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单词 GradientSystem
释义

gradient system


A gradient system in n is an autonomousMathworldPlanetmath ordinary differential equationMathworldPlanetmath

x˙=-gradV(x)(1)

defined by the gradientMathworldPlanetmath of V where V:n and VC. The following results can be deduced from the definition of a gradient system.
Properties:

  • The eigenvaluesMathworldPlanetmathPlanetmathPlanetmathPlanetmath of the linearization of (1) evaluated at equilibrium point are real.

  • If x0 is an isolated minimum of V then x0 is an asymptotically stable solution of (1)

  • If x(t) is a solution of (1) that is not an equilibrium point then V(x(t)) is a strictly decreasing function and is perpendicularMathworldPlanetmathPlanetmath to the level curves of V.

  • There does not exists periodic solutions of (1).

References

  • HSD Hirsch, W. Morris, Smale, Stephen, Devaney, L. Robert: Differential Equations, Dynamical SystemsMathworldPlanetmathPlanetmath & An Introduction to Chaos. Elsevier Academic Press, New York, 2004.
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