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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2305.14448 |
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| _version_ | 1866916347551678464 |
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| author | Graça, Daniel S. Zhong, Ning |
| author_facet | Graça, Daniel S. Zhong, Ning |
| contents | In this paper, we examine the relationship between the stability of the dynamical system $x^{\prime}=f(x)$ and the computability of its basins of attraction. We present a computable $C^{\infty}$ system $x^{\prime}=f(x)$ that possesses a computable and stable equilibrium point, yet whose basin of attraction is robustly non-computable in a neighborhood of $f$ in the sense that both the equilibrium point and the non-computability of its associated basin of attraction persist when $f$ is slightly perturbed. This indicates that local stability near a stable equilibrium point alone is insufficient to guarantee the computability of its basin of attraction. However, we also demonstrate that the basins of attraction associated with a structurally stable - globally stable (robust) - planar system defined on a compact set are computable. Our findings suggest that the global stability of a system and the compactness of the domain play a pivotal role in determining the computability of its basins of attraction. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2305_14448 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Robust non-computability of dynamical systems and computability of robust dynamical systems Graça, Daniel S. Zhong, Ning Logic Logic in Computer Science Dynamical Systems 03D78 (Primary) 37D05, 34E10 (Secondary) F.4.1; F.1.1 In this paper, we examine the relationship between the stability of the dynamical system $x^{\prime}=f(x)$ and the computability of its basins of attraction. We present a computable $C^{\infty}$ system $x^{\prime}=f(x)$ that possesses a computable and stable equilibrium point, yet whose basin of attraction is robustly non-computable in a neighborhood of $f$ in the sense that both the equilibrium point and the non-computability of its associated basin of attraction persist when $f$ is slightly perturbed. This indicates that local stability near a stable equilibrium point alone is insufficient to guarantee the computability of its basin of attraction. However, we also demonstrate that the basins of attraction associated with a structurally stable - globally stable (robust) - planar system defined on a compact set are computable. Our findings suggest that the global stability of a system and the compactness of the domain play a pivotal role in determining the computability of its basins of attraction. |
| title | Robust non-computability of dynamical systems and computability of robust dynamical systems |
| topic | Logic Logic in Computer Science Dynamical Systems 03D78 (Primary) 37D05, 34E10 (Secondary) F.4.1; F.1.1 |
| url | https://arxiv.org/abs/2305.14448 |