Equilibrium Solutions

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 $f(y_*)=0$.

Conditions and key results

The derivative test requires differentiability and a nonzero derivative for a decisive conclusion. Local stability says nothing by itself about distant initial states or other attractors. Physical feasibility may restrict the state space, so only permitted one-sided perturbations may matter at a boundary.

A reliable strategy

1. Set the autonomous right-hand side equal to zero and retain every feasible root.
2. Construct a sign chart for $f$ between roots and singularities.
3. Classify arrows toward or away from each equilibrium; use $f'(y_*)$ only when its hypotheses give a decisive sign.
4. Interpret stability within the state domain and check long-run conclusions against explicit solutions when available.

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.

Common mistakes

Verification and reasonableness

• Confirm $y(t)=y_*$ makes both sides of the differential equation zero for all $t$.
• Test signs on every adjacent interval rather than relying on a sketch.
• Perturb numerically on both feasible sides and compare with the claimed local behaviour.

Practice

1. Find equilibria of $y'=y^2-4$.
2. Classify $y=-2$.
3. What does $f'(y_*)=0$ imply?

Further deduction

Boundary equilibria need domain-aware language. In a population model restricted to $y\ge0$, the equation $y'=-y$ has an equilibrium at 0 that is asymptotically stable relative to the feasible half-line. Mathematically the same conclusion holds from both sides on $\mathbb R$, but other models may be attracting only from the physically allowed side. Stating the state space prevents an unjustified global claim.

Linearization can fail at a nonhyperbolic equilibrium where $f'(y_*)=0$. Then the first nonzero higher-order term or direct sign analysis decides stability; calling the point stable merely because the derivative is zero confuses an inconclusive test with a conclusion.

Related topics

• Autonomous Equations
• Phase Plane
• Population Models