We discuss the replacement of discrete maps by automata, algorithms for the transformation of finite length digit strings into other finite length digit strings, and then discuss what it required in order to replace chaotic phase flows that are generated by ordinary differential equations by automata without introducing unknown and uncontrollable errors. That question arises naturally in the discretization of chaotic differential equations for the purpose of computation. We discuss as examples an autonomous and a periodically driven system, and a possible connection with cellular automata is also discussed. Qualitatively, our considerations are equivalent to asking when can the solution of a chaotic set of equations be regarded as a machine, or a model of a machine.
Dynamics and Bifurcations on the Normally Hyperbolic Invariant
Manifold of a Periodically Driven System with Rank-1 Saddle