Realizing semicomputable simplices by computable dynamical systems

Coronel, Daniel; Frank, Alexander; Hoyrup, Mathieu; Rojas, Cristóbal

doi:10.1016/j.tcs.2022.09.001

Cited by 1 publication

(1 citation statement)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Furthermore, non-computability results about dynamical systems are not restricted only to basins of attraction for differential equations (some examples can be found in e.g. [PER79], [PER81] [CFHR22]). These discoveries highlight the need to understand the limitations of numerical methods in the analysis of dynamical systems.…”

Section: Introductionmentioning

confidence: 99%

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

Graça,

Zhong

2024

Logical Methods in Computer Science

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

confidence: 99%