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1.
Phase portraits of the Higgins-Selkov system / Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Mousavi, Marzieh (Isfahan University of Technology. Department of Mathematical Sciences (Iran))
In this paper we study the dynamics of the Higgins-Selkov system x˙=1−xyγ,y˙=αy(xyγ−1−1), where α is a real parameter and γ > 1 is an integer. We classify the phase portraits of this system for γ = 3,4,5,6, in the Poincaré disc for all the values of the parameter α. [...]
2021 - 10.3934/dcdsb.2021039
Discrete and continuous dynamical systems. Series B, Vol. 26 (2021)  
2.
Dynamics of the FitzHugh-Nagumo system having invariant algebraic surfaces / Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Tian, Yuzhou (Sun Yat-sen University. School of Mathematics (China))
In this paper, we study the dynamics of the FitzHugh-Nagumo system x˙=z,y˙=b(x-dy),z˙=x(x-1)(x-a)+y+cz having invariant algebraic surfaces. This system has four different types of invariant algebraic surfaces. [...]
2021 - 10.1007/s00033-020-01450-1
Zeitschrift fur Angewandte Mathematik und Physik, Vol. 72, Issue 1 (February 2021) , art. 15  
3.
17 p, 725.3 KB On the global dynamics of a three-dimensional forced-damped differential system / Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Martínez, Y. Paulina (Centre de Recerca Matemàtica) ; Valls, Clàudia 1973- (Universidade Técnica de Lisboa. Instituto Superior Técnico. Departamento de Matemática (Portugal))
In this paper by using the Poincaré compactification of ℝ³ we make a global analysis of the model xʹ = ax + y + yz, yʹ = x - ay + bxz, zʹ = cz - bxy. In particular we give the complete description of its dynamics on the infinity sphere. [...]
2020 - 10.1080/14029251.2020.1757232
Journal of Nonlinear Mathematical Physics, Vol. 27, Issue 3 (May 2020) , p. 414-428  
4.
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
5.
Bounded polynomial vector fields in R2 and Rn / Barreira, Luis (Universidade de Lisboa. Instituto Superior Técnico. Departamento de Matemática) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Valls, Clàudia 1973- (Universidade de Lisboa. Instituto Superior Técnico. Departamento de Matemática)
We characterize the bounded polynomial vector fields in R2. Additionally we provide a necessary condition but not sufficient which must be satisfied by bounded polynomial vector fields in Rn.
2020 - 10.1016/j.jde.2019.10.023
Journal of differential equations, Vol. 268, Issue 8 (April 2020) , p. 4416-4422  
6.
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
The European Physical Journal. C, Particles and Fields, Vol. 80, Issue 2 (February 2020) , art. 137
2 documents
7.
17 p, 345.5 KB Phase portraits of the quadratic polynomial Liénard differential systems / Gouveia, Marcio 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)  
8.
47 p, 886.5 KB 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  
9.
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  
10.
22 p, 806.6 KB 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  

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