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Investigating Hamiltonian dynamics by the method of covariant Lyapunov vectors
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...
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| Format: | Thesis |
| Language: | English |
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Department of Mathematics and Applied Mathematics
2025
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| _version_ | 1867614360990384128 |
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| access_status_str | Open Access |
| author | Du Plessis, Jean-Jacq |
| author2 | Skokos, Charalampos, Hillebrand, Malcolm |
| author_browse | Du Plessis, Jean-Jacq Skokos, Charalampos, Hillebrand, Malcolm |
| author_facet | Skokos, Charalampos, Hillebrand, Malcolm Du Plessis, Jean-Jacq |
| author_sort | Du Plessis, Jean-Jacq |
| collection | Thesis |
| description | 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. |
| format | Thesis |
| id | oai:open.uct.ac.za:11427/40889 |
| institution | University of Cape Town (South Africa) |
| language | eng |
| last_indexed | 2026-06-10T12:50:49.028Z |
| license_str | Not specified — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from UCTD — University of Cape Town Open Access Repository |
| publishDate | 2025 |
| publishDateRange | 2025 |
| publishDateSort | 2025 |
| publisher | Department of Mathematics and Applied Mathematics |
| publisherStr | Department of Mathematics and Applied Mathematics |
| record_format | dspace |
| source_str | UCTD — University of Cape Town Open Access Repository |
| spelling | oai:open.uct.ac.za:11427/40889 Investigating Hamiltonian dynamics by the method of covariant Lyapunov vectors Du Plessis, Jean-Jacq Skokos, Charalampos, Hillebrand, Malcolm applied mathematics 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. 2025-02-07T11:44:09Z 2025-02-07T11:44:09Z 2024 2025-02-07T11:41:26Z Thesis / Dissertation Masters MSc http://hdl.handle.net/11427/40889 eng application/pdf Department of Mathematics and Applied Mathematics Faculty of Science University of Cape Town University of Cape Town |
| spellingShingle | applied mathematics Du Plessis, Jean-Jacq Investigating Hamiltonian dynamics by the method of covariant Lyapunov vectors |
| thesis_degree_str | Master's |
| title | Investigating Hamiltonian dynamics by the method of covariant Lyapunov vectors |
| title_full | Investigating Hamiltonian dynamics by the method of covariant Lyapunov vectors |
| title_fullStr | Investigating Hamiltonian dynamics by the method of covariant Lyapunov vectors |
| title_full_unstemmed | Investigating Hamiltonian dynamics by the method of covariant Lyapunov vectors |
| title_short | Investigating Hamiltonian dynamics by the method of covariant Lyapunov vectors |
| title_sort | investigating hamiltonian dynamics by the method of covariant lyapunov vectors |
| topic | applied mathematics |
| url | http://hdl.handle.net/11427/40889 |
| work_keys_str_mv | AT duplessisjeanjacq investigatinghamiltoniandynamicsbythemethodofcovariantlyapunovvectors |