Speaker: Gian-Italo Bischi (Istituto di Scienze Economiche, University of Urbino, Italy)
Discrete time dynamical systems represented by the iteration of noninvertible maps may exhibit attracting sets and basins of attraction with topological structures which are more complex than the ones arising in other kinds of dynamical systems, such as those represented by the iteration of diffeomorphisms or those in continuous time, represented by ordinary differential equations.
Noninvertible map means "many-to-one". Geometrically, this can be expressed by saying that the phase space is "folded" by the application of the map, so that distinct points are mapped into the same point. This is equivalently stated by saying that a point has several distinct preimages, i.e. several inverses are defined, and these inverses ``unfold'' the phase space.
The mathematical treatment of these dynamical systems is still not well developed, and a significant part of the rich dynamic phenomena numerically observed is not well understood. Their study, often motivated by the problems arising in applications, have not yet converged to a systematic theory, and numerical experiments are crucial in both exploration and understanding of the rich dynamical phenomena and global bifurcations observed. In recent years, interesting results in this field have been obtained by the method of critical sets, a powerful tool through which several global properties and bifurcations, which are typical of such type of maps, are explained (see [1-3]).
The creation, destruction and the qualitative changes of the boundaries of chaotic attractors, as well as the qualitative changes in the structure of the basins' boundaries, are often explained in terms of contacts between repelling invariant sets and critical sets. These contact bifurcations may be important to understand two different routes to complexity: one related to the creation of more and more complex attractors, and one related to more and more complex structures of the basins. Both these routes to complexity may be important in the study of dynamical models met in applications (see e.g. [4-6]).
The study of such contact bifurcations is generally based on both theoretical and computational methods, and the graphical visualization becomes crucial in the discovery and explanation of new dynamic scenarios and their parameter dependence.
References
- [1] I. Gumowski and C. Mira 1#1 Chaotique, Cepadues Editions, Toulose 1980..
- [2] C. Mira, L. Gardini, A. Barugola and J.C. Cathala Chaotic Dynamics in Two-Dimensional Noninvertible Maps, World Scientific, Singapore, 1996.
- [3] R. Abraham, L. Gardini and C. Mira Chaos in Discrete Dynamical Systems (a visual introduction in two Dimensions) Springer-Verlag, 1997.
- [4] G.I. Bischi, L. Stefanini and L. Gardini ``Synchronization, intermittency and critical curves in duopoly games'', Mathematics and Computers in Simulations, 44, 559-585 (1998).
- [5] G.I. Bischi and L. Gardini ``Role of invariant and minimal absorbing areas in chaos synchronization'', Physical Review E, 58, 5710-5719 (1998).
- [6] G.I. Bischi, L. Gardini and M. Kopel ``Analysis of Global Bifurcations in a Market Share Attraction Model'', Journal of Economic Dynamics and Control (in press)