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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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  • $\begingroup$ What is $\phi$? $\endgroup$ – anomaly May 2 '16 at 5:37
  • $\begingroup$ @anomaly: I added some context, sorry for being unclear. $\endgroup$ – Leo May 2 '16 at 5:44
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    $\begingroup$ This is the rough definition of "characteristic time scale" - the solution changes significantly ($O(1)$) over a time-scals $\Delta t \sim O(T)$. $\endgroup$ – Winther May 2 '16 at 5:45
  • $\begingroup$ 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. $\endgroup$ – Chill2Macht May 2 '16 at 5:46
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    $\begingroup$ Related: What is the general definition of time-scale for a differential equation? $\endgroup$ – Winther May 2 '16 at 5:57

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