Asbroek, Tygo (2023) The Hénon Map. Bachelor's Thesis, Mathematics.
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Abstract
The H ́enon map is a two-dimensional quadratic map that, for some parameter values, has a strange attractor. This strange attractor has comparable properties to the Lorenz attractor and the strange attractors of many other systems. In this project, we discuss the map’s history and its relation to the Lorenz system. We classify the stability of the fixed points of the H ́enon map and identify curves in the parameter plane where period doubling bifurcations occur. Upon observing numerical evidence of a period doubling cascade, we identify the possibility for regions in the parameter plane where the map is chaotic. To further classify such regions, we rely on Lyapunov exponents. By means of a Lyapunov diagram, we then classify the system’s stability and attractor type in a parameter plane, where we observe numerical evidence of a chaotic attractor. Lastly, we discuss the H ́enon attractor and some of its properties. In particular, to further investigate its fractal properties, we give numerical estimates of its box-counting dimension and Lyapunov dimension. We conclude with a discussion regarding the relation that the H ́enon attractor and the unstable manifold of one its saddle fixed points seem to have.
| Item Type: | Thesis (Bachelor's Thesis) |
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| Supervisor name: | Sterk, A.E. and Luppes, R. |
| Degree programme: | Mathematics |
| Thesis type: | Bachelor's Thesis |
| Language: | English |
| Date Deposited: | 02 Aug 2023 13:57 |
| Last Modified: | 02 Aug 2023 13:57 |
| URI: | https://fse.studenttheses.ub.rug.nl/id/eprint/31065 |
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