fixed point

Picture a marble resting at the very bottom of a bowl, or balanced precariously on the top of a dome. In either case it just sits there: released from exactly that spot, it never moves. A fixed point of a dynamical system is the mathematical version of such a resting place, a state that, once occupied, is occupied forever.

For a continuous system written as dx/dt = f(x), where x is the state vector, a fixed point x* is any state at which the flow vanishes: f(x*) = 0. Because the rate of change is zero there, the trajectory that starts at x* stays at x* for all time. For a discrete map x_{n+1} = F(x_n) the analogous condition is F(x*) = x*: the map sends the point back to itself. Fixed points are the skeleton of the phase portrait, and every other trajectory is organized around them.

A fixed point can be stable (nearby states are drawn in, like the bottom of the bowl), unstable (nearby states run away, like the top of the dome), or a saddle (attracting along some directions, repelling along others). Which one it is is not decided by f(x*) = 0 alone, which only locates the point; you must examine the flow nearby, which is the job of linear stability analysis. A common slip is to assume every equilibrium is stable, but unstable equilibria are perfectly real, they are just never observed to persist.