Unpredictability and undecidability in dynamical systems
Physical Review Letters
We show that motion with as few as three degrees of freedom (for instance, a particle moving in a three-dimensional potential) can be equivalent to a Turing machine, and so be capable of universal computation. Such systems possess a type of unpredictability qualitatively stronger than that which has been previously discussed in the study of low-dimensional chaos: Even if the initial conditions are known exactly, virtually any question about their long-term dynamics is undecidable. PACS numbers:
... able. PACS numbers: 05.45.+b, 02.50.+s, 05.40.+j Traditionally, physicists studied integrable systems where a formula could be found for all time describing a system's future state. When we widened our scope in order to study the so-called "chaotic" systems, we were forced to relax our definition of what constitutes a solution to a problem, since no such formula exists. Instead, we content ourselves with measuring and describing the various statistical properties of a system, its scaling behavior, and so on: We can do this because the individual trajectories are essentially random. In this paper we introduce a class of dynamical systems, with as few as two or three degrees of freedom, in which even these modest goals are impossible: rather than being merely random, the dynamics is highly complex. We will show that these systems have a stronger kind of unpredictability than a typical chaotic system, and that as a consequence almost nothing can be said about their long-term behavior.