Computational Dynamical Systems

TL;DR

This work elucidates what it means for one ‘machine’ to simulate another, and emphasizes the necessity of defining low-complexity 'encoders' and 'decoders' to translate between the dynamics of the simulation and the system being simulated.

Abstract

We study the computational complexity theory of smooth, finite-dimensional dynamical systems. Building off of previous work, we give definitions for what it means for a smooth dynamical system to simulate a Turing machine. We then show that 'chaotic' dynamical systems (more precisely, Axiom A systems) and 'integrable' dynamical systems (more generally, measure-preserving systems) cannot robustly simulate universal Turing machines, although such machines can be robustly simulated by other kinds of dynamical systems. Subsequently, we show that any Turing machine that can be encoded into a structurally stable one-dimensional dynamical system must have a decidable halting problem, and moreover an explicit time complexity bound in instances where it does halt. More broadly, our work elucidates what it means for one 'machine' to simulate another, and emphasizes the necessity of defining low-complexity 'encoders' and 'decoders' to translate between the dynamics of the simulation and the system being simulated. We highlight how the notion of a computational dynamical system leads to questions at the intersection of computational complexity theory, dynamical systems theory, and real algebraic geometry.

Figures (6)

• Figure 1: For Example \ref{['example:uncon1']} we depict $M = [0,1]^2$, and shade in regions that encode configurations of the Turing machine. Each such region is labelled by the configuration into which it is decoded by the $\mathcal{D}$ given in the example.
• Figure 2: A sketch of the Poincaré section of a map. We consider time-dependent dynamics $f_t : M \to M$ with $f_0 = \text{Id}$. A Poincaré section of the map is a submanifold $N$ of $M$ such that for any $x \in N$ and any $k \in \mathbb{Z}_{\geq 0}$, we can define $f^k(x)$ as the $k$th time the trajectory $f_t(x)$ intersects with $N$ (which we require to occur for each $k$).
• Figure 3: Various fundamental examples in dynamical systems theory. (a) A circle map. (b) An integrable system represented as a base manifold with tori fibered over it. The red curves are either periodic or irrational orbits, contingent on the corresponding point on the base manifold. (c) The Arnold cat map. (d) One iteration of the Smale horseshoe map. (e) A member of the quadratic family. Depicted is a cobweb diagram for an orbit. (f) A gradient flow system, with various flow lines depicted. (g) An orbit of the Lorenz system, tending towards the Lorenz attractor. (h) An instance of the standard map, with various orbits displayed.
• Figure 4: An illustration of the some of the sets $C_s$ in the square $[-\frac{1}{3},1]^2$. The numerical labels of the sets indicate the corresponding $s$, with the understanding that there are infinitely many zeros to the left and infinitely many zeros to the right (i.e. $1.01$ is shorthand for ${ \hbox{{\cr \hidewidth\reflectbox{$\m@th\vec{}\mkern4mu$}\hidewidth\cr {} $\m@th0$\cr }}}1.01\vec{0}$). The sets are organized according to a thickened version of a Cantor encoding in two dimensions.
• Figure 5: A depiction of how the map $\textsf{S}$ in \ref{['E:Shift1']} acts on two subsets of the square $[-\frac{1}{3},1]^2$. Some of the sets $C_s$ are depicted (akin to Figure \ref{['fig:Cantor1']}) to clarify the action of the mapping $\textsf{S}$. The red rectangle is shifted to the right and nonlinearly compressed in the $x$-direction, as well as nonlinearly stretched in the $y$-direction. The blue rectangle is also nonlinearly compressed in the $x$-direction as well as nonlinearly stretched in the $y$-direction. Hence $\textsf{S}$ is a nonlinear version of the Baker's map, and indeed acts as a shift map since it takes $C_s$ to $C_{\sigma(s)}$.

Theorems & Definitions (143)

• Definition 1.1: informal; see Definition \ref{['def:CDS1']}
• Remark 1.2
• Definition 1.3
• Theorem 1.4
• Theorem 1.5: see Corollary \ref{['corr:measurepreserve']}
• Theorem 1.6
• Remark 1.7
• Theorem 1.8: see Theorem \ref{['thm:measurepreserve']}
• Conjecture 1.9
• Theorem 1.10