How do I plot a particular solution (with given initial conditions) of 3 differential equations with 3 variables?

The equations are the Lorentz equations:

x(dot) = sigma(y-x)

y(dot) = rx - y - xz

z(dot) = xy - bz

where sigma, r and b are fixed. I need to simulate them for various initial conditions. First off, I'm kind of sketchy on what simulate even means. I've asked around elsewhere and gotten as far as I need to use one of the built in ode solvers? like ode45 or ode1 or something. I'm just not sure how to define all 3 functions at once, or whether I even need to do that? Someone said that it wasn't necessary but I didn't understand why.

How do I tell matlab to solve all of these equations at once and give me an x-y-z plot of the trajectory obtained from a given initial (x,y,z)?

Answer 1

[Mischa Kim]

Andrew, use ode45 and something like

 function Lorentz()
 sigma = 1;
 r     = 1;
 b     = 1;
 [~,S] = ode45(@EOM,[0 50],[1 2 3],[],sigma,r,b);
 x = S(:,1);
 y = S(:,2);
 z = S(:,3)
 plot3(x,y,z)
 box
 end
 function dS = EOM(t,s,sigma,r,b)
 x  = s(1);
 y  = s(2);
 z  = s(3);
 dS = [sigma*(y - x);...
       r*x - y - x*z;...
       x*y - b*z];
 end

Note, the three equations are coupled, therefore they need to be solved simultaneously.

[Andrew Davies]

First of all, thanks for helping out. I'm quite a novice with Matlab so forgive me if some of these questions are silly.

What does EOM mean? and the @ symbol, i'm confused about the function of the @ smybol.

Also, [~,S]? And it says Error: Function definitions are not permitted at the prompt or in scripts.

Sorry if I sound like a retard, and thanks again.

[Mischa Kim]

EOM is simply the name of the function that defines the differential equations. The ode45 command returns time as the first argument (in addition to state vector). The ~ is typically used to point out that a variable - in this case time - is not further used in the simulation.

[Andrew Davies]

What about the first line, function Lorentz (), what is that for?

[Mischa Kim]

That's called a function. Simply copy-paste the entire code into one file, save it as Lorentz.m and run it from the command window.

[Star Strider]

Actually, that line defines the file as a function rather than as a main script file. A function file has special properties.