Discrete Linear Dynamical Systems

Abstract

The theory of dynamical systems is concerned with describing and studying the evolution of systems over time, where a ‘system’ is represented as a vector of variables, and there is a fixed rule governing how the system evolves. Dynamical systems originate in the development of Newtonian mechanics, and have widespread applications in many areas of science and engineering. Systems that evolve in a piecewise continuous manner (typically via differential equations) are known as continuous dynamical systems, whereas systems exhibiting discrete transitions (commonly via difference equations) are known as discrete dynamical systems.

The theory of dynamical systems comprises a rich body of techniques and results, mainly geared towards the analysis of long-term qualitative behaviour of systems: existence and uniqueness of attractors, fixed points, or periodic points, sensitivity to initial conditions, etc. From a computer-science perspective, it is somewhat surprising to note that the literature is largely devoid of work on decision problems concerning dynamical systems, e.g., whether a fixed point or a particular region will actually be reached in finite time, whether a variable will assume negative values infinitely often, etc. Such questions, in turn, have numerous applications in a wide array of scientific areas, such as theoretical biology (analysis of L-systems, population dynamics), microeconomics (stability of supply-and-demand equilibria in cyclical markets), software verification (termination of linear programs), probabilistic model checking (reachability in Markov chains, stochastic logics), quantum computing (threshold problems for quantum automata), as well as combinatorics, term rewriting, formal languages, cellular automata, the study of generating functions, etc.

In this tutorial, I will first briefly introduce the main elements of the theory of (both continuous and discrete) dynamical systems, using several illustrative examples. I will then present various interesting decision problems, mainly in the context of discrete linear dynamical systems, for which there already are many open questions. Finally, I will survey some of the main known results and techniques, which draw on a wide array of mathematical tools, including linear algebra, algebraic and analytic number theory, real algebraic geometry, and model theory.