Dynamical Systems

The general subject of Dynamics is the study of systems whose state evolves in time, such as we encounter them in the various branches of Science and of human activity: Physics, Ecology, Metheorology, Biology, Economics, and many other disciplines. The evolution law make take either of several forms: iterations, ordinary or partial differential equations, stochastic maps or flows. The goal is to build a mathematical theory of such dynamical processes allowing to understand and predict their evolution, with an eye on the innumerous pratical applications. In doing this one uses methods from the various areas of Mathematics, such as Geometry, Analysis, Algebra, Topology, and Probability.

Team and Interlocutors

C. Camacho, M. J. Dias Carneiro, C. Gutierrez, A. Lins Neto, W. de Melo, H. Munhoz, M. J. Pacifico, J. Palis, M. Peixoto, P. Sad, M. Soares, M. A. Teixeira, M. Viana

M. Lyubich (State Univ. of New York), J. Milnor (State Univ. of New York), R. Moussu (University of Dijon), J.-C. Yoccoz (Collège de France)

Real Dynamics and Ergodic Theory

W. de Melo, C. Moreira, J. Palis, M. Viana (IMPA), L. J. Diaz, (PUC-Rio), M. J. Carneiro (UFMG), C. Morales, M. J. Pacifico, E. Pujals (UFRJ), A. Lopes (UFRGS), E. de Faria (USP) have been studying dynamical systems from a global point of view, geometric as well as probabilistic, aiming at developing a general theory of so-called chaotic systems, whose behavior is highly complex and unpredictable.

Some specific research topics: existence of attractors and invariant measures; homoclinic phenomena and fractal dimensions; dynamics of interval and circle maps; attractors of flows and the Lorenz phenomenon; Lagrangean and Hamiltonian systems; dynamical robustness and partial hyperbolicity.

The group keep intense collaboration with researchers abroad, specially in France, USA, and Sweden: J. C. Yoccoz, J. Milnor, M. Lyubich, M. Benedicks, C. Bonatti. It has a strong international presence, through communications to the main scientific meetings, participation in international scientific organizations, and serving in editorial boards of several journals. It is also very active in the training of talented young researchers (Ph.D.s), with a global impact in the Brazilian and Latin American mathematical scenario. Several of its graduates are being absorbed by emergent institutions around Brazil, like UF Ceará, UNESP S Jose do Rio Preto, UF Fluminense, and UF Bahia.

Complex Dynamics and Foliations

C. Camacho, A. Lins Neto, P. Sad (IMPA), M. Soares (UFMG), L. G. Mendes, I. Pan, M. Sebastiani (UFRGS) have been devoted to studying Complex Foliations. This is an area born from Dynamical Systems which uses techniques from Complex Analysis and Algebraic Geometry to analyse problems of global and local nature, often arising from questions of real Dynamical Systems.

Some specific lines of research: analytic invariants of foliations; limit sets of foliations; stratification of the space of foliations; actions tangent to algebraic foliations; dynamics transverse to real subvarieties.

The group keeps close collaborations with other researchers around the world, specially in France, Spain, Italy, and Japan, working in related questions, including M. Brunella, D. Cerveau, R. Moussu, among others.

Its researchers participate as invited speakers in several main international meetings, specially in Europe. The group is incorporating to its activities a number of talented young students who are currently graduating at IMPA.

On focal Decomposition

The concept of focal decomposition ? the object of study ? was introduced in the 1980?s in the context of the 2-point boundary value problem for ordinary second order differential equation. It turned out to be naturally related to such diverse objects as the semi-classical quantization via the Feynman path integral method, the Brillouin zones of solid state physics, the arithmetic of positive definite quadratic forms. This research is going to be carried out by M. Peixoto in collaboration with I. Kupka, C. Pugh, A.R. da Silva.

The immediate aim of the intended research is to prove a pointwise index theorem and a uniform index theorem. The latter means that for all metrics g G, given any two points p,q M then there are at most 2m+2 geodesics of equal length connecting p to q. Besides this estimate is sharp. The proof will involve the multi-transversal theory of J. Mather and the bumpy-metric theorem of R. Abraham as proved by D. Anosov.

Dynamical Systems and Differential Equations

The following problems will be addressed by C. Gutierrez and collaborators :

• With J. Llibre (Barcelona), M. Cobo (UNICAMP), A. Sarmiento (UFMG): injectivity of C^1 planar maps and the relation of this subject with that of assymptotic stability, at infinity, of C^1 planar vector fields.
  
