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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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2025-02-24
17:40
25 p, 1.6 MB The transition between two ecological limit cycles in one predator-two competitive prey model / Castellanos, Víctor (Universidad Juárez Autónoma de Tabasco. División Académica de Ciencias Básicas (Mèxic)) ; Chan-López, Eduardo (Universidad Juárez Autónoma de Tabasco. División Académica de Ciencias Básicas (Mèxic)) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
In this paper we analyze an ecological model with a single predator and two competitive prey species. This model incorporates several key elements, including logistic growth dynamics for the prey populations, a Holling type II functional response governing predator-prey interactions, and the inclusion of intraspecific competition among predators. [...]
2024
Dynamics of continuous, discrete and impulsive systems, Vol. 31, Issue 6b (2024) , p. 377-404  
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2025-02-24
15:46
32 p, 381.7 KB Punts d'equilibri globalment atractors / Cimà, Anna (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Un punt d'equilibri d'un sistema dinàmic continu o discret és un atractor global si l'òrbita de qualsevol punt tendeix a aquest punt d'equilibri quan el temps tendeix a infinit. En aquest article tractem el problema de donar condicions suficients perquè un punt d'equilibri d'un sistema dinàmic sigui un atractor global. [...]
2021 - 10.2436/20.2002.01.99
Butlletí de la Societat Catalana de Matemàtiques, Vol. 36, Núm. 2 (2021) , p. 121-152  
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2025-02-07
11:43
18 p, 366.9 KB Parallel vector fields and global injectivity in two dimensions / Braun, Francisco (Universidade Federal de São Carlos. Departamento de Matemática) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Let U be simply connected open subset of R2, and let f: U → R2 be a local diffeomorphism. We study the global injectivity of f using the planar vector fields of type annular, radial or strip. Our main result enables the unification of proofs for classical results on global injectivity, such as the Hadamard global invertibility theorem and the condition related to the connectedness of the levels sets of one of the coordinates of f.
2024 - 10.2140/pjm.2024.333.1
Pacific Journal of Mathematics, Vol. 333, Issue 1 (November 2024) , p. 1-15  
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2025-02-07
11:43
Limit cycles in a class of planar discontinuous piecewise quadratic differential systems with a non-regular line of discontinuity (I) / He, Dongping (Sichuan University. School of Mathematics) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
In this paper we study the limit cycles which bifurcate from the periodic orbits of the quadratic uniform isochronous center ẋ = −y + xy, ẏ = x + y, when this center is perturbed inside the class of all discontinuous piecewise quadratic polynomial differential systems in the plane with two pieces separated by a non-regular line of discontinuity, which is formed by two rays starting from the origin and forming an angle α = π/2. [...]
2025 - 10.1016/j.matcom.2024.10.016
Mathematics and computers in simulation, Vol. 229 (March 2025) , p. 743-757  
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2025-02-07
11:43
25 p, 3.2 MB Global phase portraits of kukles differential systems with homogenous polynomial nonlinearities of degree 5 having a center / Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; da Silva, Mauricio Fronza (Universidade Federal de Santa Maria. Departamento de Matemàtica)
We provide 22 different global phase portraits in the Poincaré disk of all centers of the so called Kukles polynomial differential systems of the form ẋ = −y, ẏ = x + Q5(x, y), where Q5 is a real homogeneous polynomial of degree 5 defined in ℝ2.
2016 - 10.12775/TMNA.2016.049
Topological Methods in Nonlinear Analysis, Vol. 48, Issue 1 (September 2016) , p. 257-282  
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2025-02-07
11:43
The Uniform Isochronous Centers with Homogeneous Nonlinearities of Degree 6 / Dong, Guangfeng (Jinan University. Department of Mathematics) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
In this paper we study the topological phase portraits of polynomial differential systems with a uniform isochronous center, whose nonlinear parts are homogeneous polynomials of degree 6. We obtain all the distinct topological phase portraits in the Poincaré disc. [...]
2024 - 10.1007/s10883-024-09703-2
Journal of Dynamical and Control Systems, Vol. 30, Issue 3 (September 2024) , art. 34  
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2025-02-07
11:43
On the Global Nilpotent Centers of Cubic Polynomial Hamiltonian Systems / 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)
A global center for a vector field in the plane is a singular point p having R2 filled of periodic orbits with the exception of the singular point p. Polynomial differential systems of degree 2 have no global centers. [...]
2024 - 10.1007/s12591-022-00606-x
Differential Equations and Dynamical Systems, Vol. 32, Issue 4 (October 2024) , p. 1001-1011  
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2025-02-07
11:43
The Global Dynamics of a 3-Dimensional Differential System in ℝ3 via a Darboux Invariant / Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Valls, Clàudia 1973- (Universidade de Lisboa. Instituto Superior Técnico. Departamento de Matemática)
The differential system ẋ = ax − yz, ẏ = −by + xz, ż = −cz + x2, where a, b and c are positive real parameters, has been studied numerically due to the big variety of strange attractors that it can exhibit. [...]
2025 - 10.1007/s10473-025-0204-9
Acta Mathematica Scientia, Vol. 45, Issue 2 (March 2025) , p. 338-346  
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2025-02-07
11:43
Limit cycles and chaos in planar hybrid systems / Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Santana, Paulo Henrique Reis (Universidade Estadual Paulista "Júlio de Mesquita Filho". Instituto de Biociências, Letras e Ciências Exatas)
In this paper we study the family of planar hybrid differential systems formed by two linear centers and a polynomial reset map of any degree. We study their limit cycles and also provide examples of these hybrid systems exhibiting chaotic dynamics.
2025 - 10.1016/j.cnsns.2024.108382
Communications in nonlinear science and numerical simulation, Vol. 140, Part 1 (January 2025) , art. 108382  
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2025-02-07
11:43
Correction to : Limit Cycles in a Class of Planar Discontinuous Piecewise Quadratic Differential Systems with a Non-regular Line of Discontinuity (II) / He, Dongping (Sichuan University. School of Mathematics) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
In the original publication, the definition of the angle sector Σ of equation (28) is missing, and it is given in this erratum as follows: (Formula presented. ) The original article has been corrected.
2024 - 10.1007/s00009-024-02730-0
Mediterranean Journal of Mathematics, Vol. 21, Issue 7 (November 2024) , art. 192  
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