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Dynamical systems

Dynamical systems is a vast area of research covering description of deterministic processes. These systems usually model evolutionary situations coming from natural sciences, but also social sciences. The time variable can be of continuous or discrete nature and mathematical formulation of corresponding dynamics is done through the systems PDE, ODE or difference equations. There is a big spectrum of methods used: analytical, topological, algebraic, geometric... According to physical assumptions, dynamical systems are classical or quantum. The type of constraints, holonomic or nonholonomic, govern the underlying geometry. Among the systems with one-side constraints, the billiard systems are very important. According to their general behaviour and at least theoretical possibility for solvability, the dynamical systems are of nonintegrable or integrable type.

Some GFM members have recently done works in the subfields of spatially extended monotone mappings, discrete time piecewise affine models of genetic regulatory networks and symbolic dynamics for piecewise affine models.

Spatially extended monotone mappings have been studied from a wide abstract viewpoint: the general properties of discrete time infinite dimensional systems that arise from the monotony. Several specific systems belong to this class of monotone maps: time discretizations of action-reaction PDEs, bistable coupled map lattices, dissipative dynamics of Frankel-Kontorova models...

Symbolic dynamics for piecewise affine models can be applied to several concrete systems; this theory consist in using the linear dependence on the codes to develop explicit formulas for the dynamics, reducing the analysis of those systems to the study of an admissibility condition on the codes. These methods have been applied to simple interval maps but also to coupled map lattices and to discrete time piecewise affine models of genetic regulatory networks.

In this later particular kind of models the interaction with others fields (computational biology, theoretical biology) is an important source of problems and motivation.