Below we list some of the groups which we collaborate with. This list is not complete or exhaustive and the links may not be active.
Research Groups
- Barcelona UB-UPC Dynamical Systems Group (Barcelona, SPAIN) [www]
- Dynamical Systems Group (Warwick, UK) [www]
- Dynamical Systems Group at Hasselt University (Diepenbeek, BELGIUM) [www]
- Dynamical Systems group of UFG (Goiania, BRAZIL) [www]
- Espacios de Banach y Sistemas Dinámicos (Huelva, SPAIN) [www]
- Grup de Sistemes Dinàmics UIB (Illes Balears, SPAIN) [www]
- Grup de Sistemes Dinàmics UPC (Barcelona, SPAIN) [www]
- Grupo de Sistemas Dinámicos (Oviedo, SPAIN) [www]
- Grupo de Sistemas Dinâmicos da UNESP (Sâo José do Rio Preto, BRAZIL) [www]
- Research group on Differential Equations (Granada, SPAIN) [www]
- Seminari de Sistemes Dinàmics (Lleida, SPAIN) [www]
- Sistemas Dinàmicos (Murcia, SPAIN) [www]
- The Dynamical Systems Group At Boston University (Boston, USA) [www]
Networks
The recently published publications are accessible online in DDD UAB ("Dipòsit Digital de Documents de la UAB") in this link.
DDD is a server that collects, manages, preserves and disseminates scientific, educational, and institutional materials in a digital library at UAB. It brings together a wide range of media, themes, and types of documents. DDD always includes metadata describing the resource and access to the full text.
P4 and P5 are computer programs that study specific Polynomial Planar Vector Fields (or Differential Systems) of any degree. Both plot the phase portrait of a vector field in the Poincaré disk. Part of the calculations are done through an interface with the computer program Maple (giving the option of treating a vector field using symbolic manipulations), and part of them by the program itself. A piecewise vector field is a finite number of vector fields defined on a domain that is bounded by a finite number of algebraic curves (bifurcation lines). Such vector fields are often found in control theory. The piecewise polynomial vector fields can be discontinuous along the bifurcation lines. P5 is an extension of P4 and is introduced to be able to treat such piecewise vector fields. Much of the functionality of P5 is inherited by P4.
Bibliography
Chapters 9 and 10 of the book Qualitative Theory of Planar Differential Systems (F. Dumortier, J. Llibre, J. C. Artés. Springer-Verlag, Berlin, 2006 ISBN 9783540328933) contain a complete description of how P4 works and what you can do with it.
Authors
- Joan C. Artés. Universitat Autònoma de Barcelona.
- Peter de Maesschalck. Hasselt University.
- Freddy Dumortier. Hasselt University.
- Chris Herssens. Hassel University.
- Jaume Llibre, Universitat Autònoma de Barcelona.
- Oscar Saleta. Universitat Autònoma de Barcelona.
- Joan Torregrosa. Universitat Autònoma de Barcelona.
Download
P4&P5 were initially built to run on Microsoft Windows. The programs are however built in the multi-platform interface QT, allowing an easy port to Mac/OSX or Linux. On any platform, P4&P5 need an interface with Maple. Although we did not test the program with every version of Maple, it should work from Maple version 10 or higher. The last version works better with Maple 2015 or higher.
To install the programs P4&P5, please be sure to have Maple installed. Download and follow the installation instructions. Maybe you also need to install the QT software development kit beforehand. You need the open-source edition of Qt that you can download from the QT website (together with QtCreator).
- You can download P4 from https://github.com/oscarsaleta/P4/releases/latest
Functionality
- The programs draw the phase portrait on a compact part of the plane, on the Poincaré sphere, or on one of the charts at infinity. The ability to draw on the Poincaré sphere and hence to present a view of the phase portrait on the global compactified phase space is one of the novelties of this program.
- The program Determines all singular points (finite and infinite ones) of the vector field. P5 is able to present a detailed study of each singularity. It distinguishes between the different types of singularities (saddle, node, focus, center, semi-hyperbolic saddle-node, and degenerate). Singularities with purely imaginary eigenvalues are studied by means of Lyapunov coefficients to distinguish (in some cases) between center behavior and weak focus behavior.
- Degenerate singularities are studied by means of quasi-homogeneous blow-ups. It is shown that a finite number of such blow-ups unfolds the singularity in a number of hyperbolic or semi-hyperbolic singularities, each of which is studied in detail by the program.
- For each singularity, invariant separatrices are calculated (formally, up to a degree that can be specified). The formal expansion is tested for its numeric accuracy.
- A detailed report can be prepared containing a list of singularities, their separatrices, and their sectors (parabolic, hyperbolic, or elliptic). The program can plot all or individual separatrices. To speed up the integration of the separatrices of degenerate singularities (where the flow is generally very slow), it uses the rescaled (blown up) versions of the vector field in a neighborhood of the singular point.
