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:
“That doctrine [of peace at any price] has done more mischief than any I can well recall that have been afloat in this country. It has occasioned more wars than any of the most ruthless conquerors. It has disturbed and nearly destroyed that political equilibrium so necessary to the liberties and the welfare of the world.”
—Benjamin Disraeli (1804–1881)
“...there is a difference between being convinced and being stubborn. I’m not certain what the difference is, but I do know that if you butt your head against a stone wall long enough, at some point you realize the wall is stone and that your head is flesh and blood.”
—Maya Angelou (b. 1928)