Dynamical Systems




Homepage: www.ime.unicamp.br/~sdimecc


Research Topics

Our main purpose is to integrate within the Mathematics Institute at UNICAMP, the different research groups interested in investigating geometric and topological aspects of dynamical phenomena.  There is frequent interaction among these groups due to similar methods and techniques that are applicable to solving problems in these areas.


Piecewise Smooth Dynamical Dystems

Piecewise Smooth Dynamical Systems This line of research has been developed vary fast these last years, mainly due to its mathematical beauty and its strong relationship with other branches of science. In addition, establishing definitions that are consistent with applications represents a great challenge in this area. The theory of piecewise smooth dynamical systems becomes one of the common frontier between mathematics, physics, and engineering. Models in control theory, mechanical systems with impact, and nonlinear oscillations are the main sources of motivation for this line of research. Our goal consists in the study of local and global aspects of these systems that contemplate existing models.


Reversible and Equivariant Dynamical Systems

Most of the dynamical systems coming from physical problems have symmetry properties in their solutions. The study of these symmetries, as well as the systems admitting them, is the main goal of this line of research. In particular, we are interested in systems with symmetries that can be transformed into Hamiltonian/Integrable systems. In these cases, the problem of existence and persistence of invariant tori for such systems is also considered.


Topological Dynamical Systems

The main interest in this field lies in describing via differential and algebraic topology, dynamical aspects of a system or of a one-parameter family of dynamical systems under continuation. In the first case, one hopes to establish invariants that give us information on the components of the chain recurrent set (e.g. singularities, periodic orbits, SSFT) and their connections e.g. unstable manifolds. In the latter case, we consider bifurcation behaviour, birth and death of connections caused by cancellations of singularities within a continuation. One fruitful technique has been to use homotopical and homological tools found in the Conley Index Theory. Conley’s theory permits to rise above the differentiability requirements of the phase space as well as to consider richer isolated invariant sets. With these tools, one can consider as an important algebraic apparatus, a spectral sequence of a given filtered chain complex generated by the invariant sets. The unfolding of the spectral sequence exhibits rich algebraic information and provides much insight into dynamical properties (bifurcation, cancellation phenomena etc.) of a continuation of the dynamical system being studied. These techniques apply well to continuous systems such as Morse-Smale, Morse-Bott, Novikov and Gutierrez- Sotomayor flows as well as discontinuous systems. Also it can be used in the discrete case by considering Morse-Smale and Smale diffeomorphisms.


Ergodic Optimization

Ergodic optmization Ergodic theory is the branch of dynamical systems interested in the study of the iterations of a measurable map which is invariant by a given measure. The evolution of the study of maximizing invariant probabilities have given origin to an instigating research field in this branch, known as ergodic optimization. This new area is interested in analise the invariant probabilities with the maximal or the minimal ergodic average with respect to a given potential defined on the fase space of the dynamical system.


Smooth Ergodic Theory, Anosov Diffeomorphisms and Partially Hyperbolic Diffeomorphisms

The main goal is to study the dynamics of partially hyperbolic diffeomorphisms and Anosov diffeomorphisms using technics from abstract ergodic theory and the regular invariant structures associated to the given dynamics such as the stable and unstable manifolds. Among the main tools in this direction are the technics on measure disintegration along invariant foliations and Lyapunov exponents.


Morse Homology for the heat flow

The theory of hyperbolic dynamical systems allows the construction of a Morse complex associated with the gradient in closed, smooth and finite-dimensional manifolds M. The corresponding homology is called Morse homology and represents the singular homology of M. The flow of this gradient is called the heat flow. We are interested in the global analysis involving elliptic and parabolic PDEs that define Morse and Floer homologies. In addition, we are interested in the relations between Hamiltonian systems and closed geodesics in Riemannian manifolds.




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