Autonomous equations and stability

Chapter 6 - Read feedback from a phase line

An autonomous equation has no explicit time dependence:

$$y'=f(y).$$

The same state always receives the same rate. This makes long-run behavior visible on a one-dimensional phase line, often before any solution formula appears.

The phase-line procedure

1. Solve $f(y)=0$ for equilibrium states.
2. Use those states to divide the line into intervals.
3. Test the sign of $f(y)$ on each interval.
4. Draw an arrow up for $f(y)\gt 0$ and down for $f(y)\lt 0$.
5. Classify each equilibrium by the arrows around it.

An equilibrium is:

• asymptotically stable when nearby arrows point toward it (called stable in the phase-line examples below);
• unstable when nearby arrows point away; and
• semistable when one side points in and the other points out.

Lab: logistic feedback

Questions to test in the lab:

• Which parameter changes the destination?
• Which parameter changes how quickly the solution travels?
• Does a positive initial state above $K$ blow up, cross $K$, or approach $K$?

Worked example: stability before formula

Fade the phase line

For $y'=(y+1)(y-2)^2$:

• equilibria: $y=\underline{\hspace{4em}}$;
• sign on $( -\infty,-1)$: $\underline{\hspace{2em}}$;
• sign on $(-1,2)$: $\underline{\hspace{2em}}$;
• sign on $(2,\infty)$: $\underline{\hspace{2em}}$;
• stability at $-1$: $\underline{\hspace{5em}}$; at 2: $\underline{\hspace{5em}}$.

Feedback language

Near an equilibrium $y_*$, examine whether a small displacement triggers a correcting or amplifying rate.

• Negative feedback: a positive displacement produces a negative rate, and a negative displacement produces a positive rate. The equilibrium restores itself.
• Positive feedback: a displacement produces a rate in the same direction. The equilibrium repels.

When $f$ is differentiable and $f'(y_*)\ne0$:

• $f'(y_*)\lt 0$ suggests local stability;
• $f'(y_*)\gt 0$ suggests local instability.

The sign chart remains safer when $f'(y_*)=0$.

Parameters can change the portrait

Consider $y'=y(r-y)$. When $r\gt 0$, $y=0$ is unstable and $y=r$ is stable. When $r\lt 0$, their order swaps: $y=r$ is unstable and $y=0$ is stable. At $r=0$, the equilibria meet.

This qualitative change as a parameter passes a threshold is a bifurcation. It matters because parameter changes can alter not only speed, but the number and stability of long-run states.

Error clinic

Practice

1. Practice - phase line

   6.1 Draw and classify the phase line for $y'=y(y-1)(y-3)$.

   Detailed solution

2. Practice - solution behavior

   6.2 For the same equation, describe the long-run behavior for $y(0)=-1, 1/2, 2, 4$. Do not solve.

   Detailed solution

3. Integrated - logistic interpretation

   6.3 In $P'=rP(1-P/K)$, with $r,K\gt 0$, show that the growth rate $P'$ is largest at $P=K/2$. Distinguish "largest growth rate" from "largest population."

   Detailed solution

4. Challenge - threshold

   6.4 Analyze $y'=y^2-a$ for $a\gt 0$, $a=0$, and $a\lt 0$. Track how the equilibria and their stability change.

   Detailed solution

5. Practice - extension

   6.5 Analyze $y'=-(y-2)^3$. Classify the equilibrium using a sign chart, then compute $f'(2)$. What does this example show about concluding instability from a zero linearization derivative? Predict the limits for $y(0)=1$ and $y(0)=3$.

   Detailed solution

6. Challenge - transfer

   6.6 A harvested population follows $P'=P(1-P/8)-H$, where $H\ge0$ is a constant removal rate. Find and classify equilibria for $0\le H\lt 2$, $H=2$, and $H\gt 2$. For $0\lt H\lt 2$, explain the role of the smaller equilibrium. Treat $P=0$ as extinction: stop the physical model there instead of allowing negative population.

   Detailed solution

Retrieve and connect

Without looking, answer:

1. Why must separation record equilibria before division?
2. How does an integrating factor use the product rule?
3. What do inward, outward, and one-sided arrows mean at an equilibrium?

I drew each phase line before doing algebra and retrieved the two earlier methods.

Next: Turn second-order equations into the language of natural modes and oscillation.