Robust non-computability of dynamical systems and computability of robust dynamical systems

• Published in: arXiv.org
• Main Authors: Graça, Daniel S, Zhong, Ning
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 25.06.2024
• Subjects: Attraction; Basins; Computer Science - Logic in Computer Science; Dynamic stability; Dynamical systems; Equilibrium; Mathematics - Dynamical Systems; Mathematics - Logic
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2305.14448

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.