- All the above calculations can be done either in numeric mode (with given precision), or in symbolic mode (using Maple as a symbolic math manipulator).
- The program does not deal with bifurcations. The program does offer the possibility of introducing a vector field with user-defined parameters. Each time you want to study the vector field (examine singularities/draw phase portraits), you just have to enter the value of the parameters.
- The program is able to find periodic orbits up to a certain degree of precision (in particular limit cycles, i.e. isolated periodic orbits), cutting a transverse section that is specified by the user.
- Images can be exported to XFIG, Encapsulated Postscript, or JPEG.
- P5 calculates the domains bounded by algebraic curves (up to some precision). These bifurcation lines are drawn on the phase portrait.
- P5 studies singularities of the several vector fields that are used to define a piecewise polynomial vector field. Singularities that lie on a bifurcation line deserve special attention: singularities of mixed type are possible, and only those separatrices that start in the relevant domain are to be drawn on the phase portrait.
- An integration parameter allows to control the integration of orbits near bifurcation lines. Phenomena such as sliding vector fields and sewing can be observed when using the program.
Dynamical systems are, and it always has been, one of the main lines of research in Mathematics. The interest of all human civilizations to understand the movements of the planets, the evolution of populations, or the discovery of chaotic dynamics on robust deterministic systems, makes dynamical systems a main goal of study. After many years of development, the area of dynamical systems has undergone various branches to provide answers to questions of diverse nature.
Our group of research is an example of all these bifurcations that occurred in dynamical systems over the years. Precisely the starting point can be situated in 1979 when Jaume Llibre presented his Ph. D. "Evoluciones finales y movimientos quasialeatorios en el problema restringido de tres cuerpos", under the supervision of Carles Simó. Although celestial mechanics was the main field of research of the group, we can assert that currently there are three more lines in which the group is active. This growth (in fields of interest and active research) explains that after more than 40 years of activity, scientific leadership is now shared by different people. As a consequence of the particular history of the group, most of the researchers work in more than one research line. This transversality gives richness and coherence to our research project.
We hope that these pages give you enough information about our research activity in dynamical systems. We encourage everybody interested to sail on them.
If you have any trouble getting information on the pages please contact with us.
- Qualitative Theory of Differential Equations: Although during the years, we have worked on so many different problems related to the qualitative theory of differential equations, during the last few years we have centered most of our interest in problems surrounding the well-known 16th Hilbert′s problem and the Jacobian Conjecture. We bring out our work on the Markus-Yamabe conjecture, Liapunov constants, center-focus problem, Hamiltonian perturbations, Abelian integrals, isochronicity, uniqueness of limit cycles, etc. Other kinds of problems we are also interested in are the general theory of polynomial differential systems, control theory, the study of singularities via blow-ups, nondifferentiable dynamical systems, structural stability, quadratic polynomial systems, etc. Some collaborators are C. Christopher, F. Dumortier, J.P. Françoise, H. Giacomini, C. Li, D. Shafer, J. Sotomayor, M.A. Teixeira, and X. Zhang.
- Discrete Dynamical Systems: We are mainly interested in the study of the dynamics of continuous maps of one-dimensional topological spaces. Our main subjects of interest in this field are Periodic structure, Combinatorial Dynamics, Topological dynamics, and Topological entropy. In some special cases (including higher dimensional spaces), when studying the periodic structure, we also use the general theory of Lefschetz and Nielsen. We are also concerned about the study of the minimal dynamics in a homotopy class. Some colaborators are J. Guaschi, S. Kolyada, J. Los, M. Misiurewicz, J. A. Rodríguez, L. Snoha, and X. Ye.
- Celestial Mechanics: The main goal of our research is the description of the global flow for certain restricted three-body problems by using symbolic dynamics. We emphasize the dynamics near homoclinic and heteroclinic orbits. We also contribute to the study of the number and distribution of the central configurations for the n-body problem. Indeed we use all this information to study the dynamics and the global flow associated with the n-body problem. Some collaborators are M. Falconi, E.A. Lacomba, and E. Perez-Chavela.
- Complex Dynamics: Our general goal is to describe the dynamical and parameter plane given by the iteration of holomorphic maps of the complex plane or the Riemann sphere. In particular, we focus our attention on transcendental entire or meromorphic families (standard map, complex exponential family, Newton method applied to entire maps, etc) or rational iteration. In any of those settings, our approach is mostly topological including the description of the Julia set, the study of the boundary of Siegel discs, the distribution of Herman rings, the classification of the Baker domains, the properties of the bifurcation locus, etc. The main tools we use are quasiconformal surgery, symbolic dynamics, and complex analysis. Some collaborators are K. Baranski, X. Buff, B. Branner, R. Devaney, L. Geyer, and M. Moreno-Rocha.