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GSD (Grup de Sistemes Dinàmics)

Els sistemes dinàmics són, i sempre han estat, una de les principals línies d'investigació en Matemàtiques. És interès de totes les civilitzacions humanes comprendre qüestions importants, com el moviment dels planetes, l'evolució de les poblacions, o l'estudi de la dinàmica en sistemes deterministes, de manera que els sistemes dinàmics s'han convertit en un objectiu important d'estudi. Després de molts anys d'evolució, l'àrea dels sistemes dinàmics ha sofert diverses transformacions i ha desenvolupat diverses branques que han permès respondre preguntes de diversa índole.

Les linies principals d'investigació del Grup de Sistemes Dinàmics de la UAB (GSD-UAB) són: Mecànica celeste, Dinàmica complexa, Sistemes Dinàmics discrets i Teoria qualitativa d'equacions diferencials.

Els membres del nostre grup treballen principalment a les universitats catalanes (UAB, UB, UdG, UPC, URV, UVic), tot i que alguns dels nostres investigadors treballen en altres universitats d'Espanya i a l'estranger. El GSD-UAB col·labora asiduament amb diversos grups de recerca nacionals i internacionals.

Pàgina web: http://www.gsd.uab.cat
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2026-07-01
14:14
54 p, 2.6 MB Quadratic systems with two invariant real straight lines and an invariant parabola / Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Ou, Huaxin (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
After the linear differential systems in the plane the easiest ones are the quadratic polynomial differential systems. Due to their nonlinearity and also to their many applications these systems have been studied by many authors. [...]
2025 - 10.14232/ejqtde.2025.1.66
Electronic Journal of Qualitative Theory of Differential Equations, Num. 66 (2025) , p. 1-54  
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2026-07-01
13:17
49 p, 8.6 MB Phase Portraits of (2;1) Reversible Vector Fields of Low Codimension / Buzzi, Claudio (Universidade Estadual Paulista "Júlio de Mesquita Filho". Instituto de Biociências, Letras e Ciências Exatas) ; Llibre, Jaume (Universitat Autònoma de Barcelona) ; Santana, Paulo Henrique Reis (Universidade Estadual Paulista "Júlio de Mesquita Filho". Instituto de Biociências, Letras e Ciências Exatas)
In this paper we study the phase portraits and bifurcation diagram of the symmetric singularities of codimensions zero, one and two of planar reversible vector fields having a line of reversibility.
2026 - 10.1007/s12591-026-00756-2
Differential Equations and Dynamical Systems, March 2026  
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2026-06-30
20:12
13 p, 484.2 KB From Generalized Controlled Nonlinear Oscillators to Lorenz-Like Systems / Ginoux, Jean-Marc (Centre National de la Recherche Scientifique. Aix Marseille University. Université de Toulon) ; Meucci, Riccardo (Consiglio Nazionale Delle Ricerche. Istituto Nazionale di Ottica) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Chen, Guanrong (City University of Hong Kong. Department of Electrical Engineering) ; Chua, Leon O. (University of California. Department of Electrical Engineering and Computer Sciences)
Damped and driven oscillators are generally modeled with a nonautonomous second-order nonlinear ordinary differential equation including a sinusoidal driving forcing term, such as the forced Duffing equation and the forced Holmes-Rand equation. [...]
2026 - 10.1142/S0218127426300223
International journal of bifurcation and chaos in applied sciences and engineering, Vol. 36, Num. 10 (August 2026) , art. 2630022  
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2026-06-25
21:12
11 p, 375.2 KB Periodic orbits of two particular classes of planar continuous piecewise linear differential systems with three zones separated by two parallel straight lines / Fonseca, A. F. (Universidade Federal de Itajubá. Instituto de Matemática e Computação) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Mello, Luis Fernando (Universidade Federal de Itajubá. Instituto de Matemática e Computação)
We study the existence and non-existence of periodic orbits in the planar continuous piecewise linear differential systems with three zones separated by two parallel straight lines. First with the additional hypothesis that the three vector fields have no equilibrium points neither real nor virtual we prove that these differential systems have no periodic orbits. [...]
2026 - 10.1007/s40590-026-00903-3
Boletin de la Sociedad Matematica Mexicana, Vol. 32, Num. 2 (June 2026) , art. 69  
