In mathematics, the point is an equilibrium point for the differential equation
if for all .
Similarly, the point is an equilibrium point (or fixed point) for the difference equation
if for .
Equilibria can be classified by looking at the signs of the eigenvalues of the linearization of the equations about the equilibria. That is to say, by evaluating the Jacobian matrix at each of the equilibrium points of the system, and then finding the resulting eigenvalues, the equilibria can be categorized. Then the behavior of the system in the neighborhood of each equilibrium point can be qualitatively determined, (or even quantitatively determined, in some instances, by finding the eigenvector(s) associated with each eigenvalue).
An equilibrium point is hyperbolic if none of the eigenvalues have zero real part. If all eigenvalues have negative real part, the equilibrium is a stable equation. If at least one has a positive real part, the equilibrium is an unstable node. If at least one eigenvalue has negative real part and at least one has positive real part, the equilibrium is a saddle point.
Famous quotes containing the words equilibrium and/or point:
“They who feel cannot keep their minds in the equilibrium of a pair of scales: fear and hope have no equiponderant weights.”
—Horace Walpole (1717–1797)
“Film is more than the twentieth-century art. It’s another part of the twentieth-century mind. It’s the world seen from inside. We’ve come to a certain point in the history of film. If a thing can be filmed, the film is implied in the thing itself. This is where we are. The twentieth century is on film.... You have to ask yourself if there’s anything about us more important than the fact that we’re constantly on film, constantly watching ourselves.”
—Don Delillo (b. 1926)