Abstract
Asking and answering the right questions about computability and computational complexity of dynamical systems is more important than many may realize. In this note, I discuss the principal challenges, their implications, and some instructive open problems.
 
Highlights
Numerical studies have informed the general paradigm of the expected behaviour of typical dynamical systems
In this note I aim to describe the modern paradigm of numerical modeling of dynamical systems, and its limitations from the point of view of computational complexity and computability, posing questions on what can and cannot be computed in theory
The formal model of computation widely accepted in computer science is a Turing machine (TM); it is provably equivalent to a program in any common programming language running on a computer with unlimited storage
Summary
Techniques of numerical modeling and theoretical study of computational complexity of the corresponding problems go hand-in-hand. Numerical studies have informed the general paradigm of the expected behaviour of typical dynamical systems. In a way, this has become a self-fulfilling prophecy – we look to model natural phenomena using mathematics which was developed to interpret the results of the modeling; MICHAEL YAMPOLSKY (II) numerical experiments of this nature seem to “work” – that is, they produce plausible results which agree with the established paradigm. In this note I aim to describe the modern paradigm of numerical modeling of dynamical systems, and its limitations from the point of view of computational complexity and computability, posing questions on what can and cannot be computed in theory. I speculate on what happens when computer simulations of complex natural processes are performed, why the results appear to make sense, and what they describe
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