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I am trying to reduce the number of parameters of the ODE system: \begin{align} \label{eq:Iprime} I'(t)&= \sigma B-\mu I\\ B'_1(t)&=r_1 B_1\left(1-\frac{B_1 }{K_1}\right)-d_1IB_1-m (B_1-B_2) \\ \label{eq:B2prime} B'_2(t)&=r_2 B_2\left(1-\frac{B_2 }{K_2}\right)-d_2IB_2-m (B_2-B_1) \end{align} (where $B=B_1+B_2$) by applying nondimensionalization. In all the examples I have seen this reduced the number of parameters. However if I apply the technique to my problem the number of parameters remain the same! Why is this the case?

Work thus far

We assume the following scaling: $$I=\widetilde{I}\widehat{I} \quad B_1=\widetilde{B}_1\widehat{B}_1 \quad B_2=\widetilde{B}_2\widehat{B}_2 \quad t=\tilde{t} \hat{t} $$

where $\widetilde{I}, \widetilde{B}_1, \widetilde{B}_2, \tilde{t}$ are constants (dimension-carrying), to be chosen, and $\widehat{I}, \widehat{B}_1, \widehat{B}_2, \hat{t}$ are the dimensionless variables. Subbing these into our ODEs leads to \begin{align} \frac{d(\widetilde{I}\widehat{I})}{d(\tilde{t} \hat{t} )}&=\sigma ( \widetilde{B}_1\widehat{B}_1 + \widetilde{B}_2\widehat{B}_2)-\mu \widetilde{I}\widehat{I}\\ \frac{d \widehat{I}}{d \hat{t}}&=\frac{\tilde{t}\sigma}{\widetilde{I}} ( \widetilde{B}_1\widehat{B}_1 + \widetilde{B}_2\widehat{B}_2)-\frac{\mu \tilde{t}}{\widetilde{I}} \widehat{I} \\ \frac{d \widehat{I}}{d \hat{t}}&=\left[\frac{\tilde{t}\sigma}{\widetilde{I}} \widetilde{B}_1\right] \widehat{B}_1 + \left[ \frac{\tilde{t}\sigma}{\widetilde{I}} \widetilde{B}_2 \right] \widehat{B}_2-\left[\frac{\mu \tilde{t}}{\widetilde{I}}\right] \widehat{I} \\ \end{align}

\begin{align} \frac{d(\widetilde{B}_1\widehat{B}_1)}{d(\tilde{t} \hat{t} )}&=r_1 \widetilde{B}_1\widehat{B}_1\left(1-\frac{\widetilde{B}_1\widehat{B}_1 }{K_1}\right)-d_1\widetilde{I}\widehat{I}\widetilde{B}_1\widehat{B}_1-m (\widetilde{B}_1\widehat{B}_1-\widetilde{B}_2\widehat{B}_2) \\ \frac{d \widehat{B}_1}{d \hat{t}}&=r_1 \tilde{t}\widehat{B}_1\left(1-\frac{\widetilde{B}_1 }{K_1}\widehat{B}_1\right)-d_1\tilde{t }\widetilde{I} \widehat{I}\widehat{B}_1-m\frac{\tilde{t}}{\widetilde{B}_1} (\widetilde{B}_1\widehat{B}_1-\widetilde{B}_2\widehat{B}_2) \\ \frac{d \widehat{B}_1}{d \hat{t}}&=\left[ r_1 \tilde{t}\right]\widehat{B}_1\left(1-\left[ \frac{\widetilde{B}_1 }{K_1}\right]\widehat{B}_1\right)-\left[ d_1\tilde{t } \widetilde{I}\right]\widehat{I}\widehat{B}_1-\left[m \tilde{t} \right] \widehat{B}_1+\left[m \frac{\tilde{t} \widetilde{B}_2}{\widetilde{B}_1} \right]\widehat{B}_2 \end{align} (the procedure for $B_2$ is the same)

The square brackets are to indicate the number of independent choices for the scales. In all of the examples I have seen so far the number of independent choices for the scales are the same as the number of the orginal parameters for the system. At maximum I can only choose four of these choices to be $1$. Meaning that I will have $13-4=9$ parameters remaing. The same as what I started with. What am I doing wrong/why can't I reduce the number of parameters in my system? If you need more clarification on this problem please ask.

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$\def\d{\mathrm{d}}$Take$$ \widetilde{t} = \frac{1}{m},\ \widetilde{I} = μ \widetilde{t} = \frac{μ}{m},\ \widetilde{B}_1 = \widetilde{B}_2 = \frac{\widetilde{I}}{σ \widetilde{t}} = \frac{μ}{σ}, $$ then the system is reduced to\begin{align*} \frac{\d \widehat{I}}{\d \widehat{t}} &= \widehat{B}_1 + \widehat{B}_2 - \widehat{I},\\ \frac{\d \widehat{B}_1}{\d \widehat{t}} &= \frac{r_1}{m} \widehat{B}_1 \left( 1 - \frac{μ}{σK_1} \widehat{B}_1 \right) - \frac{μd_1}{m^2} \widehat{I}\widehat{B}_1 - (\widehat{B}_1 - \widehat{B}_2),\\ \frac{\d \widehat{B}_2}{\d \widehat{t}} &= \frac{r_2}{m} \widehat{B}_1 \left( 1 - \frac{μ}{σK_2} \widehat{B}_1 \right) - \frac{μd_2}{m^2} \widehat{I}\widehat{B}_1 - (\widehat{B}_2 - \widehat{B}_1), \end{align*} which has six parameters.

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  • $\begingroup$ Can you explain the process you used to get to this point? I want to better understand the process of how you got to where you did. $\endgroup$ – AzJ Mar 15 '18 at 23:52
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    $\begingroup$ @AzJ First, $\widetilde{B}_2$ should be taken the same as $\widetilde{B}_1$ because they're quantities of the same dimensions. Next, since $r_1$ may not be equal to $r_2$ (the same situation for $K_1, K_2$ and $d_1, d_2$), I tried to normalize the rest parameters. $\endgroup$ – Saad Mar 15 '18 at 23:57
  • $\begingroup$ @AzJ If you find my answer useful, please accept it :) $\endgroup$ – Saad Mar 16 '18 at 0:16
  • $\begingroup$ Okay one more question before I award the bounty is there any greater reason (some kind of structure in ODEs) that this any other ODEs like it can not be reduce to systems of n-m parameters, where n is the number of original parameters, and m is the number of state variables? Or is there just an unusual case that has no pattern? $\endgroup$ – AzJ Mar 16 '18 at 0:21
  • $\begingroup$ @AzJ Actually I don't know much insight on nondimensionalization, but in my opinion, systems like yours can be reduced because variables only appear in polynomial fashion, i.e. $\widehat{B}_1$ and $\widehat{B}_1^2$. If it has terms like $\widehat{B}$ and $\sin\widehat{B}_1$ simultaneously, nondimensionalization doesn't seem to work. $\endgroup$ – Saad Mar 16 '18 at 0:47

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