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2026-06-25
21:12
32 p, 360.2 KB Averaging Approach to the Limit Cycles of A Class of Discontinuous Piecewise Differential Systems / Li, Jie (Southwest Petroleum University) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
These last years an increasing interest appeared for studying the discontinuous piecewise differential systems motivated by the rich applications in modelling real phenomena. One of the difficulties for understanding the dynamics of these systems is the study of their limit cycles. [...]
2026 - 10.1007/s10114-026-2316-0
Acta Mathematica Sinica. English Series, Vol. 42, Num. 4 (April 2026) , p. 940-960  
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2026-06-29
19:12
19 p, 509.6 KB On the dynamics of the Moon-Rand systems / Cao, Chen (Shanghai Jiao Tong University. School of Mathematical Sciences) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
This paper investigates the dynamics of the Moon-Rand system, a 3-dimensional differential system coming from the parametric stiffness control of flexible space structures. First we characterize the phase portraits of the system on the sphere of infinity by employing the Poincaré compactification in R3. [...]
2026 - 10.1016/j.cnsns.2026.110232
Communications in nonlinear science and numerical simulation, Vol. 161, Num. Part 3 (October 2026) , art. 110232  
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2026-06-23
17:12
15 p, 644.6 KB Plane Curves with Proportional Affine and Euclidean Curvatures / Garcia, Ronaldo A. (Universidade Federal de Goiás. Instituto de Matemática e Estatística) ; Gasull, Armengol (Universitat Autònoma de Barcelona. Department de Matemàtiques)
In this paper, we obtain all the locally convex plane curves with proportional curvatures (Euclidean and affine). The analysis is performed from a special second order differential equation that, in appropriated coordinates, and using the support function the problem is reduced to the properties of a periodic linear nonhomogeneous second differential equation of the type y'' + y = a (t), a (t + 2π) = a (t).
2026 - 10.1007/s12346-026-01475-3
Qualitative theory of dynamical systems, Vol. 25, Num. 2 (April 2026) , art. 59  
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2026-07-01
19:12
23 p, 748.4 KB On the dynamics of the autonomous Duffing-Holmes oscillator / Diz-Pita, Érika (Universidade de Santiago de Compostela) ; Llibre, Jaume (Universitat Autònoma de Barcelona) ; Otero-Espinar, M. Victoria (Universidade de Santiago de Compostela)
The autonomous Duffing–Holmes oscillator ˙x = y, ˙y = x − x3 + by − kz, ˙z = w (y − z), depending on the three parameters b, k, and w, has been studied previously by several authors who showed that for certain parameter values, it exhibits chaotic motion, or that numerically it has two periodic orbits coming from a Hopf bifurcation, or that it can have three equilibria for some given values of the parameters. [...]
2026 - 10.15388/namc.2026.31.46613
Nonlinear Analysis: Modelling and Control, Vol. 31, Num. 3 (2026) , p. 699-721  
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2026-06-23
18:13
7 p, 283.9 KB Limit Cycles and Invariant Algebraic Curves / Gasull, Armengol (Universitat Autònoma de Barcelona) ; Santana, Paulo Henrique Reis (Universidade Estadual Paulista "Júlio de Mesquita Filho". Instituto de Biociências, Letras e Ciências Exatas)
We give lower bounds in terms of n for the number of limit cycles of polynomial vector fields of degree n having any prescribed invariant algebraic curve. By applying them when the ovals of this curve are also algebraic limit cycles, we obtain a new recurrent property for the Hilbert numbers. [...]
2026 - 10.3934/cpaa.2026012
Communications on pure & applied analysis, Vol. 28 (April 2026) , p. 214-220  
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2026-06-29
17:12
11 p, 609.5 KB Invariant meridians and parallels on the hyperboloid of one sheet / Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Murza, Adrian (University of Texas at Dallas. Department of Mathematical Sciences)
We analyze the polynomial vector fields in R3 that have the hyperboloid of one sheet as an invariant surface. We characterize the maximum number of invariant meridians and parallels of this hyperboloid that such polynomial vector fields can have, as a function of the degree of the polynomial vector field.
2026 - 10.1016/j.difgeo.2025.102326
Differential Geometry and its Application, Vol. 102 (February 2026) , art. 102326  
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