6.2 Stable and Unstable Equilibrium Points

Equilibrium Points of Differential Equations

• Defined as points where the derivative, represented as ( x' ), is zero.

• Found by setting the vector field equal to zero: ( x' = 0 ).

• Critical for understanding systems described by differential equations.

Connection to Scientific Equilibrium

• Equilibrium points relate to broader concepts of equilibrium in science (e.g., chemical equilibrium, ecosystem balance).

• Correspond to states where the system is unchanging: ( x' = 0 ).

Examples of Equilibrium Points

• Example from logistic equation with two equilibrium points:

  • ( x = 0 ) (unstable)

  • ( x = K ) (stable)

• Change arrows around these points dictate their behavior:

  • Arrows away from ( x = 0 ) indicate instability.

  • Arrows towards ( x = K ) indicate stability.

Types of Equilibrium Points

Stable Equilibrium

• Definition: Nearby points move toward the equilibrium point.

• Example: ( x = K ) in the logistic model.

• Visual: Ball at the bottom of a bowl - perturbations lead back to the equilibrium.

Unstable Equilibrium

• Definition: Nearby points move away from the equilibrium point.

• Example: ( x = 0 ) in the logistic model.

• Visual: Ball on top of a hill - slightest push causes it to roll away from equilibrium.

Semi-Stable Equilibrium

• Definition: Stability in one direction and instability in another.

• Example: ( x = 0 ) in the vector field ( x' = x^2 ) - stable on the left, unstable on the right.

• Visual: Ball on a ridge - return on one side but roll away on the other.

Technical Note on Equilibrium Points

• For the function ( x' = x^2 ):

  • Has technically two equilibrium points at ( x = 0 ) (both roots at zero).

  • The character of this equilibrium is that it is neither strictly stable nor unstable (semi-stable).

Robustness of Equilibrium Points

• Important to recognize stability under perturbations.

• Systems modeled by differential equations should exhibit similar behavior despite small changes.

• Example: ( x' = x^2 ) is not robust because small changes can alter the number of equilibrium points.

Simple Linear Functions in 1D

• Definition of linear functions:

  1. ( f(x+y) = f(x) + f(y) )

  2. ( f(ax) = a f(x) )

• Two forms in 1D:

  • ( f(x) = Kx ) (where ( K > 0 )) leads to a stable equilibrium.

  • ( f(x) = -Kx ) leads to an unstable equilibrium.

Visualizing Linear Vector Fields

• ( x' = Kx ) produces outward arrows from the origin (unstable equilibrium).

• ( x' = -Kx ) produces inward arrows toward the origin (stable equilibrium).