Integrability and Chaos - algebraic and geometric approach

TL;DR

This thesis explores complexity in dynamical systems through algebraic and geometric methods, focusing on integrability, chaos, Lyapunov exponents, and variational equations, with practical algorithms and examples.

Contribution

It introduces a combined algebraic and geometric framework for analyzing dynamical systems' complexity and provides an algorithm for computing Lyapunov spectra.

Findings

Differential Galois group theory constrains system integrability

Geometric formulation aids in understanding chaos and stability

Algorithm successfully computes Lyapunov spectra in examples

Abstract

This thesis presents two descriptions of complexity in dynamical systems. The algebraic approach deals with the differential Galois group theory and its restrictions on integrability. The geometric part is a formulation of dynamics in the language of differential geometry with particular application to Lyapunov exponents and variational equations. The algorithm for calculating the Lyapunov spectrum is illustrated with three examples.