**Abstract**

The main interests of the group of researchers involved in this project are the infinite dimensional dynamical systems. They have spent a considerable amount of energy towards the development of a geometric theory for such dynamical systems. Finite or infinite dimensional dynamical systems are mathematical models for many problems in physics, biology, economy, engineering and so on. These dynamical systems are usually associated to differential equations that can be ordinary differential equations, partial differential equations, functional differential equations, partial-functional differential equations and discrete systems. Examples of mathematical models that give rise to dynamical systems are the Wave Equation, the Fitz-Hugh Nagumo Equation, the Hodgkin-Huxley Equation, the Equation for superconductivity of liquids, models of population growth, the Navier-Stokes Equation, Reaction Diffusion Equations, Heat Equations, the Kortweg-de Vries Equation, Cahn-Hilliard Equation, Schroedinger Equation, Benjamin-Ono Equations among others. If we understand how these dynamical systems behave we may be able to give some information about the modelled phenomenon. In order that a mathematical model reproduces the behavior of the modelled system, we must know a complete system of laws that governs the system. It is clear that some influences that the system suffers are so small that may be forgotten or simply neglected during the modelling. Besides this, all parameters in the model are determined with some error. AlI this is saying that practical models are only approximations of an ideal model and errors are unavoidable. With this in mind it is of fundamental importance that the models enjoy some stability with respect to alI kind of perturbations. A way to ensure such stability is to prove that the asymptotic dynamics is stable under perturbations. After these considerations, in the study of such dynamical systems we identify some very important stages: (i) modelling: using the empiric laws that govern the phenomenon one derives a mathematical model that takes into account all the important influences in the phenomenon described, (ii) the local well posedness: the model obtained must be consistent and its solutions must exist, be unique and behave continuously with respect to initial conditions, (iii) the study of the dynamics and its roughness with respect to perturbations: this step is crucial to the survival of modelling since everything in the mathematical model is determined approximately and therefore the solutions (for large times) must be stable under perturbations of alI parameters in the model, and (iv) feedback to the modelled phenomenon: the understanding of the model may give important information about the modelled phenomenon or may allow us to understand how to interfere in it in such a way to produce desired outputs. Within this general framework this project proposes to consider several questions related to the Geometric Theory of Dynamical Systems such as (i) Local and global well posedness; (ii) Asymptotic behavior of solutions; (iii) Invariant sets, its properties and roughness; (iv) Existence and regularity of special solutions. One specially interesting problem in which the group has been working is related to the semilinear equations and deals with the attempt to transfer information from the linear part of equation to the asymptotic behavior of the whole equation. To be more specific, we try to show that if the linear (unbounded) part of the equation behaves "continuously" (e.g. the resolvent behaves continuously) then, the nonlinear dynamics behaves continuously. The projects related to this are called upper semicontinuity and lower semicontinuity of attractors or more generally invariant sets and smooth linearization. Another question related to the study of partial differential semilinear models is how to obtain, for each initial data, the existence of a unique solution and to prove that this solution depends continuously of the initial data. To prove such results for the widest possible class of nonlinearities and spaces is perhaps the most fundamental question in differential equations. In infinite dimension spaces many models have this property only for a restricted class of nonlinearities and it is our goal to study, for several models, which is the widest class possible for which the model is locally well posed. Projects related to this problem are called semilinear equations with critical exponents. The special solutions that will be studied are periodic orbits, traveling waves, homoclinic orbits, heteroclinic orbits and more generally bounded orbits. In the applications of differential equations these solutions play a fundamental role. Some more specific questions such as to show that the dynamics of a infinite dimensional dynamical system can be understood studying a finite dimensional dynamical system are also of great interest. Such projects are called reduction to finite dimension and make strong use of the invariant manifold theory. We also search for estimates on the size of global attractors and to analyse the synchronization in reaction diffusion equations that involves time delays, nonlinear boundary conditions and other phenomenon with applications to biological models. In another research line, the bifurcation of solutions in reaction diffusion equations with delayed logistic boundary conditions is being studied. In this case it may appear caotic behavior coming from bifurcations with higher codimensions. These phenomenon are very interesting for applications. We are also interested in the study of dynamical systems defined by partial differential equations in graphs to obtain a better understanding of the structure of the spectrum of the linear operators involved in this problems, that may not be self-adjoint, and analyse the possibility of occurrence of Hopf bifurcation. Another topic of interest is to study the dynamics of the Chafee-Infante problem in unbounded domains. The results obtained have application in Mechanics, Electrical Engineer, Fluid Dynamics, Population Growth, Nerve-Impulse transmission, etc. The right environment to produce new researchers in Mathematics; that is, a good graduate program is a lively, vibrating and productive research environment. To obtain such research environment the main goals of this project are: (i) To promote greater interaction between participants of the project working in different institutions; (ii) To contribute to consolidation of emerging research groups in institutions associated to the project; (iii) To intensify the realization of specialized seminars; (iv) To stimulate the young members of the research team to participate of pos-doctor programs in highly recognized centers and generate conditions so that the more experienced members of the team may maintain and increase their collaboration with other institutions and researchers; (v) To increase the number researchers in the area attracting a greater number ofgraduate students pos-doctors in the area; (vi) To intensify the flux of visiting researchers to the several institutions in which there are team members, optimizing the expenses with such visits promoting scientific excursions; (vii) To promote once a year the meeting ICMC .Summer Meeting in Differential Equations; (viii) To make it easier for the participants of the project to attend to meetings held in Brazil or abroad, with the aim to divulge the results obtained; (ix) Acquire modern computational equipment and softwares to optimize the productivity of those researchers whose work requires the use of computers, and (x) To improve the bibliography of our library in the area of the project buying specialized books. The specific problems proposed in the project are grouped in the research lines enumerated below. (i) Obtain results that enable us to show the existence of patterns through the variations in the reaction or in the nonlinear boundary conditions for domains with higher dimension. (ii) Study the Conley index as a tool in the study of qualitative properties of dynamical systems. The property of continuation in the Conley index theory, a principIe of singular continuation and the equivariant Conley index (iii) Smooth linearization around a hyperbolic equilibrium in infinite dimensional Banach spaces. Applications to hyperbolic partial differential equations. (iv) The study of dynamics of reaction diffusion equations with time delay and nonlinear boundary conditions: attractors, synchronization, bifurcations, blow-up and applications to biological problems. (v) The study of dynamical systems defined by partial differential equations in graphs. Hopf bifurcation. Generic simplicity of eigenvalues. (vi) The investigation of the continuity of attractors and invariant sets for parabolic and hyperbolic semilinear problems relatively to parameters. (vii) The study of the local well posed problems for semilinear and quasi-linear parabolic equations in the critical growth case. (viii) The study of the existence, regularity and dependence upon parameters of invariant manifolds with applications to reduction to finite dimensions, to the existence of patterns and to discretization. (ix) Attractors in unbounded domains. The Chafee-Infante equation in unbounded domains. (x) The investigation of the dissipativeness and synchronization of parabolic, hyperbolic and electric power systems. (xi) The study of the uniform dissipativeness and synchronization of discrete systems. (xii) The study of chaotic systems in problems of segmentation of images. (xii) The study of some equations with non local diffusion terms. (xiv) The investigation of some generic properties by boundary perturbations. (xv) The study of the compact concentration in elliptic systems. (xvi) The study of partial differential equations in fluid dynamics. (AU)

Scientific publications
(14)

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)