stability


stability
/steuh bil"i tee/, n., pl. stabilities.
1. the state or quality of being stable.
2. firmness in position.
3. continuance without change; permanence.
4. Chem. resistance or the degree of resistance to chemical change or disintegration.
5. resistance to change, esp. sudden change or deterioration: The stability of the economy encourages investment.
6. steadfastness; constancy, as of character or purpose: The job calls for a great deal of emotional stability.
7. Aeron. the ability of an aircraft to return to its original flying position when abruptly displaced.
8. Rom. Cath. Ch. a vow taken by a Benedictine monk, binding him to residence for life in the same monastery in which he made the vow.
[1400-50; < L stabilitas, equiv. to stabili(s) STABILE + -tas- -TY; r. late ME stablete < OF < L, as above]
Syn. 6. steadiness, strength, soundness, poise, solidity, balance.

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In mathematics, a condition in which a slight disturbance in a system does not produce a significant disrupting effect on that system.

A solution to a differential equation is said to be stable if a slightly different solution that is close to it when x = 0 remains close for nearby values of x. Stability of solutions is important in physical applications because deviations in mathematical models inevitably result from errors in measurement. A stable solution will be usable despite such deviations.

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▪ solution of equations
      in mathematics, condition in which a slight disturbance in a system does not produce too disrupting an effect on that system. In terms of the solution of a differential equation, a function f(x) is said to be stable if any other solution of the equation that starts out sufficiently close to it when x = 0 remains close to it for succeeding values of x. If the difference between the solutions approaches zero as x increases, the solution is called asymptotically stable. If a solution does not have either of these properties, it is called unstable.

 For example, the solution y = ce-x of the equation y′ = -y is asymptotically stable, because the difference of any two solutions c1e-x and c2e-x is (c1 - c2)e-x, which always approaches zero as x increases. The solution y = cex of the equation y′ = y, on the other hand, is unstable, because the difference of any two solutions is (c1 - c2)ex, which increases without bound as x increases. A given equation can have both stable and unstable solutions. For example, the equation y′ = -y(1 - y)(2 - y) has the solutions y = 1, y = 0, y = 2, y = 1 + (1 + c2e-2x)-1/2, and y = 1 - (1 + c2e-2x)-1/2 (see Graph—>). All these solutions except y = 1 are stable because they all approach the lines y = 0 or y = 2 as x increases for any values of c that allow the solutions to start out close together. The solution y = 1 is unstable because the difference between this solution and other nearby ones is (1 + c2e-2x)-1/2, which increases to 1 as x increases, no matter how close it is initially to the solution y = 1.

      Stability of solutions is important in physical problems because if slight deviations from the mathematical model caused by unavoidable errors in measurement do not have a correspondingly slight effect on the solution, the mathematical equations describing the problem will not accurately predict the future outcome. Thus, one of the difficulties in predicting population growth is the fact that it is governed by the equation y = axce, which is an unstable solution of the equation y′ = ay. Relatively slight errors in the initial population count, c, or in the breeding rate, a, will cause quite large errors in prediction, even if no disturbing influences occur.

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Universalium. 2010.

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