Equilibrium Solutions

A precise treatment of equilibria, stability, linearization, and the distinction between local and global behaviour.

A direction field supplies a local slope at every point; a solution curve must follow those slopes. A grid of short oriented arrows is crossed by a smooth solution curve that follows their local slopes. A dashed equilibrium line separates regions where the indicated slope directions differ.

Precise definition

For an autonomous equation $y'=f(y)$, an equilibrium solution is the constant function $y(t)=y_$ where $f(y_)=0$. It is stable in the Lyapunov sense if solutions starting sufficiently close remain close; asymptotically stable if they also converge to $y_*$; unstable if the stability condition fails.

Notation and mathematical language

Perturb the state as $y=y_+u$. If $f$ is differentiable, the linearization is $u'\approx f'(y_)u$. A negative derivative predicts local exponential decay, a positive derivative predicts growth, and zero makes the linear test inconclusive. Phase-line signs provide a direct one-dimensional classification.

Conceptual picture

An equilibrium is a balance of rates, not absence of underlying activity. In a mixing tank, inflow and outflow solute rates may be nonzero but equal. Stability asks what happens after a small displacement from balance, a different question from solving $f(y_*)=0$.

Fully worked example

Interpretation and application

Equilibria describe steady populations, chemical balances, market adjustment models, and mechanical rest states. A stable equilibrium in the equations need not be desirable, and estimated parameters can move or eliminate equilibria; sensitivity analysis belongs with interpretation.