  
• With M. Cobo (UNICAMP), A. López (ICMC-USP), R. de la Llave (University of Texas): generalized interval exchange transformations and smooth foliations on two-manifolds.
  
  
• With R. Oliveira (UE Maringa) and R. Garcia (UF Goiás): phase portrait determination of binary differential equations.
  
  
• With B. Svaiter (IMPA), C. Biasi (ICMC-USP): local and global injectivity problems for maps from R^n into itself.
  
  
• The C^r Closing Lemma.

Qualitative Theory of Differential Equations of Classical Geometry

The following research problems are being studied by J. Sotomayor and his collaborators:

• Description of bifurcation diagram of umbilic points of codimension one and two on surfaces immersed in R^3. Collaboration with C. Gutierrez (ICMC) and R. Garcia (UFG).
  
  
• Study of curvature lines and axial curvature lines of surfaces immersed in R^4. Collaboration with R. Garcia (UFG).
  
  
• Stability and bifurcation of vector fields and implicit differential equations. Collaboration with R. Garcia (UFG).
  
  
• Study of stability and bifurcation of special classes of vector fields (piecewise linear, Lipschitz, reversible). Collaboration with D. Henry (IME-USP) and R. Garcia (UFG).
  
  
• Structural Stability and Bifurcation Analysis of dos Discontinuous / Vector Fields and their relations with Filippov conventions, using the method regularization method. Collaboration with D. Panazzolo (IME-USP), R. Garcia (UFG), A. L. Machado (UNIP) e A. Guzmán (IME-USP).

Completely Integrable Systems

Given a (partial or ordinary) equation ? with discrete or continuous time, one searches for a linearizing change of variables. When this is possible, additional bonuses frequently pop up: an explicit solution to the original equation (with a proof of its existence), which in turns allows for the study of special properties (asymptotics, conserved quantities, phase space behavior), the occasional interpolation of orbits in the discrete time case. The changes of variables usually arise as "inverse problems", of independent interest. The subject combines a number of mathematical techniques - Lie groups, symplectic geometry, algebraic geometry, functional analysis, complex variables.

The following specific directions are considered by C. Tomei : Differential equations: The applications of the direct and inverse theory of nth order differential operators on the line, developed by Beals (Yale), Deift (NYU) and Tomei are far from exhausted.

Spectral theory and matrix numerical analysis: The linearizing change of variables frequently makes use of spectral information, either of operators or matrices. This links differential equations and spectral theory, with applications to numerical analysis (eigenvalue and singular value computation, stability of large matrices arising in hydro planning.

The nonlinearities in integrable systems are, in a sense, false - they disappear under the appropriate change of variables. Certain situations instead are genuinely nonlinear, and their study requires other techniques.

Local and topological aspects of nonlinear functions: By making use of singularity theory and extensions of degree theory, with heavy use of infinite dimensional topology, Malta, Saldanha and Tomei (PUC-Rio) study the global geometry of noninear functions. Their work started with functions from the plane to the plane, then considered periodic differential equations of the first order, and now considers nonlinear Sturm-Liouville operators.

Optimization : Practice with nonlinear equations made possible the study of certain nonconvex optimization problems in hydroelectric planning (subject of a recent PhD thesis directed by C. Tomei, with an extended paper recently submitted). Research continues, in collaboration with engineers.

Asymptotic Behaviour and Bifurcation of Non-linear Equations

In the study of stability and robustness with respect to parameter variation, we often need uniform estimates of attractors, which are independent of parameters, when they vary in a specified set. Therefore, we try to obtain abstract results, in general using Lyapunov like functions, that allow us to get concrete estimates in certain applied problems that involve systems with complex behaviour. These results are used in the analysis of stability, control and synchronization. H. Munhoz, in collaboration with N. Bretas and L. F. C. Alberto, apply the above ideas to the study of Stability of Power Systems and have already obtained two good papers that appeared in excellent international journals. J. G. Ruas Filho is also a collaborator in this subject.

Munhoz is developing an interesting application of synchronization to communication systems. with collaborators M. F. Gameiro, M. G. Simões, E. C. de Souza, they apply the above ideas to a problem of codification-decodification of signals. A prototype of that product has been built using chaotic systems and DSP.