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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:

$\dot{J} = -\frac{1}{\epsilon} (I^3-(C-C_0) I-F)-\frac{1}{\epsilon} \gamma J \\ \dot{I}=\frac{1}{\epsilon} J \\ \dot{C}= a_0J+a_1CF+a_4J^2 \\ \dot{F}=a_2JF-a_3C$

lala_12
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  • If $C$ is a 'proportion' (i.e. a ratio) then it's dimensionless. Similarly, $I$ is computed as a ratio (of market capitalisation) and is also dimensionless. And so $J$ will be of dimension per-unit-time. Which doesn't seem very helpful... – lemon Apr 18 '16 at 09:17
  • It makes sense, that $C$ must be dimensionless, if it is a proportion. I have updated my question and listed the 4 differential equations under 'EDIT'. – lala_12 Apr 18 '16 at 09:36
  • I don't see how this is supposed to be a physics question. – ACuriousMind Apr 18 '16 at 09:41

1 Answers1

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$C$ 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:

$$ [a_0]=...=[a_4]=T^{-1} $$

lemon
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