Analysis Methods of Nonlinear Mappings with Applications to Physics of Plasmas

Author(s)

Kai Ullmann

• Document type: Doctoral Thesis
• Press: São Paulo.
• Institution: Universidade de São Paulo (USP). Instituto de Física (IF/SBI)
• Defense date: 1998-02-13

Examining board members

Ibere Luiz Caldas; Ricardo Egydio de Carvalho; Sylvio Ferraz de Mello; Felipe Barbedo Rizzato; Jose Carlos Sartorelli

• Advisor: Ibere Luiz Caldas

Abstract

Initially, we derive an unidimensional dissipative mapping with two parameters, which represents a prototype for soft oscillators perturbed periodically by an external force of constant intensity, direction, and periodicity. We procede then to analyze the possible trajectory classes of this system, using Lyapunov exponents, spectral analysis, and winding numbers, among other algorithms. We also introduce a new analysis tool, called \"bifurcation diagrams in the frequency space\". Then we perform a system analysis on the parameter plane, identifying the different regions of dynamical behavior (periodical, quasi-periodical, and chaotical). We also perform a more detailed analysis of the transitions between these regions, specially their main characteristics in the frequency space. An algorithm to find unstable periodic orbits of this mapping, which are of great importance for studying chaos control with small perturbations, is also introduced, and a statistical study of these orbits presented. Finally, we study the behavior of this system with time-dependent parameters, both regularly, which leads us to the detection of nonchaotic strange attractors and the problems involved in their characterization, and irregularly, reproducing small random fluctuations, always present in experimental situations. (AU)