Investigating Hamiltonian Dynamics by the Method of Covariant Lyapunov Vectors

TL;DR

This thesis applies covariant Lyapunov vectors to analyze Hamiltonian systems, improving computational methods, revealing system hyperbolicity features, and exploring DNA bubble dynamics through Lyapunov-based measures.

Contribution

It introduces an efficient algorithm for computing CLVs, develops methods for convergence analysis, and applies these to Hamiltonian systems and DNA models, revealing new dynamical insights.

Findings

Convergence rates of CLVs can be effectively measured.

Splitting subspaces become tangent during sticky regimes, indicating hyperbolicity.

Size of DNA bubbles correlates with ILV distributions, not CLV spatial distribution.

Abstract

In this thesis, we review the theory of Lyapunov exponents and covariant Lyapunov vectors (CLVs) and use these objects to numerically investigate the dynamics of several autonomous Hamiltonian systems. The algorithm which we use for computing CLVs is the one developed by Ginelli and collaborators (G&C), which is quite efficient and has been used previously in many numerical investigations. Using two low-dimensional Hamiltonian systems as toy models, we develop a method for measuring the convergence rates of vectors and subspaces computed via the G&C algorithm, and we use the time it takes for this convergence to occur to determine the appropriate transient time lengths needed when applying this algorithm to compute CLVs. The tangent dynamics of the centre subspace of the H\'enon-Heiles system is investigated numerically through the use of CLVs, and we propose a method that improves the accuracy of the centre subspace computed with the G&C algorithm. As another application of the method of CLVs to the H\'enon-Heiles system, we find that the splitting subspaces (which form a splitting of the tangent space and define the CLVs) become almost tangent during sticky regimes of motion, an observation which is related to the hyperbolicity of the system. Additionally, we investigate the dynamics of bubbles (i.e. thermal openings between base pairs) in homogeneous DNA sequences using the Peyrard-Bishop-Dauxois lattice model of DNA. For the purpose of studying short-lived bubbles in DNA, the notions of instantaneous Lyapunov vectors (ILVs) are introduced in the context of Hamiltonian dynamics. While we find that the size of the opening between base pairs has no clear relationship with the spatial distribution of the first CLV at that site, we do observe a distinct relationship with various ILV distributions.