(Georgia Institute of Technology, 2007-08-07)
Lessard, Jean-Philippe
Studying the zeros of a parameter dependent operator F defined on a Hilbert space H is a fundamental problem in mathematics. When the Hilbert space is finite dimensional, continuation provides, via predictor-corrector algorithms, efficient techniques to numerically follow the zeros of F as we move the parameter. In the case of infinite dimensional Hilbert spaces, this procedure must be applied to some finite dimensional approximation which of course raises the question of validity of the output. We introduce a new technique that combines the information obtained from the predictor-corrector steps with ideas from rigorous computations and verifies that the numerically produced zero for the finite dimensional system can be used to explicitly define a set which contains a unique zero for the infinite dimensional problem F: HxR->Im(F).
We use this new validated continuation to study equilibrium solutions of partial differential equations, to prove the existence of chaos in ordinary differential equations and to follow branches of periodic solutions of delay differential equations. In the context of partial differential equations, we show that the cost of validated continuation is less than twice the cost of the standard continuation method alone.
(Georgia Institute of Technology, 2005-07-19)
Moeller, Todd Keith
We introduce a new class of Conley-Morse chain maps for the purpose of comparing the qualitative structure of flows across multiple scales.
Conley index theory generalizes classical Morse theory as a tool for studying the dynamics of flows. The qualitative structure of a flow, given a Morse decomposition, can be stored algebraically as a set of homology groups (Conley indices) and a boundary map between the indices (a connection matrix). We show that as long as the qualitative structures of two flows agree on some, perhaps coarse, level we can construct a chain map between the corresponding chain complexes that preserves the relations between the (coarsened) Morse sets. We present elementary examples to motivate applications to data analysis.
(Georgia Institute of Technology, 2005-07-19)
Gameiro, Marcio Fuzeto
We use computational homology to characterize the geometry of complicated
time-dependent patterns. Homology provides very basic topological (geometrical)
information about the patterns, such as the
number of components (pieces) and
the number of holes. For 3-dimensional patterns it also provides the number of
voids. We apply these techniques to patterns generated by experiments on spiral
defect chaos, as well as to numerically simulated patterns in the Cahn-Hilliard
theory of phase separation and on spiral wave patterns in excitable media.
These techniques allow us to distinguish patterns at different parameter values,
to detect complicated dynamics through the computation of positive Lyapunov
exponents and entropies, to compare experimental data with numerical simulations,
to quantify boundary effects on finite size domains, among other things.