Xn+1 = sin(3.24Yn) - Zn cos(0.43Xn)) + 3
Yn+1 = Zn sin(-0.45Xn+1) - cos(2.43Yn))
Zn+1 = sin(Xn+1)
developed by myself based on Pickover of choascope in Jul. 2015.
Xn+1 = 2sin(Xn) - 4.7 / (2 + sin(Yn))
Yn+1 = -2Xn
developed by myself in Mar. 2015.
Xn+1 = Yn - 0.97Xn + 5 / (1 + Xn2) - 5
Yn+1 = -0.997Xn
Gumowski-Mira attractor.
Xn+1 = 2.6sin(Xn + Yn) - 0.97Xn + 5 / (1 + Xn2) - 5
Yn+1 = -0.97Xn
developed by myself in Feb. 2015, based on the Gumowski-Mira attractor.
Xn+1 = Yn + aXn + 2(1-Xn)/(1+Xn2)
Yn+1 = -Xn + aXn+1 + 2(1-Xn+1)/(1 + Xn+12)
developed by I. Gumowski and C. Mira.

Equation of Chemical Equilibrium

A very important equation on chemical reaction is learnt in high school, but the equation is frequently derived using a wrong manner even in books. Here, detail of the issue is introduced as simply as possible.

Let we have an equilibrium reaction as follows:
aA + bB <=> cC + dD   (1),
where a, b, c, and d are numbers, and A, B, C, and D are compounds.
Then a critical equation holds true as shown below.
[C]c[D]d / [A]a[b]b = K   (2),
where K is equilibrium constant.
How can we develop the equation (2)? For this purpose, it is believed that a way connecting kinetics and thermodynamics is true; however, it is incorrect.

This paragraph shows the wrong method for developing eq. (2). The rate of left- to right-hand side is shown as
r = k[A]a[b]b   (3),
where k is a rate constant.
Similarly, the opposite reaction rate is
r' = k'[C]c[D]d   (4),
where k' is a rate constant.
When the reaction (1) is in equilibrium,
r = r'   (5)
k[A]a[b]b = k'[C]c[D]d   (6)
<=> [C]c[D]d / [A]a[b]b = k / k' = K   (2')
Eq. (2') is well known as the law of mass action (質量作用の法則). Many people easily accept this proof as it is very simple, but, unfortunately, this proof is based on an idealized story.

(5) has no problem, but (3) and (4) are incorrent in general. A rate does not always follow such a formula. For example, a reaction between tertiarybutyl chloride and iodide ion (7), which is a typical SN1 reaction, provides a rate equation of (8).
C(CH3)3Cl + I- -> C(CH3)3I + Cl-   (7)
r = k[(CH3)3Cl]   (8)
C(CH3)3Cl is first converted to C(CH3)3+, and subsequently I- attacks the cation. The former is the rate-determining step. Thus, this reaction follows the equation (8).

Consequently, the equilibrium equation (2) should be derived from another basis. That is chemical potential, the detail of which is omitted here as it is slightly complicated for general readers. However, roughly speaking, the meaning of equation (2) is that energy of the denominator and the numerator molecules (= left- and right-hand-side molecules in eq. (1)) is equal each other, when their ratio is K. Hence, it can be equilibrium.

1. If the reaction (7) is in equilibrium, a more proper equation may be as follows:
r = k[(CH3)3Cl]{[I-] / ([I-] + x[Cl-])}, where x is a coefficient.
2. Accurately, K is dimensionless quantity, and concentration should be transformed into activity.