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

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?

• Are there any restrictions to the general form of a differential equation, for example do you want to limit to linear, autonomous and/or homogeneous differential equations, if so are you then interested in higher order, or coupled differential equations? – Kwin van der Veen Oct 2 '15 at 10:13
• @fibonatic I don't have any restriction, I just like to know what is the general definition of this concept, but even if you know this concept for a specific class of differential equations then I will be happy to hear that. – AmirHosein Sadeghimanesh Oct 2 '15 at 10:16
• Namely some non-autonomous have a polynomial solution, so you can't characterize a half-life/time-scale. I would think that this can only be done for exponential and periodic solutions. – Kwin van der Veen Oct 2 '15 at 10:21
• @fibonatic no problem with that, so do you mean time-scale only is defined for differential equations with exponential and periodic solutions? I should mention that I don't know the definition of time-scale, I only know in the case I wrote in my question, it is defined using half-life time. What I need is the main definition of time-scale or the concept that can do those two jobs I talked about them at end of my question and be identified with time-scale on the specific example I said. – AmirHosein Sadeghimanesh Oct 2 '15 at 10:26

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 $2\log(2)t_2$.