Describing chaotic systems
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866914866271354880 |
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| author | Le, Brandon |
| author_facet | Le, Brandon |
| contents | In this paper, we discuss the Lyapunov exponent definition of chaos and how it can be used to quantify the chaotic behavior of a system. We derive a way to practically calculate the Lyapunov exponent of a one-dimensional system and use it to analyze chaotic behavior of the logistic map, comparing the $r$-varying Lyapunov exponent to the map's bifurcation diagram. Then, we generalize the idea of the Lyapunov exponent to an $n$-dimensional system and explore the mathematical background behind the analytic calculation of the Lyapunov spectrum. We also outline a method to numerically calculate the maximal Lyapunov exponent using the periodic renormalization of a perturbation vector and a method to numerically calculate the entire Lyapunov spectrum using QR factorization. Finally, we apply both these methods to calculate the Lyapunov exponents of the Hénon map, a multi-dimensional chaotic system. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_07919 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Describing chaotic systems Le, Brandon General Mathematics In this paper, we discuss the Lyapunov exponent definition of chaos and how it can be used to quantify the chaotic behavior of a system. We derive a way to practically calculate the Lyapunov exponent of a one-dimensional system and use it to analyze chaotic behavior of the logistic map, comparing the $r$-varying Lyapunov exponent to the map's bifurcation diagram. Then, we generalize the idea of the Lyapunov exponent to an $n$-dimensional system and explore the mathematical background behind the analytic calculation of the Lyapunov spectrum. We also outline a method to numerically calculate the maximal Lyapunov exponent using the periodic renormalization of a perturbation vector and a method to numerically calculate the entire Lyapunov spectrum using QR factorization. Finally, we apply both these methods to calculate the Lyapunov exponents of the Hénon map, a multi-dimensional chaotic system. |
| title | Describing chaotic systems |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2407.07919 |