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If any of you have a copy of Strogatz this is problem 3.5.5 part a. It reads:

While considering the bead on the rotating hoop we used phase plane analysis to show that the equation $$\epsilon \frac{d^2\phi}{d\tau^2}+\frac{d\phi}{d\tau}=f(\phi)$$ has solutions that rapidly relax to the curve where $\frac{d\phi}{d\tau}=f(\phi)$. Recall, $\epsilon = \frac{m^2 g r}{b^2}$.

a.)Estimate the time scale $T_f$ for this rapid transient in terms of $\epsilon$ and also in terms of the original quantities.

So I know I am supposed to dimensionalize the equation in terms of time or something, but I really cannot figure this out. Please help!

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The dynamics of the bead on the rotating hoop are governed by the equation $$\epsilon \phi_{\tau\tau} + \phi_{\tau} = f(\phi).$$ Since $\epsilon$ is small, we often neglect the first term and look for solutions that satisfy $$ \phi_{t} = f(\phi) .$$ This is perfectly fine, and works for almost all values of $t$. We are interested in finding the values of $t$ where this isn't perfectly fine, where throwing away the $\epsilon\phi_{tt}$ term is problematic.

Strogatz speaks about a "rapid transient". What's going on physically is that there's some sort of behaviour that happens for only a short time period (perhaps a few very rapid oscillations) which is then followed by a long period of some other behaviour (perhaps exponential decay).

The same sort of thing happens in spatial dimensions as well. If you model fluid flowing through a wide pipe there is a very thin layer next to the boundary of the pipe where viscous drag between the fluid and wall is important, but in the center of the pipe it makes no difference.

It is this rapid transient that is governed by the $\epsilon \phi_{tt}$ term. To find the time scale where this terms can't be neglected we make a change of variable $\tau = \epsilon^{p} s$. In our original equation you can think of the $\phi_{t}$ and $f(\phi)$ terms balancing each other (they're the same size). We make this change of variable in the hope that we can find some other way to balance the terms in the equation.

$$\tau = \epsilon^{p} s $$ $$\Rightarrow \frac{\partial \phi}{\partial \tau} = \frac{\partial \phi}{\partial s}\epsilon^{-p}$$ Substitute into the original differential equation. $$\epsilon^{1-2p} \phi_{ss} + \epsilon^{-p}\phi_{s} = \epsilon^{0}f(\phi).$$ This differential equation can only be solved if:

  1. All three terms are the same size, or
  2. Two of the terms are the same size and the third is negligible.

The first option is impossible - there's no way for us to choose $p$ such that $1-2p = -p = 0$. What about the second option?

We can try $p = 0$, then the $\phi_{s}$ and $f(\phi)$ terms balance and the first term is negligible. This is our original differential equation.

We can try $p = \frac{1}{2}$. Then the equation becomes $$ \phi_{ss} + \frac{1}{\sqrt{\epsilon}}\phi_{s} = f(\phi).$$ This is no good; the $\phi_{ss}$ and $f(\phi)$ terms are balanced, but the $\frac{1}{\sqrt{\epsilon}}$ term is enormous.

Finally, we can try $p = 1$. Then $$ \phi_{ss} + \phi_{s} = \epsilon f(\phi).$$ Great! The $\phi_{ss}$ and $\phi_{s}$ terms are the same size and the $\epsilon f(\phi)$ term can be neglected.

We still need to decide where the dynamics are governed by this equation and where they are governed by the original equation. We found this equation by making the substitution $\tau = \epsilon^{1} s$. Equation (1) on page 61 of Strogatz was reduced to the non-dimensional form $$\epsilon \phi_{\tau\tau} + \phi_{\tau} = f(\phi)$$ through the scaling $\frac{b}{mg} \tau = t$. Putting this together: $$t = \frac{b}{mg}\tau = \frac{b}{mg} \epsilon^{1} s.$$ We have an expression for $\epsilon$. Substitution gives $$t = \frac{b}{mg} \frac{m^{2}gr}{b^{2}} s = \frac{mr}{b} s.$$ Thus the fast time scale is $T_{\textrm{fast}} = \frac{mr}{b}$, which has units of time as required. If you make this substitution as Strogatz asks in part b of the question you will recover the equation $$\phi_{ss} + \phi_{s} = \epsilon f(\phi)$$ from above.

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I was able to get the result you mentioned, but I do not understand why we are looking at the first term when it is not negligible, could you give an intuitive idea of what this means (in the context of this question)? Also, how is the time scale obtained from this? I literally learned about nondimensionalization within a day or two, so I don't have a deep understanding of what is occurring and how we obtain results from it. In addition, what does it mean to be of order $\epsilon$ or order 1, etc.? – KF Gauss Oct 1 '12 at 7:55
I don't think the units match up, $\epsilon$ does not have units, but $T_{fast}$ should have units of time. I guess we have to use the fact that $$\tau = \frac{t}{T_{slow}}$$. How do we formally state the time scale (i.e. what is the formal meaning of a time scale?) – KF Gauss Oct 2 '12 at 21:27
Yeah, you're right - I rushed the last part of the answer. The $T_{\textrm{fast}}$ I've given is a scaling between the two non-dimensional parameters $\tau$ and $s$ instead of $t$ and $s$. I'll edit in the correction. – in_wolframAlpha_we_trust Oct 3 '12 at 7:12

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