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I Linearizing a System of ODE's

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  1. Dec 12, 2016 #1
    I am given the equations of Lorenz with respect to deterministic non-periodic flow:

    [tex] \frac{dX}{dt} = Pr(Y-X), X(0)=X_{0} [/tex]
    [tex] \frac{dY}{dt} = -XZ + rX - Y, Y(0) = Y_{0} [/tex]
    [tex] \frac{dZ}{dt} = XY-bZ, Z(0) = Z_{0} [/tex]

    where Pr is the Prandtl number, r = Ra/Rac is the ratio of the Rayleigh number to its critical value, and b is a parameter that characterize the wave-number. I am told for the question that none of the derivation or mathematics behind it matter.

    So the "question" (not really a question but merely a statement for me to figure out) is: A nonlinear quantity YZ may be linearized by replacing with Yn or nZ, where one of the original variables becomes a free parameter n. The equations above (Lorenz') can now be converted into a vector equation of the form

    [tex] \dot x = A(n)x [/tex]

    Where [itex] x = [X,Y,Z]^{T} [/itex] and [itex] \dot x [/itex] means [itex]\frac{dx}{dt}[/itex]

    I don't think I'm interpreting the question correctly. It says I can replace any variable with a free parameter n? However I do that for instance with letting Y be a free parameter but equations two and three of Lorenz will not be linear as two variables will still be present. I'm under assumption that 'linearization' means having one variable with respect to the other, so I guess I'm at a conceptual loss here and would like to be put in the right direction.
     
  2. jcsd
  3. Dec 12, 2016 #2
    In simple words, a mathematical expression is linear if it does not contain products of any form between the unknown variables/functions/vectors/matrices (so say if x and y are unknowns the expression 5x+2y is linear but the expression 5xy+y or 5x^2+y are not linear).

    In your case your equations 2 and 3 are not linear because they contain the terms XZ and XY respectively. I believe you are asked to replace these terms with nZ and nY, leaving X as it is in every other term of the equations.
     
    Last edited: Dec 12, 2016
  4. Dec 12, 2016 #3
    That would seem to be convenient, however I must point out that it says Yn or nZ . Would it be appropriate to consider this an "inclusive or" statement? I was under the assumption given the context that one or the other but not both, but what you suggest makes more sense. I just need to be certain before continuing.
     
  5. Dec 12, 2016 #4
    It is an exclusive or statement. But what I do does not violate the exclusive or. I am linearizing two different terms(the term XY to nY and the term -XZ to -nZ), not the same term in two different ways.
     
  6. Dec 12, 2016 #5
    I think I see now. Many thanks.
     
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