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As Strogatz writes in his book Nonlinear Dynamics And Chaos (p. 64)

There are often several ways to nondimensionalize an equation, and the best choice might not be clear at first. Therefore we proceed in a flexible fashion. We define a dimensionless time $\tau$ by $$\tau = \frac{t}{T}$$ where $T$ is a characteristic time scale to be chosen later. When $T$ is chosen correctly, the new derivatives $d\phi/d\tau$ and $d^2\phi/d\tau^2$ should be $\mathcal O(1)$, i.e., of order unity.

Why should the new derivatives be bounded?

Added context: The system he discusses in that section is an overdamped bead on a rotating hoop where $\phi$ is the angle between the bead and the downward vertical direction.

Added: as Joriki writes

The characteristic time is usually defined to be the time in which a quantity decreases by $1/\mathrm e$.

Is this somehow behind the choice of $\mathcal O(1)$?

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Leo
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  • What is $\phi$? – anomaly May 02 '16 at 05:37
  • @anomaly: I added some context, sorry for being unclear. – Leo May 02 '16 at 05:44
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    This is the rough definition of "characteristic time scale" - the solution changes significantly ($O(1)$) over a time-scals $\Delta t \sim O(T)$. – Winther May 02 '16 at 05:45
  • I think the idea is that $\tau$ should be dimensionless, and that this constitutes being $\mathscr{O}(1)$ - see the "nondimensionalize an equation" part. Note that when he says that T is a time scale, it doesn't have to be a constant, it just has to be a monotone increasing function in general. – Chill2Macht May 02 '16 at 05:46
  • It depends what you want to know, there can be more than one time scale in a problem, for example an oscillator with a small damping term has a fast time scale of the period of an oscillation, and a slow time scale, the time over which the amplitude decreases. By rescaling the equation appropriately you can focus on each of these time-scales separately. – David May 02 '16 at 05:54
  • @David: and what exactly makes us choose $O(1)$? – Leo May 02 '16 at 05:55
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    Related: What is the general definition of time-scale for a differential equation? – Winther May 02 '16 at 05:57
  • In this case $\phi$ is an angle so $O(1)$ consitutes a significant angle change: if $\Delta\phi \ll1$ the angle has not changed much and if $\Delta\phi = \pi$ it has changed the maximum. In a more general context (say if $\phi$ was a temperature or the distance traveled) then $\phi$ would have to be rescaled for us to say that $O(1)$ to be considered a significant change. – Winther May 02 '16 at 06:03
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    @Leo you want to end up with a dominant balance, and generally we write the largest terms as being $O(1)$, because you can always multiply an equation by a constant to make the largest terms $O(1)$. Here is an example of an ODE with two natural length scales that might show the idea more clearly. – David May 02 '16 at 06:05
  • It's supposedly explained in Strogatz's lecture: https://youtu.be/eZmzmQW-fAA?t=3075 – Cheng Nov 23 '22 at 14:11

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