What is the general definition of time-scale for a differential equation?

Question

For a differential equation like $\dot{x}=ax$ where $x$ is a function on an independent variable $t$, and $\dot{x}=\frac{dx}{dt}$, and $a$ is a constant, we define the time-scale $\frac{1}{a}$, which if $a$ be a negative real number we can look at this concept as the half-life time of $x$ over $\ln 2$. And you know half-life time means the time you need to loose half of your initial amount of $x$. As you can see it at the following calculation.

Let $a=-k$ where $k\in\mathbb{R}^{>0}$. From our differential equation we have $x=x_0e^{-kt}$ so if I show the half-life time by $\tau_{\frac{1}{2}}$ then $$\frac{x_0}{2}=x_0e^{-k\tau_{\frac{1}{2}}}\Longrightarrow \ln 2=k\tau_{\frac{1}{2}}\Longrightarrow \tau_{\frac{1}{2}}=\frac{\ln 2}{k}$$ With this concept time-scale we do many things, one is choosing the step size for numerical simulation of our differential equation using Euler method for example. Or if we have a system of differential equations in the form above, we say which one has a faster effect on our populations by comparing time-scales of each present differential equation.

But what should I define time-scale for a general form of a differential equation which at least do those two works for us that I mentioned?

Answer 1

I think what you introduced above would be meaningful in any kind of differential equation coming from evaluating some real event, whether it be in biology or chemistry or physics ..., but for example, it would have a regularity in periodic DE. So that if it's negative then you loose and if it's positive you get. But if you wanna define it, I think the possible right definition can be: "the effectiveness of constant coefficient of time depending on what's being evaluated"

Answer 2

The term time-scale w.r.t ODEs is often loosely defined as the amount of time for the system (the solution $y(t)$) to change 'significantly'. This is about as precise of a definition as you are going to get as there are so many fundamentally different types of time-scales we can talk about depending on the system in question.

Lets take some examples: for the ODE $\dot{y} = -ky\implies y=e^{-kt}$ a useful time-scale is the half-life; the time it takes for $y$ to half its value. For the ODE $\ddot{y} + \omega^2 y \implies y = \sin(\omega t)$ a half-life is not a useful concept, however we can talk about a period $T = \frac{2\pi}{\omega}$ of oscillations. For the ODE $\ddot{y} + \omega^2y + k\dot{y} = -k\omega e^{-kt}\cos(\omega t) \implies y = e^{-ky}\sin(\omega t)$ we can talk about both a period and a half-life so there are two useful time-scales associated with this ODE. One can come up with many other examples and because of this it is hard to give a very precise definition of the term.

When discussion ODEs describing physical systems (where quantities have units) we can use dimensional analysis to search for time-scales in the problem. This is often very useful for getting intuition about how solutions will behave without having to solve the acctual ODE. For example for the ODE $\ddot{y} + a^2\dot{y} + b^2 y = 0$ we have two dimension-full parameters $[a] = s^{-1/2}$ and $[b]=s^{-1}$ so we can expect that the solution can be charactherized by two time-scales $t_1 \propto \frac{1}{b}$ and $t_2 \propto \frac{1}{a^2}$ and depending on the relative size of $t_1$ and $t_2$ we can deduce the rough behavior of the system. It's however only a full solution of the ODE that can reveal that the true period of oscillations is given by the more complicated expression $\frac{4\pi t_1t_2}{\sqrt{4t_2^2-t_1^2}}$ and the half-life is given by $2\log(2)t_2$.

I guess what I'm trying to say with all the example above is that time-scales is a very useful concept without it needing to have a very precise definition.