Daniel S. Graça ; Ning Zhong - Robust non-computability of dynamical systems and computability of robust dynamical systems

lmcs:11381 - Logical Methods in Computer Science, June 26, 2024, Volume 20, Issue 2 - https://doi.org/10.46298/lmcs-20(2:19)2024
Robust non-computability of dynamical systems and computability of robust dynamical systemsArticle

Authors: Daniel S. Graça ; Ning Zhong

    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.


    Volume: Volume 20, Issue 2
    Published on: June 26, 2024
    Accepted on: April 23, 2024
    Submitted on: May 26, 2023
    Keywords: Mathematics - Logic,Computer Science - Logic in Computer Science,Mathematics - Dynamical Systems,03D78 (Primary) 37D05, 34E10 (Secondary),F.4.1,F.1.1

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