Mathematical Study of Chaos
In general, a situation where things are confused and incomprehensible is described as "chaos," but "chaos" is also a long-standing academic term referring to the "impossibility of long-term prediction due to sensitive dependence on initial conditions." Everyone has likely had the experience of oversleeping slightly in the morning, catching a train one later than usual, and having this small delay ripple through to the next connection, amplifying the delay and ultimately resulting in being significantly late for the target time. As symbolized by this example, cases where even a tiny difference in initial values leads to a major change in the state much later are called "chaos" because future prediction is difficult.
Natural phenomena that change over time according to certain laws are described by differential equations. Most differential equations are non-linear and chaotic. While chaos likely did not appear in the Newtonian mechanics studied in high school, the "three-body problem," where three point masses attract each other through universal gravitation, is well known as an example of chaos. Another commonly cited example of chaos is weather forecasting. A weather forecast for three days from now is reasonably accurate, but accurately predicting the weather one month from now is difficult. This is because the fundamental equations describing the airflow that governs weather exhibit chaos. Humanity seems to have had the concept of chaos since ancient times, as seen in folk songs and other sources, but it was not until the latter half of the 20th century that chaos began to be clearly recognized as an object of science. The development of computers is heavily related to this.
I will introduce a model that captures part of the essence of chaos. As shown in Figure 1 below, a square is decomposed into two parts, named A and B. Consider a transformation that maps A to A' by doubling it horizontally and halving it vertically, and maps B to B' by doubling it horizontally and halving it vertically. Have you ever kneaded flour dough to make bread? You likely repeated the process of stretching it thin, folding it, stretching it thin again, and folding it. The above transformation mathematically represents the change in flour dough during bread making and is called the baker's transformation (or pie-kneading transformation). Any two points on the square that are very close together will eventually move apart by repeating the transformation. In other words, small errors in initial values are gradually amplified, leading to large changes later. Part of the essence of chaos lies in the geometric properties of "stretching" and "folding" shapes, and the reason pie dough mixes well is because the "baker's transformation is chaotic."
As the mathematical study of chaos progressed, it became clear that there are types of chaos that cannot be modeled by the baker's transformation. I will introduce a new model for studying such chaos. As shown in Figure 2 below, a square is decomposed into four parts, named A, B, C, and D. Consider a transformation that maps A to A' by tripling it horizontally and halving it vertically, B to B' by tripling it horizontally and halving it vertically, C to C' by tripling it horizontally and doubling it vertically, and D to D' by tripling it horizontally and doubling it vertically. C' and D' cover the entire square. In this transformation, the balance between the influence received from regions A and B and the influence received from C and D is key. Figure 3 plots periodic points where the influence from A and B is greater. Figure 4 plots periodic points where the influence from C and D is greater. The difference in colors represents the difference in periods. As the period increases, both types of periodic points fill the square, but their distribution patterns differ. These facts were recently proven with mathematical rigor. Because two different types of chaos coexist in an inseparable form, this transformation is called the hetero-chaotic baker's transformation. Although it appears to be an extremely simple model, there is still much that is not understood, attracting the interest of many researchers.
Figures 3 and 4 are courtesy of Yoshitaka Saiki (Professor, Hitotsubashi University).