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Linearizing Lorenz Equations using the Implicit Euler Method

I need help to linearize the Lorenz equations so that I can use Matlab to create the butterfly effect, etc. We were given the linearized equations but a couple of students pointed out that one of them was wrong. I don't know which equation is wrong, so if someone could show me how to at least linearize the first Lorenz equation using implicit Eulers, then I can reproduce the method to check the other two equations. I would like to see step by step how the method is employed. We are to only use implicit Eulers. I need them linearized so that I can use Gauss-Seidel iteration in Matlab to create the butterfly effect (we are not to use ode45 or other solvers, hence the need for GS iteration and linearized equations).

here is what the instructor gave us:

dt = s(y-x)

dt = x(r-z)-y

dt = -bz + xy

s = 10, r = 28, b = 8/3

s = sigma, r = rho, b = beta
dt = 10^-4

initial conditions:

x = 2, y = 2, z = 2

x^n+1- x^n
dt = s*x^n+1 + s*y^n+1 → x^n+1 = (1+s*y^n+1)/(1+dt * s)

y^n+1 - y^n
dt = -y^n+1 + x^n+1(r-z^n+1) → y^n+1 = (1 + dt * x^n+1(r-zn+1))/(1 +dt)

z^n+1 - z^n
dt = -b*z^n+1 + x^n+1 * y^n+1 → zn+1 = (1 + dt * x^n+1 * yn+1)/(1 + b*dt )

Thank you for any help possible.

Solution Summary

The expert examines linearizing Lorenz Equations using the Implicit Euler Method.