an equilibrium point

An equilibrium point is a place in the phase plane where the system, once there, stays forever. It is a point of perfect balance: a pendulum hanging straight down, a population at exactly its carrying capacity, two species at numbers where neither grows nor shrinks. Nothing is pushing the state in any direction, so it simply rests. In the wind-map picture, it is a point of dead calm where every arrow shrinks to zero.

Mathematically, for the system x' = f(x, y), y' = g(x, y), an equilibrium is a point (x0, y0) where BOTH rates vanish at once: f(x0, y0) = 0 and g(x0, y0) = 0. You find them by solving those two equations simultaneously. There the velocity field is exactly the zero vector, so a state starting at the equilibrium has zero velocity and never leaves — the constant function (x(t), y(t)) = (x0, y0) is a genuine solution, a trajectory that is a single point.

Equilibria are the skeleton of every phase portrait. You always locate them first, because the long-term behaviour of the whole system is organised around them: trajectories are pulled toward stable equilibria, repelled from unstable ones, and steered past saddles. The next question — what the nearby trajectories DO — is the classification problem (node, saddle, spiral, center), and it is decided by the system's behaviour in the immediate neighbourhood of the equilibrium.