Consideration of a Primitive Chaos

Abstract

A new concept, primitive chaos, was proposed, as a concept closely related to the fundamental problems of sciences themselves such as determinism, causality, free will, predictability, and time asymmetry [J. Phys. Soc. Jpn. 2014, 83, 1401. This concept is literally a primitive chaos in such a sense that it leads to the characteristic properties of the conventional chaos under natural conditions. Then, two contrast concepts, nondegenerate Peano continuum and Cantor set, are known as the origins of the primitive chaos. In this study, the relation of these origins is investigated with the aid of a mathematical method, topology. Then, we can see the emergence of interesting concepts such as the relation of whole and part, and coarse graining, which imply the essence of our intrinsic recognition for phenomena.

A Structure behind Primitive Chaos

“…can be defined, and f has the chaotic properties such as the existence of a dense orbit, the density of periodic points, and topological transitivity [3]. As described in the previous section, the new primitive chaos…”

“…At first, let us suppose that the primitive chaos (X, {X λ , λ ∈ Λ}, {f X λ , λ ∈ Λ}) satisfies the following conditions (2)-( 5) which play an important role in leading the primitive chaos to the properties of the conventional chaos [3].…”

“…, f Xm } of the primitive chaos. At first, note that the condition (5) leads to the uniqueness of the initial point x 0 of the property (P) [3]. Therefore, we can consider the map…”

“…Then, these results seem to be an answer of the question "Why we are surrounded by diverse chaotic behaviors?" [2,3].…”

“…In the primitive chaos, each set X λ implies an event or a selection, each sequence (ω n ) ∞ n=0 implies a time series, and each map f X λ implies a law or causality [1,2]. Under natural conditions, the primitive chaos leads to the characteristic properties of the conventional chaos [3,4], such as the existence of a nonperiodic orbit, the existence of the periodic point whose prime period is n for any n ∈ N, the existence of a dense orbit, the density of periodic points, sensitive dependence on initial conditions, and topological transitivity. In this sense, this primitive chaos is literally a primitive chaos.…”