Nonlinear dynamics

Abstract

The aim of this project is to apply the Nonlinear Dynamical Systems and Chaos Theory to the theoretical and experimental studies of a broad range of problems: Turbulence and Hamiltonian Chaos in Plasma, Classical and "Quantum " Chaos, Biological Neural Networks / Biological Rhythms. We will study the mixing and the lagrangean transport in chaotic systems described by hamiltonian quasi-integrable systems with few degrees of freedom. Simplectic maps will be used to precisely approach the properties of trapping and escape of flow tines in degenerate systems with non-monotonic angular frequency profiles. Systems describing the turbulence in magnetic confined plasma will be considered and the chaotic transport of confined particles will be investigated. We will also study the transport properties of chaotic magnetic field lines in a Tokamak with magnetic limiters or divertors. Electric probes will be used to study the coupling between the modes of the turbulent oscillations at the Tokamak edge; we will apply spectral analysis and wavelets techniques to evaluate the anomalous transport of particles due to the turbulence. To introduce new techniques for communications with chaos, we will develop experimental tools and systems to study the control and synchronization of chaotic electric circuits coupled with themselves or connected with plasma discharge tubes. We will study the pattern formation in vibration-forced granular systems, the experimental bifurcations and the dynamical properties of chaotic attractors from electric circuits and hydrodynamic systems (formation of bubbles and drops). The properties of these systems will be investigated by applying both metrical and topological characterization methods: spectrum of Lyapunov exponents, construction of parameter spaces, isoperiodic diagrams, symbolic planes, etc. Nonlinear Dynamical Systems techniques will be also extensively used to the computational and experimental study of biological and hybrid neural networks (networks produced by coupling biological and electronic artificial neurons). We will use biological neurons that control the movements of the stomach of the Brazilian crab Callinectes sapidus and mathematical models of the electrical neural activity implemented in analog electronic circuits. Both systems will be used to study the formation of spatio-temporal patterns, which are both reliable and flexible from the connection of chaotic biological neurons, as well as to the study of how information is transmitted and processed in a small biological (or hybrid) neural network.We will also use nonlinear models and techniques to analyze experimental data from higher levels of organization in biological systems and to investigate, by computational modeling and experimentally, various circadian rhythms present in the behavior of honeybee colonies and hamsters. (AU)