Using Lyapunov Exponents to Explain the Dynamics of Complex Systems

Abstract

Complex systems often display chaotic dynamics, characterised by being exponentially sensitive to changes in initial conditions. Such systems are in general difficult to analyse, due to the large number of nonlinearly interacting degrees of freedom. Dynamical-systems theory provides a framework for analysing such systems. One of the tools from this theory is the Lyapunov exponent, which quantifies the rate at which initially nearby trajectories converge or diverge over time. The exponent can be used to study how the stability of a complex system depends on different system parameters. The finite-time Lyapunov exponent can be used to reveal organising structures in the phase space of the system that separate it into different characteristic regions. These structures are referred to as Lagrangian coherent structures. In this thesis, the Lyapunov exponent and Lagrangian coherent structures are used to explore the properties of complex systems. In the two presented papers, artificial neural networks are analysed, which are machine-learning algorithms with a large number of interconnected nonlinear computational nodes. We show that these systems can be analysed as complex dynamical systems, and show, among other things, how this perspective helps shedding light on how the neural networks learn to perform classification tasks. Additionally, a project on how microswimmers can escape through transport barriers in flows using orientational diffusion is presented, where the transport barriers are Lagrangian coherent structures.