/2017094
Journal of applied analysis and computation, Vol. 7 Núm. 4 (2017) , p. 1549-1569  
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8.
6 p, 556.2 KB Analytic nilpotent centers as limits of nondegenerate centers revisited / García, Isaac (Universitat de Lleida. Departament de Matemàtica) ; Giacomini, Hector (Université de Tours(France). Laboratoire de Mathématiques et Physique Théorique) ; Giné, Jaume (Universitat de Lleida. Departament de Matemàtica) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
We prove that all the nilpotent centers of planar analytic differential systems are limit of centers with purely imaginary eigenvalues, and consequently the Poincaré-Liapunov method to detect centers with purely imaginary eigenvalues can be used to detect nilpotent centers.
2016 - 10.1016/j.jmaa.2016.04.046
Journal of mathematical analysis and applications, Vol. 441 (2016) , p. 893-899  
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9.
27 p, 1007.0 KB Bifurcation diagram and stability for a one-parameter family of planar vector fields / García-Saldaña, Johanna Denise (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Gasull, Armengol (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Giacomini, Hector (Université de Tours(France). Laboratoire de Mathématiques et Physique Théorique)
We consider the 1-parameter family of planar quintic systems, ˙x = y3−x3, y˙ = −x + my5, introduced by A. Bacciotti in 1985. It is known that it has at most one limit cycle and that it can exist only when the parameter m is in (0. [...]
2014 - 10.1016/j.jmaa.2013.11.047
Journal of mathematical analysis and applications, Vol. 413 (2014) , p. 321-342  
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10.
14 p, 421.0 KB Explicit upper and lower bounds for the traveling wave solutions of Fisher-Kolmogorov type equations / Gasull, Armengol (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Torregrosa, Joan (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Giacomini, Hector (Université de Tours(France). Laboratoire de Mathématiques et Physique Théorique)
It is well-known that the existence of traveling wave solutions for reaction-diffusion partial differential equations can be proved by showing the existence of certain heteroclinic orbits for related autonomous planar differential equations. [...]
2013 - 10.3934/dcds.2013.33.3567
Discrete and continuous dynamical systems. Series A, Vol. 33 Núm. 8 (2013) , p. 3567-3582  
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