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1.
37 p, 678.9 KB Z2-symmetric planar polynomial Hamiltonian systems of degree 3 with nilpotent centers / Dias, Fabio Scalco (Universidade Federal de Itajubá. Instituto de Matemática e Computação (Brazil)) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Valls, Clàudia 1973- (Universidade de Lisboa. Instituto Superior Técnico. Departamento de Matemàtica)
We provide the normal forms and the global phase portraits in the Poincaré disk of all Z2-symmetric planar polynomial Hamiltonian systems of degree 3 having a nilpotent center at the origin.
2019
Electronic Journal of Differential Equations, Vol. 2019, Issue 82 (2019) , p. 1-29
2 documents
2.
29 p, 568.3 KB Global dynamics of Hořava-Lifshitz cosmology with non-zero curvature and a wide range of potentials / Gao, Fabao (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
The global dynamics of a cosmological model based on Hořava-Lifshitz gravity in the presence of curvature is described by using the qualitative theory of differential equations.
2020 - 10.1140/epjc/s10052-020-7714-3
European Physical Journal C, Vol. 80, Issue 2 (February 2020) , art. 137
2 documents
3.
Phase portraits of the quadratic polynomial Liénard differential systems / Gouveia, Márcio R.A. (Universidade Estadual Paulista. Instituto de Biociências, Letras e Ciências Exatas. Departamento de Matemática (Brazil)) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Roberto, Luci Any (Universidade Estadual Paulista. Instituto de Biociências, Letras e Ciências Exatas. Departamento de Matemática (Brazil))
We classify the global phase portraits in the Poincaré disc of the quadratic polynomial Liénard differential systems ˙ x = y, ˙ y = (ax + b)y + cx2 + dx + e, where (x,y) ∈R2 are the variables and a,b,c,d,e are real parameters.
2020 - 10.1017/prm.2020.10
Proceedings of the Royal Society of Edinburgh Section A: Mathematics, (March 2020)  
4.
Global phase portraits of ℤ2-symmetric planar polynomial Hamiltonian systems of degree three with a nilpotent saddle at the origin / Corbera Subirana, Montserrat (Universitat de Vic. Universitat Central de Catalunya) ; Valls, Clàudia 1973- (Universidade de Lisboa. Instituto Superior Técnico. Departamento de Matemàtica)
We characterize the phase portraits in the Poincaré disk of all planar polynomial Hamiltonian systems of degree three with a nilpotent saddle at the origin and ℤ2-symmetric with (x,y)→(-x,y).
2020 - 10.1142/S0218127420500066
International Journal of Bifurcation and Chaos, Vol. 30, Issue 1 (January 2020) , art. 2050006  
5.
Global qualitative dynamics of the Brusselator system / Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Valls, Clàudia 1973- (Universidade de Lisboa. Instituto Superior Técnico. Departamento de Matemática)
We study the dynamics of the Brusselator model by analysing the flow of this differential system in the Poincaré disc.
2020 - 10.1016/j.matcom.2019.01.017
Mathematics and computers in simulation, Vol. 170 (April 2020) , p. 107-114  
6.
Global dynamics of the Hořava-Lifshitz cosmological system / Gao, Fabao (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Using the qualitative theory of the differential equations we describe the global dynamics of the cosmological model based on Hořava-Lifshitz gravity in a Friedmann-Lemaître-Robertson-Walker space time with zero curvature and without the cosmological constant term.
2019 - 10.1007/s10714-019-2635-x
General Relativity and Gravitation, Vol. 51, Issue 11 (November 2019) , art. 152  
7.
Global topological configurations of singularities for the whole family of quadratic differential systems / Artés, Joan Carles (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Schlomiuk, Dana (Université de Montréal. Département de Mathématiques et de Statistiques) ; Vulpe, Nicolae (Academy of Science of Moldova. Institute of Mathematics and Computer Science)
In Artés et al. (Geometric configurations of singularities of planar polynomial differential systems. A global classification in the quadratic case. Birkhäuser, Basel, 2019) the authors proved that there are 1765 different global geometrical configurations of singularities of quadratic differential systems in the plane. [...]
2020 - 10.1007/s12346-020-00372-7
Qualitative Theory of Dynamical Systems, Vol. 19, Issue 1 (April 2020) , art. 51  
8.
Global dynamics of a virus model with invariant algebraic surfaces / Dias, Fabio Scalco (Universidade Federal de Itajubá. Instituto de Matemática e Computação (Brazil)) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Valls, Clàudia 1973- (Universidade de Lisboa. Instituto Superior Técnico. Departamento de Matemàtica)
In this paper by using the Poincaré compactification in R3 we make a global analysis for the virus system x˙=λ-dx-βxz,y˙=-ay+βxz z˙=ky-μz with (x, y, z) ∈ R3, β> 0, λ, a, d, k and μ are nonnegative parameters due to their biological meaning. [...]
2019 - 10.1007/s12215-019-00417-0
Rendiconti del Circolo Matematico di Palermo, (May 2019)  
9.
17 p, 570.2 KB Normal forms and global phase portraits of quadratic and cubic integrable vector fields having two nonconcentric circles as invariant algebraic curves / Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Messias, Marcelo (Universidade Estadual Paulista. Departamento de Matemática e Computação (Brazil)) ; Reinol, Alisson C. (Universidade Estadual Paulista. Departamento de Matemática e Computação (Brazil))
In this paper, we give the normal form of all planar polynomial vector fields of degree d ≤ 3 having two nonconcentric circles C and C as invariant algebraic curves and the function H=C C , with α and β real values, as first integral. [...]
2017 - 10.1080/14689367.2016.1263600
Dynamical Systems, Vol. 32, Issue 3 (2017) , p. 374-390  
10.
15 p, 434.9 KB New Family of Centers of Planar Polynomial Differential Systems of Arbitrary Even Degree / Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Mousavi, Marzieh (Isfahan University of Technology. Department of Mathematical Sciences) ; Nabavi, Arefeh (Isfahan University of Technology. Department of Mathematical Sciences)
The problem of distinguishing between a focus and a center is one of the classical problems in the qualitative theory of planar differential systems. In this paper, we provide a new family of centers of polynomial differential systems of arbitrary even degree. [...]
2019 - 10.1007/s10883-019-09432-x
Journal of Dynamical and Control Systems, Vol. 25, issue 4 (Oct. 2019) , p. 619-630  

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