- #1
Euler2718
- 90
- 3
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.
[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.