Stability & Equilibria

An equilibrium point (or fixed point) of a differential equation is a constant solution where the system does not change. The central question of stability theory is: if we start near an equilibrium, do solutions stay nearby or drift away? This distinction between stable and unstable equilibria has profound implications across physics, biology, engineering, and economics.

An equilibrium is stable if nearby solutions remain close for all future time, and asymptotically stable if they actually converge to the equilibrium. It is unstable if arbitrarily small perturbations can cause solutions to move far away.

For a one-dimensional autonomous ODE dx/dt = f(x), equilibria occur where f(x) = 0. The sign of f(x) determines the direction of flow on the phase line. Watch how particles flow toward stable equilibria (filled circles) and away from unstable ones (open circles).

Key insight: An equilibrium x* is stable when f'(x*) < 0. At such points, f(x) is positive to the left and negative to the right, creating flow toward x*. Conversely, when f'(x*) > 0, the flow pushes solutions away from the equilibrium.

For higher-dimensional systems, we classify equilibria by linearizing the system. Given a nonlinear system dx/dt = F(x) with equilibrium at x*, we compute the Jacobian matrix J = DF(x*). The eigenvalues of J determine the local behavior.

The Hartman-Grobman Theorem guarantees that if no eigenvalue has zero real part (a hyperbolic equilibrium), then the nonlinear system is topologically equivalent to its linearization near the equilibrium. This means the linearized phase portrait faithfully represents the qualitative behavior of the full nonlinear system in a neighborhood of the fixed point.

Compare the full nonlinear phase portrait (left) with the linearized approximation (right) near each equilibrium of a Lotka-Volterra variant. Notice how the two portraits agree closely near the equilibrium but can diverge farther away.

Key insight: At (0, 0) the eigenvalues are +1 and -1, giving a saddle point -- trajectories approach along one direction and flee along another. At (1, 1) the eigenvalues are purely imaginary, producing a center in the linearization. Since this is a non-hyperbolic case, the Hartman-Grobman theorem does not apply, and higher-order terms determine the true behavior.

When linearization is inconclusive (e.g., eigenvalues on the imaginary axis), we can use Lyapunov's direct method. The idea is to find an energy-like function V(x) that decreases along solutions. If V is positive definite and its time derivative dV/dt is negative semi-definite, then the equilibrium is stable. If dV/dt is strictly negative definite, the equilibrium is asymptotically stable.

Think of V as measuring the "energy" of the system. If energy always decreases, the system must settle down to equilibrium. The level curves of V act as nested barriers that trajectories can only cross inward, never outward.

Watch a trajectory cut inward through the level curves of V(x,y) = x² + y². The right panel shows V decreasing monotonically over time, confirming that the origin is asymptotically stable. Adjust the initial angle to see how every starting direction converges.

Key insight: The Lyapunov function V = x² + y² has level curves that are circles. Trajectories of the system cross these circles inward, and the time series of V(t) confirms it is strictly decreasing. This proves asymptotic stability of the origin without solving the ODE explicitly.