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1.
17 p, 1.6 MB Global dynamics and bifurcation of periodic orbits in a modified Nosé-Hoover oscillator / Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Messias, Marcelo (Universidade Estadual Paulista. Departamento de Matemática e Computação (Brazil)) ; Reinol, Alisson C. (Universidade Tecnológica Federal do Paraná. Departamento Acadêmico de Matemática (Brazil))
We perform a global dynamical analysis of a modified Nosé-Hoover oscillator, obtained as the perturbation of an integrable differential system. Using this new approach for studying such an oscillator, in the integrable cases, we give a complete description of the solutions in the phase space, including the dynamics at infinity via the Poincaré compactification. [...]
2020 - 10.1007/s10883-020-09491-5
Journal of Dynamical and Control Systems, vol. 17 (June 2020) p. 491-506  
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2.
15 p, 297.8 KB Quadratic three-dimensional differential systems having invariant planes with total multiplicity nine / 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. Instituto de Biociências, Letras e Ciências Exatas. Departamento de Matemática (Brazil))
In this paper we consider all the quadratic polynomial differential systems in R having exactly nine invariant planes taking into account their multiplicities. This is the maximum number of invariant planes that these kind of systems can have, without taking into account the infinite plane. [...]
2018 - 10.1007/s12215-018-0338-x
Rendiconti del Circolo Matematico di Palermo, Vol. 67, Issue 3 (December 2018) , p. 569-580  
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3.
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  
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4.
11 p, 614.5 KB Darboux invariants for planar polynomial differential systems having an invariant conic / Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Messias, Marcelo (FCT-UNESP(Brazil). Departamento de Matemática e Computacâo) ; Reinol, Alisson C (FCT-UNESP(Brazil). Departamento de Matemática e Computacâo)
We characterize all the planar polynomial differential systems with a unique invariant algebraic curve given by a real conic and having a Darboux invariant.
2014 - 10.1007/s00033-013-0390-5
ZAMP. Journal of Applied Mathematics and Physics, Vol. 65 (2014) , p. 1127-1136  
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5.
21 p, 502.4 KB Normal Forms for Polynomial Differential Systems in R^3 Having an Invariant Quadric and a Darboux Invariant / Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Messias, Marcelo (FCT-UNESP (Brazil). Departamento de Matemática e Computacâo) ; Reinol, Alisson C (FCT-UNESP (Brazil). Departamento de Matemática e Computacâo)
We give the normal forms of all polynomial differential systems in R^3 which have a nondegenerate or degenerate quadric as an invariant algebraic surface. We also characterize among these systems those which have a Darboux invariant constructed uniquely using the invariant quadric, giving explicitly their expressions. [...]
2015 - 10.1142/S0218127415500157
International journal of bifurcation and chaos in applied sciences and engineering, Vol. 25 Núm. 1 (2015) , p. 1550015  
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