Determining flow of a function with three fixed points

Question

This question has left me dumbfounded for quite a while:

Let […] $P\left ( x \right )=\left ( x-a \right )\left ( x-b \right )\left ( x-c \right )$.

Identify the fixed points and the flow for $\dot{x}=P\left ( x \right )$, where $a<b<c$.

The fixed points are $x=a$, $x=b$, $x=c$.

I cannot proceed further. How should I think about this?

Answer 1

Without loss of generality, assume that $P(x)= (x-a)(x-b)(x-c)$, where $a\leq b\leq c$. Since $\dot{x}=0$ when $x=a$, $x=b$, or $x=c$, we see that the points $a$, $b$, $c$ are fixed points.

Now, let's sketch its phase portrait by graphing $\dot{x} = P(x)$ on the $xy$-plane, where $y=\dot{x}$.

$\textbf{Case 1}$. Assume $a< b< c$. Then we have the following:

• in the interval $(-\infty,a)$, $\dot{x}$ takes a negative value, i.e., $\dot{x}<0$,
• in the interval $(a,b)$, $\dot{x}>0$,
• in the interval $(b,c)$, $\dot{x}<0$,
• in the interval $(c,\infty)$, $\dot{x}>0$.

This means $a$ is repelling, $b$ is attracting, and $c$ is repelling.

$\textbf{Case 2}$. Assume $a=b<c$. Then

• in the interval $(-\infty,a)$, $\dot{x}<0$,
• in the interval $(a,c)$, $\dot{x}<0$,
• in the interval $(c,\infty)$, $\dot{x}>0$.

So $a=b$ is neither attracting nor repelling, and $c$ is a repelling fixed point.

$\textbf{Case 3}$. Assume $a<b=c$. Then

• in the interval $(-\infty,a)$, $\dot{x}<0$,
• in the interval $(a,c)$, $\dot{x}>0$,
• in the interval $(c,\infty)$, $\dot{x}>0$.

So $a$ is repelling, while $b=c$ is neither attracting nor repelling.

$\textbf{Case 4}$. Assume $a=b=c$. Then

• in the interval $(-\infty,a)$, $\dot{x}<0$,
• in the interval $(a,\infty)$, $\dot{x}>0$.

This means $a=b=c$ is a repelling fixed point.

Answer 2

The remaining task is to characterise the flow, i.e., the evolution of states:

• Consider for example an initial condition $x_0$ with $a<x_0<b$ and answer the question how it will evolve. Do the same for all the other relevant cases.

• Alternatively, you may already know how to characterise the fixed points as stable and unstable and continue from there.

Answer 3

You now need to determine the sign of $P$ on each of the intervals, which would determine raising or falling behavior of the solution. You might also use that close to the fixed points, the solutions $x$ behave similar to $a+u$,... with $u$ a solution of $\dot u =P'(a)u$ etc.