DYNAMICAL SYSTEMS

Title: DYNAMICAL SYSTEMS

Research Question: How can we define computational universality for dynamical systems? What are the dynamical properties of a universal system?

Methodology: The researchers propose a general definition of universality that applies to arbitrary discrete time symbolic dynamical systems. They define universality as undecidability of a model-checking problem. This definition encompasses various types of dynamical systems, such as Turing machines, cellular automata, and subshifts.

Results: The researchers derive necessary conditions for undecidability and universality. They propose that universal systems must have a sensitive point and a proper subsystem. They also discuss the thesis that computation should occur at the 'edge of chaos,' and they exhibit a universal chaotic system.

Implications: This research provides a robust and universal definition of computational universality for dynamical systems. It also offers insights into the dynamical properties of universal systems and their relationship with computability and complexity. The implications of this work extend to various fields, including physics, computer science, and mathematics.

Link to Article: https://arxiv.org/abs/0404021v4 Authors: arXiv ID: 0404021v4