Investigating Hamiltonian Dynamics by the Method of Covariant Lyapunov Vectors

TL;DR

This work develops and applies covariant Lyapunov vector theory to autonomous Hamiltonian dynamics, grounding the analysis in Ginelli’s CLV algorithm and a detailed convergence framework. It systematically builds the mathematical and numerical toolkit (QR/SVD, principal angles, subspace distances) and then validates these methods on low-dimensional systems (Hénon–Heiles and a 3-D model), uncovering centre-subspace shear dynamics and the need for centre corrections to accurately recover CLVs. Extending to a high-dimensional DNA lattice (Peyrard–Bishop–Dauxois), the thesis introduces instantaneous Lyapunov exponents and vectors, and reveals localized ILV distributions correlated with site displacements and bubbles, along with weighted ILV measures that connect local instability to dynamic structural features. The findings illuminate convergence rates, centre-subspace geometry, and hyperbolicity variations in Hamiltonian systems, and demonstrate how CLVs and ILVs can yield practical, fine-grained insights into complex, high-dimensional dynamics with potential applications to biomolecular systems. Overall, the work advances both methodology and application of CLV-based stability analysis in physics and biology, providing concrete guidelines for transient times and centre-subspace treatment in CLV computations.

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 Hénon-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 Hénon-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.

Figures (57)

• Figure 1: Diagram of QR decomposition. The blue stripes denote the orthonormal columns of $Q$, the red stripe is the positive diagonal of $R$, and the grey region indicates where entries of $R$ are zero.
• Figure 2: Diagram of singular value decomposition. The blue stripes denote the orthonormal columns of $U$ and $V$, the red stripe is the non-negative diagonal of $\Sigma$, and the grey regions indicate where entries of $\Sigma$ are zero.
• Figure 3: Example of a $2\times 2$ real matrix $A$ which maps a unit circle to an ellipse. The principal semi-axes of the ellipse are denoted by the vectors $\sigma_i\hat{\boldsymbol{\mathrm{u}}}_i$, $i=1,2$, and their corresponding pre-images under $A$ are denoted by $\hat{\boldsymbol{\mathrm{v}}}_i$.
• Figure 4: An example of the minimum angle $\theta_1$ between two 1-D subspaces $U$ and $V$ (drawn as red lines), where $\hat{\boldsymbol{\mathrm{u}}}_1\in U$ and $\hat{\boldsymbol{\mathrm{v}}}_1\in V$ (drawn as black arrows) are representative unit vectors with maximal dot product.
• Figure 5: Two examples of principal angles and vectors (drawn as black arrows) between subspaces $U$ and $V$ (drawn as red lines or planes) in $\mathbb{R}^3$. In the example on the left, $\dim V=1$, while $\dim V=2$ on the right. In both cases, $\dim U=2$.