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)
“Every nation ... whose affairs betray a want of wisdom and stability may calculate on every loss which can be sustained from the more systematic policy of its wiser neighbors.”
—James Madison (1751–1836)
“A fixed image of the future is in the worst sense ahistorical.”
—Juliet Mitchell (b. 1940)
“Sometimes apparent resemblances of character will bring two men together and for a certain time unite them. But their mistake gradually becomes evident, and they are astonished to find themselves not only far apart, but even repelled, in some sort, at all their points of contact.”
—Sébastien-Roch Nicolas De Chamfort (1741–1794)