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I'm not sure if I can post this here, but check this out. This is some text from Boccara's Modeling Complex Systems. The thing which confuses me is that there is a dimension reduction from 4 parameters to just 1 (as stated in the text). This dazzles me, because I still see 4 parameters when looking at the resulting equations.

Could anyone shed some light?

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onimoni
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Original equations contain four parameters $b,d,s,e$ so that in order to discuss the behavior of dynamics, you have to fix three parameters and investigate the case varying the rest one. That's so complicated of course. After introducing $h,p,\tau,\rho$, equations contain only one parameter $\rho$ so that in $(h,p;\tau)$ space, no matter how $b,d,s,e$ change, which means you don't have to fix anything, if we know the behavior of dynamics varying $\rho$, everything seems clear.

Let me give a more intuitive example. I want to investigate the shape of the region $x^2+y^2\leq1$. Instead of fix each $x$ and find $y$, we set $r^2=x^2+y^2$ and the region becomes $r^2\leq1$, in which case we don't need to consider both $x$ and $y$ and instead focus on $r$. When solving equations, variables substitution usually gives simpler view.

Shuchang
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  • But i dont understand, look at equation (2.4), there are clearly 3 parameters: $\rho, p$ and $h$...? So how can you say that there is only one parameter, namely $\rho$? – onimoni Oct 23 '13 at 09:43
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    $p,h$ are not parameters, they are dynamical variables. When we transform $(H,P)$ space to $(h,p)$ space, number of parameters reduce. – Shuchang Oct 23 '13 at 12:45