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2.2 Equilibrium Solutions and Stability

Autonomous differential eq: \[ \frac{dy}{dt} = f(y) \]

no \( t \) (or \( x \))

(always separable)

population models are autonomous:

\[ \frac{dy}{dt} = ky \]

\[ \frac{dy}{dt} = ky(M-y) \]

the values of \( y \) where \( f(y) = 0 \) (right side is zero) are called critical points

\[ \frac{dy}{dt} = y - 1 \quad \text{has critical pt } y = 1 \]

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if \( C \) is a critical pt, then \( y = C \) is called an equilibrium solution

\[ \frac{dy}{dt} = y - 1 \quad \text{has an equilibrium solution } y = 1 \]

\( \hookrightarrow y' = 0 \) so \( y \) stays the same

above or below \( y = 1 \) solutions change

Graph of y versus t showing a horizontal line at y=1 representing an equilibrium solution.

some equations are easy to solve

\[ \frac{dy}{dt} = y - 1 \quad y(0) = y_0 \]

\[ \frac{1}{y-1} dy = dt \]

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\[ \ln(y-1) = t + C \]\[ y - 1 = Ce^t \]\[ y = 1 + Ce^t \]

Given the initial condition \( y(0) = y_0 \):

\[ y_0 = 1 + C \quad \text{so} \quad C = y_0 - 1 \]

\( y = 1 + (y_0 - 1)e^t \)

  • If \( y_0 = 1 \), then \( y = 1 \) for all \( t \)
  • If \( y_0 > 1 \), then as \( t \to \infty \), \( y \to \infty \)
  • If \( y_0 < 1 \), then as \( t \to \infty \), \( y \to -\infty \)

Solutions nearby go away from this equilibrium.

This is called an unstable equilibrium.

Graph of y versus t showing solution curves diverging from the horizontal equilibrium line y=1.
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If the equation is not easy to solve, we can still understand the stability of equilibrium solutions by using a phase diagram w/o solving the differential eq.

\[ \frac{dy}{dt} = y - 1 \quad \text{equilibrium: } y = 1 \quad \left( \frac{dy}{dt} = 0 \right) \]

Then identify \( \frac{dy}{dt} \) near each critical pt.

Phase line diagram for y showing an unstable equilibrium at y=1 with arrows pointing away.

\( y < 1 \)

\( \frac{dy}{dt} = y - 1 \)

if \( y < 1 \)

\( y' < 0 \)

(decreasing)

\( y = 1 \)

\( y' = 0 \)

(equilibrium)

solutions go away from \( y = 1 \)

so it is unstable

\( y > 1 \)

\( \frac{dy}{dt} = y - 1 \) and if \( y > 1 \)

\( y' > 0 \)

(increasing)

Solutions here increase over time

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Autonomous Differential Equations: Stability Analysis

\[ \frac{dy}{dt} = y(y-1)^2 \]

Critical pts: \( y=0, y=1 \)

Phase line for y' = y(y-1)^2 showing critical points at 0 and 1 with directional arrows.

Analysis of the phase line:

  • For \( y < 0 \), \( y' < 0 \)
  • At \( y = 0 \), \( y' = 0 \)
  • For \( 0 < y < 1 \), \( y' > 0 \) (inc)
  • At \( y = 1 \), \( y' = 0 \)
  • For \( y > 1 \), \( y' > 0 \) (inc)

y = 0 is unstable

y = 1 is called a semi-stable equilibrium (one side approaches, the other leaves)

Solution Curves in the ty-plane

The following graph illustrates the behavior of solution curves relative to the equilibrium solutions \( y=0 \) and \( y=1 \).

Graph of solution curves y(t) showing divergence from y=0 and semi-stability at y=1.
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Higher Order Polynomial Autonomous Equation

\[ \frac{dy}{dt} = (y-1)(y+2)(y^2-9) \]

Critical pts: \( y = 1, -2, -3, 3 \)

Phase line for y' = (y-1)(y+2)(y^2-9) showing stability of critical points -3, -2, 1, and 3.

Stability classification from the phase line:

  • \( y = -3 \): stable (\( y' = 0 \))
  • \( y = -2 \): unstable (\( y' = 0 \))
  • \( y = 1 \): asymptotically stable (\( y' = 0 \))
  • \( y = 3 \): unstable (\( y' = 0 \))

Note: Between \( y=1 \) and \( y=3 \), \( y' < 0 \). For \( y > 3 \), \( y' > 0 \).

Solution Curves Visualization

The graph below depicts the family of solution curves and their convergence or divergence relative to the four equilibrium lines.

Graph of solution curves y(t) for multiple equilibrium points showing stable and unstable behaviors.