Stability and Equilibrium of Autonomous ODEs

Autonomous Equations

An Ordinary Differential Equation (ODE) is considered autonomous if there is no explicit dependence on the independent variable $t$:

$$\frac{dy}{dt} = f(y)$$

Common population models like exponential growth and logistic growth are examples of autonomous equations:

Exponential Growth: $\frac{dP}{dt} = kP$
Logistic Growth: $\frac{dP}{dt} = kP(M-P)$

Definition: Critical Point

A critical point of an autonomous equation is a value $y = c$ (or $P = c$) such that $f(c) = 0$. At this point, $\frac{dy}{dt} = 0$, making the constant function $y(t) \equiv c$ an equilibrium solution.

Classifying Stability

Stability describes how solution curves behave in the neighborhood of a critical point:

Unstable: Solution curves are "repelled away" from the critical point on both sides.
Stable: Solution curves move "toward" the critical point from both sides.
Semi-stable: Solution curves move "away" on one side and "toward" on the other.

Worked Examples

Example 1: Basic Analysis

Given $\frac{dy}{dt} = y - 1$:

The function is $f(y) = y - 1$.
Critical point at $y = 1$.
Behavior:
- If $y < 1 \implies y' < 0$ (decreasing away from 1).
- If $y > 1 \implies y' > 0$ (increasing away from 1).
Conclusion: The equilibrium $y \equiv 1$ is unstable.

Example 2: Semi-stability

Given $y' = y(y - 1)^2$:

Critical points at $y = 0$ and $y = 1$.
Testing Points:
- $y = -1 \implies y' = (-1)(-2)^2 = -4$ (Negative / Away from 0).
- $y = 0.5 \implies y' = (0.5)(-0.5)^2 = 0.125$ (Positive / Toward 1).
- $y = 2 \implies y' = (2)(1)^2 = 2$ (Positive / Away from 1).
Conclusion: $y = 0$ is unstable, and $y = 1$ is semi-stable.

Example 3: Logistic Form

Given $y' = 3y(2 - y)$:

Critical points at $y = 0$ and $y = 2$.
Solution curves move "toward" $y = 2$ from both sides (above and below).
Conclusion: $y = 0$ is unstable; $y = 2$ is stable.

Example 4: Complex Polynomial

Given $\frac{dy}{dt} = (y - 1)(y + 2)(y^2 - 9)$:

Critical points are found at $y = 1, -2, -3, 3$. Their classifications are as follows:

Critical Point ($y$)	Classification
3	Unstable
1	Stable
-2	Unstable
-3	Stable