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

Daniel S. Graça, Ning Zhong

TL;DR

This work addresses the computability of basins of attraction in dynamical systems and how global stability interacts with local stability. It combines a discrete-time construction (via a Turing-machine encoding) and a continuous-time ODE formulation to show that a system can have a computable, stable sink while its basin of attraction remains robustly non-computable under perturbations. A key insight is that local stability near a sink does not ensure basin computability, but global stability on a compact planar domain restores computability for basins of sinks in structurally stable systems. The results delineate a boundary between negative computability phenomena and positive computability guarantees, emphasizing the role of global dynamics and compactness for practical numerical analysis of basins.

Abstract

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.

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

TL;DR

This work addresses the computability of basins of attraction in dynamical systems and how global stability interacts with local stability. It combines a discrete-time construction (via a Turing-machine encoding) and a continuous-time ODE formulation to show that a system can have a computable, stable sink while its basin of attraction remains robustly non-computable under perturbations. A key insight is that local stability near a sink does not ensure basin computability, but global stability on a compact planar domain restores computability for basins of sinks in structurally stable systems. The results delineate a boundary between negative computability phenomena and positive computability guarantees, emphasizing the role of global dynamics and compactness for practical numerical analysis of basins.

Abstract

In this paper, we examine the relationship between the stability of the dynamical system and the computability of its basins of attraction. We present a computable system that possesses a computable and stable equilibrium point, yet whose basin of attraction is robustly non-computable in a neighborhood of in the sense that both the equilibrium point and the non-computability of its associated basin of attraction persist when 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.
Paper Structure (10 sections, 12 theorems, 44 equations, 4 figures)

This paper contains 10 sections, 12 theorems, 44 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Computable analysis
  4. Dynamical systems
  5. Proof of Theorem B -- robust non-computability in the discrete-time case
  6. Proof of Theorem A -- robust non-computability in the continuous-time case
  7. Iterating a map with an ODE
  8. Ensuring that the halting configuration is a sink
  9. Establishing robust non-computability under perturbations
  10. Proof of Theorem C -- Basins of attraction of structurally stable planar systems are uniformly computable

Key Result

Theorem A

There exists a computable $C^{\infty}$ function $f$ for which the system (eq_main) possesses a computable sink $s_0$, but the basin of attraction of $s_0$ is non-computable. Moreover, this non-computability is robust and persists under small perturbations.

Figures (4)

  • Figure 1: Example of a dynamical system having three equilibrium points $A,B,C$. The points $A$ and $B$ are sinks (i.e. stable equilibrium points) while $C$ is not (it is a so-called saddle equilibrium point). The region in orange is the basin of attraction of $A$ while the region in blue is the basin of attraction of $B$.
  • Figure 2: An example of a structurally stable system on the left. Even if perturbed the main properties of the system persist. For example, there is a connection between the sink $A$ and the saddle $C$ and similarly for $B$ and $C$ which persists under (small perturbation).
  • Figure 3: An example of a structurally unstable system on the left, known as a saddle connection. We can find a perturbation, which can be assumed to be as small as we want, that is able to break the connection between the saddles $A$ and $B$.
  • Figure 4: Result of the algorithm from GZ22 which computes hyperbolic equilibrium points and hyperbolic periodic orbits with some given (input) accuracy. The periodic orbit is surrounded by a (red) outer boundary and an inner (blue) boundary which delimitates a region approximating the periodic orbit. The orange square delimitates an equilibrium point.

Theorems & Definitions (24)

  • Theorem A
  • Theorem B
  • Theorem C
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Proposition 3.1: GZ15
  • ...and 14 more