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:
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—Anton Pavlovich Chekhov (1860–1904)
“The town is divided into various groups, which form so many little states, each with its own laws and customs, its jargon and its jokes. While the association holds and the fashion lasts, they admit nothing well said or well done except by one of themselves, and they are incapable of appeciating anything from another source, to the point of despising those who are not initiated into their mysteries.”
—Jean De La Bruyère (1645–1696)