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单词 AllSolutionOfTheLorenzEquationEnterAnEllipsoid
释义

all solution of the Lorenz equation enter an ellipsoid


If σ,τ,β>0 then all solutions of the Lorenz equationMathworldPlanetmath

x˙=σ(y-x)
y˙=x(τ-z)-y
z˙=xy-βz

will enter an ellipsoid centered at (0,0,2τ) in finite time.In addition the solution will remain inside the ellipsoid once ithas entered. To observe this we define a Lyapunov functionMathworldPlanetmath

V(x,y,z)=τx2+σy2+σ(z-2τ)2.

It thenfollows that

V˙=2τxx˙+2σyy˙+2σ(z-2τ)z˙
=2τxσ(y-x)+2σy(x(τ-z)-y)+2σ(z-2τ)(xy-βz)
=-2σ(τx2+y2+β(z-r)2-bτ2).

We then choose an ellipsoid which all the solutions will enter andremain inside. This is done by choosing a constant C>0 such thatthe ellipsoid

τx2+y2+β(z-r)2=bτ2

is strictly contained in the ellipsoid

τx2+σy2+σ(z-2τ)2=C.

Therefore all solution will eventually enter and remain inside the above ellipsoid since V˙<0 when a solution is located at the exterior of theellipsoid.

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更新时间:2025/5/4 11:40:36