Investigating Hamiltonian dynamics by the method of covariant Lyapunov vectors
| dc.contributor.advisor | Skokos, Charalampos, Hillebrand, Malcolm | |
| dc.contributor.author | Du Plessis, Jean-Jacq | |
| dc.date.accessioned | 2025-02-07T11:44:09Z | |
| dc.date.available | 2025-02-07T11:44:09Z | |
| dc.date.issued | 2024 | |
| dc.date.updated | 2025-02-07T11:41:26Z | |
| dc.description.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. | |
| dc.identifier.apacitation | Du Plessis, J. (2024). <i>Investigating Hamiltonian dynamics by the method of covariant Lyapunov vectors</i>. (). University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics. Retrieved from http://hdl.handle.net/11427/40889 | en_ZA |
| dc.identifier.chicagocitation | Du Plessis, Jean-Jacq. <i>"Investigating Hamiltonian dynamics by the method of covariant Lyapunov vectors."</i> ., University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics, 2024. http://hdl.handle.net/11427/40889 | en_ZA |
| dc.identifier.citation | Du Plessis, J. 2024. Investigating Hamiltonian dynamics by the method of covariant Lyapunov vectors. . University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics. http://hdl.handle.net/11427/40889 | en_ZA |
| dc.identifier.ris | TY - Thesis / Dissertation AU - Du Plessis, Jean-Jacq AB - 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. DA - 2024 DB - OpenUCT DP - University of Cape Town KW - applied mathematics LK - https://open.uct.ac.za PB - University of Cape Town PY - 2024 T1 - Investigating Hamiltonian dynamics by the method of covariant Lyapunov vectors TI - Investigating Hamiltonian dynamics by the method of covariant Lyapunov vectors UR - http://hdl.handle.net/11427/40889 ER - | en_ZA |
| dc.identifier.uri | http://hdl.handle.net/11427/40889 | |
| dc.identifier.vancouvercitation | Du Plessis J. Investigating Hamiltonian dynamics by the method of covariant Lyapunov vectors. []. University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics, 2024 [cited yyyy month dd]. Available from: http://hdl.handle.net/11427/40889 | en_ZA |
| dc.language.rfc3066 | eng | |
| dc.publisher.department | Department of Mathematics and Applied Mathematics | |
| dc.publisher.faculty | Faculty of Science | |
| dc.publisher.institution | University of Cape Town | |
| dc.publisher.institution | University of Cape Town | |
| dc.subject | applied mathematics | |
| dc.title | Investigating Hamiltonian dynamics by the method of covariant Lyapunov vectors | |
| dc.type | Thesis / Dissertation | |
| dc.type.qualificationlevel | Masters | |
| dc.type.qualificationlevel | MSc |