Linearizing Lorenz Equations using the Implicit Euler Method
Not what you're looking for?
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:
dx
dt = s(y-x)
dy
dt = x(r-z)-y
dz
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.
Purchase this Solution
Solution Summary
The expert examines linearizing Lorenz Equations using the Implicit Euler Method.
Purchase this Solution
Free BrainMass Quizzes
Know Your Linear Equations
Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.
Geometry - Real Life Application Problems
Understanding of how geometry applies to in real-world contexts
Exponential Expressions
In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.
Graphs and Functions
This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.
Multiplying Complex Numbers
This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.