Robust non-computability and stability of dynamical systems
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In this paper, we examine the relationship between the stability of the dynamical system x^'=f(x) and the computability of its basins of attraction. We present a computable C^∞ system x^'=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 - planar system are computable. Our findings suggest that the global stability of a system plays a pivotal role in determining the computability of its basins of attraction.
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