Gary William Flake

“If we increase r [in a logistic map] even more, we will eventually force the system into a period-8 limit cycle, then a period-16 cycle, and so on. The amount that we have to increase r to get another period doubling gets smaller and smaller for each new bifurcation. This cascade of period doublings is reminiscent of the race between Achilles and the tortoise, in that an infinite number of bifurcations (or time steps in the race) can be confined to a local region of finite size. At a very special critical value, the dynamical system will fall into what is essentially an infinite-period limit cycle. This is chaos.”
― Gary William Flake, The Computational Beauty of Nature: Computer Explorations of Fractals, Chaos, Complex Systems, and Adaptation

“Later Turing proved that Turing machines could compute exactly the same functions as lambda calculus, which proved that all three models of computation are equivalent. This is a truly remarkable result, considering how different the three models of computation are. In Church's 1941 paper he made a statement that is now known as the Church-Turing thesis: Any function that can be called computable can be computed by lambda calculus, a Turing machine, or a general recursive function.

Recall the point that was made about functions describing relationships between numbers and models of computation describing functions. Well, the Church-Turing thesis is yet another level more fundamental than a model of computation. As a statement about models of computation, it is not subject to proof in the usual sense; thus, it is impossible to prove that the thesis is correct. Once could disprove it by coming up with a model of computation over discrete elements that could calculate things that one of the other models could not; however, this has not happened. The fact that every posed model of computation has always been exactly equivalent to (or weaker than) one of the others lends strong support to the Church-Turing thesis.”
― Gary William Flake, The Computational Beauty of Nature: Computer Explorations of Fractals, Chaos, Complex Systems, and Adaptation