S.E. de S. Pinto

Periodic-orbit analysis and scaling laws of intermingled basins of attraction in an ecological dynamical system.

Chaotic dynamical systems with two or more attractors lying on invariant subspaces may, provided certain mathematical conditions are fulfilled, exhibit intermingled basins of attraction: Each basin is riddled with holes belonging to basins of the other attractors. In order to investigate the occurrence of such phenomenon in dynamical systems of ecological interest (two-species competition with extinction) we have characterized quantitatively the intermingled basins using periodic-orbit theory and scaling laws. The latter results agree with a theoretical prediction from a stochastic model, and also with an exact result for the scaling exponent we derived for the specific class of models investigated. We discuss the consequences of the scaling laws in terms of the predictability of a final state (extinction of either species) in an ecological experiment.

Influence of impurities on the dynamics and synchronization of coupled map lattices

The influence of impurities and defects on the dynamics and synchronization of coupled map lattices (CML) is studied. In the context of CML we define impurities as sites in the lattice which have another local dynamics that from the whole lattice and defects as sites in the lattice without any dynamics. We show that synchronization and spatial intermittence are obtained as a function of the number of impurities present on a one-dimensional lattice. We also derive an analytical condition for a signal to “transpose” an impurity. For open flow models, we show that not only the presence of the impurity but also its position along the lattice and its local dynamics can be used to manipulate the lattice in order to obtain a regular or irregular motion. We also show how defects can be used to store information in a lattice.

Parametric evolution of unstable dimension variability in coupled piecewise-linear chaotic maps.

In the presence of unstable dimension variability numerical solutions of chaotic systems are valid only for short periods of observation. For this reason, analytical results for systems that exhibit this phenomenon are needed. Aiming to go one step further in obtaining such results, we study the parametric evolution of unstable dimension variability in two coupled bungalow maps. Each of these maps presents intervals of linearity that define Markov partitions, which are recovered for the coupled system in the case of synchronization. Using such partitions we find exact results for the onset of unstable dimension variability and for contrast measure, which quantifies the intensity of the phenomenon in terms of the stability of the periodic orbits embedded in the synchronization subspace.