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GSD (Dynamical systems)

Dynamical systems is, and always has been, one of the main lines of research in Mathematics. It lies in the interest of all human civilizations to understand important questions such as the movement of the planets, the evolution of populations, or the discovery of chaotic dynamics in robust deterministic systems, which is why dynamical systems has become a major goal of study. After many years of evolution, the area of dynamical systems has undergone various transformations and developed branches to provide answers to questions of diverse nature.

The interests of the Dynamical Systems Group of UAB (GSD-UAB) can be described by stating our main research lines: Celestial Mechanics, Complex Dynamics, Discrete Real Dynamical Systems and Qualitative Theory of Differential Equations.

The members of our group work mainly in Catalonian universities (UAB, UB, UdG, UPC, URV, UVIC), although some of our researchers work in other universities in Spain and abroad. GSD-UAB collaborates with various national and international research groups.

Web page: http://www.gsd.uab.cat
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2026-02-19
21:13
5 p, 1.7 MB Limit Cycles of Continuous-Discontinuous Piecewise Linear Hamiltonian Systems in ℝ2 Separated by the Curve y=sinx / Chachapoyas, N. (Universidade Federal de Itajubá. Instituto de Matemática e Computação) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Meza-Sarmiento, I. S. (Universidade Federal de Itajubá. Instituto de Matemática e Computação) ; Vidarte, J. (Universidade Federal de Itajubá. Instituto de Matemática e Computação)
These last decades piecewise differential systems have been studied intensively, mainly due to their applications. Inside the study of the dynamics of these differential systems, the limit cycles, that is, the isolated periodic orbits inside the set of all periodic orbits of the system, play a main role. [...]
2026 - 10.1002/mma.70230
Mathematical methods in the applied sciences, Vol. 49, Num. 3 (February 2026) , p. 2093-2097  
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2026-02-19
20:12
18 p, 308.1 KB Global centres in a class of quintic polynomial differential systems / Da Cruz, Leonardo Pereira Costa (Universidade de São Paulo. Instituto de Ciências Matemáticas e de Computação) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
A centre of a differential system in the plane R2 is an equilibrium point p having a neighbourhood U such that U \ {p} is filled with periodic orbits. A centre p is global when R2 \ {p} is filled with periodic orbits. [...]
2026 - 10.1017/prm.2024.43
Proceedings of the Royal Society of Edinburgh Section A: Mathematics, Vol. 156, Num. 1 (February 2026) , p. 39-54  
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2026-02-19
20:12
20 p, 361.3 KB Local and global analysis of the displacement map for some near integrable systems / Braun, Francisco (Universidade Federal de São Carlos. Departamento de Matemática) ; Da Cruz, Leonardo Pereira Costa (Universidade de São Paulo. Instituto de Ciências Matemáticas e de Computação) ; Torregrosa, Joan (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
In this paper, we introduce an alternative method for applying averaging theory of orders 1 and 2 in the plane. This is done by combining Taylor expansions of the displacement map with the integral form of the Poincaré-Poyntriagin-Melnikov function. [...]
2025 - 10.1016/j.physd.2025.134932
Physica D: Nonlinear Phenomena, Vol. 483 (December 2025) , art. 134932  
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2026-02-19
19:12
19 p, 391.5 KB Double zero-Hopf bifurcation in a four-dimensional hyperchaotic system / Lima, Mauricio Firmino Silva (Universidade Federal do ABC. Centro de Matematica Computação e Cognição) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
The subject of this paper concerns with the bifurcation of limit cycles for a four-dimensional hyperchaotic system. This hyperchaotic system depends on five parameters and we restrict our study to the set of parameters where a double zero-Hopf bifurcation may occurs, that is, where an isolated equilibrium point has a double zero and a pair of purely imaginary eigenvalues. [...]
2025 - 10.1007/s40590-025-00819-4
Boletin de la Sociedad Matematica Mexicana, Vol. 31, Num. 3 (November 2025) , art. 142  
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2026-02-19
18:12
18 p, 2.0 MB Limit Cycles for Discontinuous Piecewise Differential Systems in R3 Separated by a Paraboloid / Jimenez Ruiz, Jeidy Johana (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Valls, Clàudia 1973- (Universidade de Lisboa. Instituto Superior Técnico. Departamento de Matemática)
