I am trying to do a dimensional analysis of a nonlinear system of differential equations.
I have some problems with determine the dimensions of the different variables.
The differential system is with inspiration from E.C. Zeeman's catastrophe theory. I am trying to model the unstable behaviour of stock exchanges.
The differential equations are:
$\dot{J} = f_1(I,C,F,J) \\ \dot{I}=f_2(J) \\ \dot{C}=f_3(C,F,J) \\ \dot{F}=f_4(C,F,J)$
with $f_i, i=1..4$ being functions of the listed variables.
The variable $I$ is the Dow-Jones index, the variable $J$ denotes the rate of change of the Dow-Jones index.
The variable $J$ can be regarded as a dependent variable, depending upon the rate of buying and selling of investors.
The variable $C$ is the proportion of speculative money in the market, and the variable $F$ is the excess demand for stock by fundamentalists.
By fundamentalists I mean a type of investor, who act on the basis of estimates of large economic factors such as supply and demand, money supply, etc. Before a fundamentalist invests in a firm, he/she instructs hos research team to asses its viability, it's growth potential and market potential.
What would be the right choose of dimensions for the above mentioned variables?
I think it would be suitable to choose money ($\$$) as the dimension of the variable $C$, but I don't know about the other variables.
EDIT: The 4 differential equations are:
$\dot{J} = -\frac{1}{\epsilon} (I^3-(C-C_0) I-F)-\frac{1}{\epsilon} \gamma J \\ \dot{I}=\frac{1}{\epsilon} J \\ \dot{C}= a_0J+a_1CF+a_4J^2 \\ \dot{F}=a_2JF-a_3C$
$C$ and $I$ are both computed by taking ratios of numbers of the same dimension, and are therefore dimensionless:
$$ [C]=[I]=1 $$
If you look at your first differential equation (in the edit), the fact that $I^3$ (which is dimensionless) is combined with $F$ without any coefficients means that $F$ must also be dimensionless
$$ [F]=1 $$
Combining the first two equations you see that $J$ must also be dimensionless
$$ [J]=1 $$
and the coefficients $\epsilon$ and $\gamma$ satisfy:
\begin{align} [\epsilon] &= T \\ [\gamma] &=1 \end{align}
where $T$ is the time dimension.
From here, it follows straightforwardly that the remaining coefficients are all per-unit-time:
$$ [a_0]=...=[a_4]=T^{-1} $$