Subsection 2.3.1 Stability
Section 2.2 introduced the indea of analyzing a differential equation qualitatively, often without actually knowing the solution. In qualitative analysis, the goal is often describing a reasonable narrative for the solution. A big part of that narrative can be described in the language of
stability. In this section, I want to introduce the concept of stability and how it is used to describe solutions to differential equations.
I’ll start with an autonomous first order equation.
\begin{equation*}
\frac{dp}{dt} = f(p)
\end{equation*}
For these autonomous equation, stability is a properties of the steady states. Recall that the steady states are the values of the population where the rate of change is zero; the population at a steady state doesn’t move. Stability asks: what happens when the initial value of the system is very close to the steady state?
Definition 2.3.1.
The steady states of an autonomous DE are classified by their stability.
A steady state \(P\) is stable if \(p(t)
\rightarrow P\) for any initial value close to \(P\text{.}\) Stable steady states are also called attractors. In the phase line diagram, trajectories points toward stable steady states.
A steady state is partially stable if \(p(t)
\rightarrow P\) for some initial values close to \(P\text{,}\) but \(p(t)\) diverges away from \(P\) for other initial values close to \(P\text{.}\) I can also be a bit more specific here. The state is called stable or attractive from above if \(p(t)
\rightarrow P\) for initial values slighlty larger than \(P\text{;}\) likewise, the state is called stable or attractive from below is \(p(t) \rightarrow P\) for initial values slightly smaller than \(P\text{.}\) On the phase line, a partially stable state has one trajectory pointing towards the steady state and one pointing away.
If both trajectories point away, the steady state is called unstable. Unstable steady states are also called repellors.
I’ve given the definition of stability for first order autonomous DEs. However, the concept of stability has much broader applications to a variety of differential equations (and particularly to systems of differential equations). I am not going to give a formal definition, but the general idea is the same. More complicated DEs and systems of DEs can also have steady states. They are stable, or attractors, if nearby initial values tend towards the steady steady state. They are unstable, or repellors, if neary initial values tend away from the steady states. In more complicated systems, all kinds of intermediate or partially stable behaviours are also possible.
Chapter 6 will investigate some of the interesting intermediate behaviours for systems of two linear DEs.
Subsection 2.3.2 Linearization
Once I have identified a steady state of a DE system, my interest is often limited to the behaviour of initial values close to the steady state. This is the behaviour that stability capture: whether the system approachs or diverges from the steady state when its start a small distance away.
Let me take the same setup of an autonomous equation
\begin{equation*}
\frac{dp}{dt} = f(p)
\end{equation*}
with steady state \(P\text{.}\) I’ll assume that \(f(p)\) is an analytic function. I can define a new function \(q(t) =
p(t) - P\text{.}\) I am essentially moving the steady state to zero with this new function. I can change the differential equation to reflect this new function \(q(t)\text{.}\) The derivative is the same: \(\frac{dq}{dt} = \frac{dp}{dt}\text{,}\) since the only difference is the subtraction of a constant. On the left, I can replace \(p\) by \(P+q\text{.}\) That gives a new autonomous differential equation.
\begin{equation*}
\frac{dq}{dt} = f(P + q)
\end{equation*}
As desired, \(q=0\) is now the steady state of this new DE. Now I’m going to expand the function \(f\) as a taylor series centered at \(P\text{.}\) (Why would I want to expand \(f\) as a taylor series? Well, because it accomplishes the goal of this section, as you will see.) This is a bit strange and subtle: the variable for this taylor series is \(q\text{,}\) not \(t\text{.}\) Here is the result, just showing the first two terms of the taylor series.
\begin{equation*}
\frac{dq}{dt} = f(P + q) = f(P) + f^\prime(P) (P + q - P) +
\ldots
\end{equation*}
Now \(f(P) = 0\) since \(P\) was, by definition, a steady state of the original autonomous DE. In the second term, the \(P\) cancels off. Let me make those adjustments.
\begin{equation*}
\frac{dq}{dt} = f(P + q) = f(P) + f^\prime(P) (P + q - P) +
\ldots = f^\prime(P) q + \ldots
\end{equation*}
Now I want to take an approximation to the differential equation by ignoring everything past the first taylor series term. This is a severe approximaiton, but it is still quite useful. Here is the new approximated DE.
\begin{equation*}
\frac{dq}{dt} \cong f^\prime(P) q
\end{equation*}
Definition 2.3.2.
The linearization of an autonomous DE at the steady state \(P\) is the related differential equation produced by taking only the first term of the taylor series of the right side.
\begin{equation*}
\frac{dq}{dt} = f^\prime(P) q
\end{equation*}
The solution to the linearized DE is always an exponential function, since the approximate DE always has the form of the standard percentage growth differential equation.
\begin{equation*}
q(t) = q_0 e^{f^\prime(P) t}
\end{equation*}
Indeed, I can think of the linearization as the best approximation to the solution by an exponential function.
Since it is an exponential function, this linearized solution is either exponential growth or decay, depending on the sign of \(f^\prime(P)\text{.}\) Therefore, the sign of \(f^\prime(P)\) determines the stability: positive and the solution is unstabe, since exponential growth takes the solution away from the steady state; negative and the solution is stable, since exponential decay takes the solution towards the steady state. This can also be seen in the phase line, since the sign of the derivative can indicate the trajectories on either side of the steady state. If \(f^\prime(P) = 0\text{,}\) then the stability is determined by the higher order terms of the taylor series expansion and the linearization of the differential equation is inconclusive.
For now, there isn’t much more I will do with this linearization. However, it is worth introducing here as a theme since it is so central to applied mathematics. Linear equations are almost always the first kind of DE that applied mathematicians try to use, typically since they have elegant and accessible solutions. Everything else gets simply referred to a ‘non-linear’; in many ways, ‘non-linear’ is a synonmy for annoying and complicated. However, linear models only go so far and often the non-linearity holds the key to understanding a model. Even so, I often want to understand the linear part first and then figure out how to add in the non-linearity in a reasonable fashion to add subtlety to a model.
