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unscientific
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Homework Statement
The figure below shows the path of a particle governed by the Lorenz equations with r = 28, σ = 10, b = 8/3. The x'es and boxes show points where the path crosses the plane z = r − 2σ > 0.
(a) Which indicator shows a decreasing z and which shows an increasing z?
(b) Show the length of element ## | \delta x | ## between the two paths either grows or decays exponentially if aligned with one of the eigenfunctions of jacobian ##\frac{J + J^T}{2}##.
(c) Find ##\frac{J + J^T}{2}## and its eigenvalues at (0, 0, r-2σ). Hence deduce that of ##\delta x ## grows is in the x-y plane while it decays is along the direction of z-axis.
(d) Show the volume decreases exponentially with ##\delta V = \delta V_0 e^{−(σ+1+b)t}##
Since ##\frac{(J + J^T)}{2}## is symmetric, its eigenfunctions are orthogonal. Show that for a cubic element where the three displacement directions are along the eigenfunctions in section (c) decays at the same rate.
Homework Equations
Lorentz equations are given by:
[tex]\dot x = \sigma(y-x)[/tex]
[tex]\dot y = rx - y - xz [/tex]
[tex]\dot z = xy - bz [/tex]
The Attempt at a Solution
Part (a)
[/B]
For ## \dot z < 0##, ##xy < bz \approx 21##.
So the boxes represent decreasing z, the x'es represent increasing z.
Part (b)
How do I show it either grows or decays exponentially? Do I put in z = r − 2σ and find the eigenvalues? Wouldn't it be part (c)? I think this part is simpler than it seems.
Part (c)
The matrix ##\frac{J + J^T}{2}## becomes:
I found the eigenvalues to be:
[tex]\lambda_{1,2} = -\frac{ -(\sigma + 1) \pm \sqrt{ (\sigma+1)^2 - 4(\sigma - \frac{9}{4} \sigma^2) } }{2} [/tex]
[tex]\lambda_3 = -b[/tex]
Part (d)
[tex]\nabla \cdot \vec u = -(\sigma + 1 + b) [/tex]
Then the result follows.
This is all that I have managed to do so far, would appreciate any input, thank you!