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

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Andrew Davies
Andrew Davies 2014-5-18
评论： Star Strider 2014-5-18
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)?

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回答（1 个）

Mischa Kim
Mischa Kim 2014-5-18
编辑：Mischa Kim 2014-5-18
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.
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Mischa Kim
Mischa Kim 2014-5-18
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.

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