Stability of Fixed Points
The simplest kind of an orbit is a fixed point, or an equilibrium. If a mechanical system is in a stable equilibrium state then a small push will result in a localized motion, for example, small oscillations as in the case of a pendulum. In a system with damping, a stable equilibrium state is moreover asymptotically stable. On the other hand, for an unstable equilibrium, such as a ball resting on a top of a hill, certain small pushes will result in a motion with a large amplitude that may or may not converge to the original state.
There are useful tests of stability for the case of a linear system. Stability of a nonlinear system can often be inferred from the stability of its linearization.
Read more about this topic: Stability Theory
Famous quotes containing the words stability of, stability, fixed and/or points:
“No one can doubt, that the convention for the distinction of property, and for the stability of possession, is of all circumstances the most necessary to the establishment of human society, and that after the agreement for the fixing and observing of this rule, there remains little or nothing to be done towards settling a perfect harmony and concord.”
—David Hume (1711–1776)
“...I feel anxious for the fate of our monarchy, or democracy, or whatever is to take place. I soon get lost in a labyrinth of perplexities; but, whatever occurs, may justice and righteousness be the stability of our times, and order arise out of confusion. Great difficulties may be surmounted by patience and perseverance.”
—Abigail Adams (1744–1818)
“...a fixed aim furnishes us with a fixed measure, by which we can decide whether such or such an action proposed is worth trying for or not, and as aims must vary with the individual, the decisions of any two people as to the desirableness of an action may not be the same.”
—Anna C. Brackett (1836–1911)
“When our relatives are at home, we have to think of all their good points or it would be impossible to endure them. But when they are away, we console ourselves for their absence by dwelling on their vices.”
—George Bernard Shaw (1856–1950)