Equilibrium of an ODE

An equilibrium of an autonomous ODE $y^{\prime }=f(y)$ is a value $y^{\ast }$ such that $f(y^{\ast })=0$. The system, once placed at an equilibrium, stays there forever — there’s no force driving change.

For a system ${\mathbf{x}}^{\prime }=\mathbf{F}(\mathbf{x})$ in higher dimensions, equilibria are the solutions of $\mathbf{F}({\mathbf{x}}^{\ast })=0$. Often the origin $0$ is an equilibrium for linear systems ${\mathbf{x}}^{\prime }=A\mathbf{x}$ since $A\cdot 0=0$.

Also called fixed points, steady states, or critical points depending on context.

Examples

For the Logistic model $\frac{dp}{dt}=rp(1-p/K)$:

• $p=0$ is an equilibrium (no population at all).
• $p=K$ is an equilibrium (population at carrying capacity).

For an undamped pendulum ${\theta }^{\prime \prime }+{\omega }^2\sin \theta =0$ converted to first-order form:

• $\theta =0,{\theta }^{\prime }=0$: pendulum hanging straight down.
• $\theta =\pi ,{\theta }^{\prime }=0$: pendulum balanced upside down.

Stability

An equilibrium can be stable, unstable, or somewhere in between:

• Stable: nearby trajectories stay nearby. Small perturbations don’t grow.
• Asymptotically stable: nearby trajectories converge to the equilibrium. Perturbations decay.
• Unstable: at least some nearby trajectories move away.

For 1D autonomous ODEs, you can determine stability by looking at the sign of $f$ near the equilibrium:

• If $f(y)>0$ for $y$ slightly less than $y^{\ast }$ and $f(y)<0$ for $y$ slightly greater: $y^{\ast }$ is stable (arrows point toward it).
• If $f(y)<0$ on the left and $f(y)>0$ on the right: $y^{\ast }$ is unstable (arrows point away).

For higher-dimensional systems, stability is determined by the eigenvalues of the Jacobian matrix at the equilibrium — see Stability of autonomous systems for the full theory.

Why equilibria matter

Equilibria are the organizing centers of a dynamical system. The long-term behavior of trajectories is largely determined by which equilibria they approach (or avoid):

• A population dynamics model’s equilibria are the long-term population levels.
• An RLC circuit’s equilibrium is the steady-state response.
• A chemical reaction’s equilibria are the concentrations the system tends toward.

If you can find the equilibria and classify their stability, you can predict the long-term behavior of the system without solving the ODE explicitly. This is the philosophy behind Phase plane behaviour and Stability of autonomous systems.

Phase line analysis (1D)

For a single autonomous ODE $y^{\prime }=f(y)$, draw a number line with the equilibria marked. Between consecutive equilibria, $f$ has constant sign. Mark arrows:

• $f>0$: arrow points right (toward larger $y$).
• $f<0$: arrow points left.

Then it’s visually obvious which equilibria are stable (arrows point in) and which are unstable (arrows point out).

For example, $y^{\prime }=y(2-y)$:

• Equilibria at $y=0$ and $y=2$.
• For $y<0$: $y<0$ and $2-y>0$, so the product $y(2-y)<0$. Arrow left.
• For $0<y<2$: $y>0$, $2-y>0$, product positive. Arrow right.
• For $y>2$: $y>0$, $2-y<0$, product negative. Arrow left.

So $y=0$ is unstable (arrows point away), $y=2$ is stable (arrows point in).

For higher-dimensional generalizations (more than one variable), see Phase plane behaviour.