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 According to a relatively simple determinism, Chaos is a phenomenon that impossible to predict long-term due to its irregular and complex. Chaos was discovered by a meteorologist Edward Lorenz in 1963. Lorenz found that the meteorological model by a rudimentary computer simulation greatly diverges due to slight differences of initial value input and that long-term prediction of weather is impossible. It is because the weather is chaotic so that the weather forecast is not always exactly. Various natural phenomena such as turbulence in the fluid system, the atmosphere surrounding the earth as well as a co-operative phenomena of Belousov-Zhabotinsky reaction in chemical reaction system are considered chaotic phenomena.

【Mathematical model of Chaos】

 In a deterministic system, given the initial value, we can accurately calculate the state of the future system. For example, if we give an initial value for the equation of motion, we can know exactly the position and speed of the object at any moment advance from the initial state.

 Then, how about deterministic chaos? The figure below is a deterministic system (Logistic map) that can be represented by a simple formula, starting calculation from the initial value 0.3.

Logistic Map
Logistic Map
fig.1 Logistic Map

 The figure below is a time series plot of the Logistic map. Such complicated behavior is also determined by a simple formula.
Also, comparing the time series when calculate with the initial value 0.3 and 0.30001, we can see that the two time-series show completely different behaviors with the lapse of time due to the very slight initial value difference. This is Chaos, amplifying very slight differences when time passes and making long-term prediction impossible.

Time series of Logistic maps
fig.2 Time series of Logistic maps