In planar piecewise differential systems it is known that when the discontinuity curve is a straight line and both differential systems are linear centers, these piecewise differential systems have no limit cycles but if they are separated by other types of discontinuity curves, such as parabolas, then they have limit cycles. [...]
2023 - 10.1007/s12591-023-00668-5
Differential Equations and Dynamical Systems, December 2023  
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2025-08-28
10:49
16 p, 401.5 KB Monotonous Period Function for Equivariant Differential Equations with Homogeneous Nonlinearities / Gasull, Armengol (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Rojas, David (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
We prove that the period function of the center at the origin of the Zk-equivariant differential equation z˙= iz + a(zz¯)nzk + 1, a ≠ 0, is monotonous decreasing for all n and k positive integers, solving a conjecture about them. [...]
2025 - 10.1007/s00009-025-02879-2
Mediterranean journal of mathematics, Vol. 22, Issue 5 (August 2025) , art. 112  
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2025-07-28
14:24
28 p, 687.3 KB Quadratic vector fields in class I / Artés Ferragud, Joan Carles (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Chen, Hebai (Central South University. School of Mathematics and Statistics (China)) ; Ferrer, Lluc Manel (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Jia, Man (Central South University. School of Mathematics and Statistics (China))
In [Ye et al. , Theory of Limit Cycles, 1986], quadratic systems are classified into three different normal forms (I, II and III) with increasing number of parameters. The simplest family is I and even several subfamilies of it have been studied, and some global attempts have been done, up to this paper, the full study was still undone. [...]
2025 - 10.1080/14689367.2024.2436223
Dynamical Systems, Vol. 40, Issue 2 (2025) , p. 191-222  
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2025-07-28
14:24
Polynomial differential systems with invariant algebraic curves of arbitrary degree formed by Legendre polynomials / Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Valls, Clàudia 1973- (Universidade de Lisboa. Instituto Superior Técnico. Departamento de Matemática)
In 1891 Poincaré asked: Given m ≥ 2, is there a positive integer M(m) such that if a polynomial differential system of degree m has an invariant algebraic curve of degree ≥ M(m), then it has a rational first integral? Brunella and Mendes repeated the same open question in 2000, and Lins-Neto in 2002. [...]
2025 - 10.1016/j.jpaa.2025.108001
Journal of Pure and Applied Algebra, Vol. 229, Issue 8 (August 2025) , art. 108001  
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2025-07-28
14:24
On the Limit Cycles Bifurcating from the Periodic Orbits of a Hamiltonian System / Anacona, Gerardo H. (Universidade Federal de Goiás. Instituto de Matemática e Estatística) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Freitas, Bruno (Universidade Federal de Goiás. Instituto de Matemática e Estatística)
This paper concerns the weak 16th Hilbert problem and considers the Hamiltonian center a: = -y2n-1, a: = x2n-1, and we perturb it by all polynomials of degree 2n-1 for n = 2, 3, 4, 5, 6, 7, 8. We prove that the maximum number of limit cycles that can bifurcate from the periodic orbits of this center for n = 2, 3, 4, 5, 6, 7, 8, under the mentioned perturbations and using the averaging theory of first order, is 1, 4, 3, 2, 5, 6, 7, respectively.
2025 - 10.1142/S0218127425500403
International journal of bifurcation and chaos in applied sciences and engineering, Vol. 35, Issue 4 (March 2025) , art. 2550040  
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2025-07-28
14:24
5 p, 617.4 KB Zero-Hopf bifurcation of a 5D hyperchaotic quadratic polynomial differential systems / Diab, Zouhair (Echahid Cheikh Larbi Tebessi University. Department of Mathematics and Computer Science) ; Guirao, Juan Luis Garcia (Universidad Politécnica de Cartagena. Departamento de Matemática Aplicada y Estadística) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
A zero-Hopf equilibrium of a 5-dimensional autonomous differential system is an equilibrium point for which the Jacobian matrix of the system evaluated at that equilibrium has three zero eigenvalues and a pair of purely imaginary eigenvalues. [...]
2025 - 10.1016/j.matcom.2025.06.021
Mathematics and computers in simulation, Vol. 238 (December 2025) , p. 383-387  
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