MATLAB: Phase diagram of a second-order differential equation

phase diagram of a second-order differential equation

Hello everyone

I have solved a second-order differential equation, and as a result of it I have obtained the values of an angle, phi, and its first derivative on time, phidot, assuming that a time equal to zero both are zero. Now, I would like to do a phase diagram as the one that I have attached. Which is the most suitable function to plot and what I need?

Any suggestion?

Best Answer

It's your equation

Assume you have a curve

Then to create a quiver or streamline you need ( ϕ,

, u, v )

I looked here and see that (ϕ,

) in [

]

What is the connection between u, v and

So substituting (ϕ,

) in

we have angle

I assume u=1, then v=a

I used parameters you gave

F = @(phi,dphi) -OmegaR^2/4*sin(4*phi) -2*Omegae*gamma*Hy*cos(phi) -2*Omegae*alpha*dphi;
xx = -pi:0.3:pi;
[p,dp] = meshgrid(xx);      % grid for phi and dphi
v = F(p,dp)./dp;
u = v*0 + 1;
quiver(p,dp,u,v,'b')
hold on
streamline(p,dp,u,v,xx,xx,'r')
hold off

i got this

streamline somehow didn't work

Related Solutions

MATLAB: Solve a second-order differential equation with constant parameters changing through an external parameter

It appears that what you want to do is to pass your data to your ODE function ‘fx’. See the documentation section on Passing Extra Parameters to understand how to do that.

Please do not use global variables!

function dx=fx(t,x);
     % Global variables: Hx, Hy, OmegaR, Omegae, gamma, alpha

Do this instead:

     function dx = fx(t, x, Hx, Hy, OmegaR, Omegae, gamma, alpha);

and your ode45 call then becomes:

         [T,X] = ode45(@(t,x)fx(t, x, Hx, Hy, OmegaR, Omegae, gamma, alpha), time, initial_conditions);

I assume that the additional variables are scalars that do not change during the integration, and that you want to run several integrations, each with different additional parameters. You can do this with a for loop, saving ‘T’ and ‘X’ as cell arrays in each iteration (easiest), then plotting them individually at the end (likely also with a loop).

MATLAB: Ode45 inside a nested loop

I want to show you a simple example of how ode45 works. I wrote simple solver like ode45

function main
clc
F = @(t,x) [x(2); x(1)-2*sin(x(1))];
ts = [0 5];
x0 = [0.5 1];
[t1,x] = ode45(F,ts,x0);                    % standard ode45 solver
[t2, res] = new_ode45_solver(F,ts,x0);      % solve equations with new solver
h1 = plot(t1,x,'r');
hold on
h2 = plot(t2,res,'b');
hold off
legend([h1(1) h2(2)],'ode45 solver','my new ode45 solver')
end
function [t, res] = new_ode45_solver(F,ts,x0)
n = 100;                            % number of steps
dt = (ts(2)-ts(1))/n;               % time step
res = zeros(n,2);                   
t = linspace(ts(1),ts(2),n)';
x = x0;
for i = 1:n
    res(i,:) = x;
    x = x + dt*F(t(i),x)';          % standard Euler scheme/method

end
end

Look at this line

x = x + dt*F(t(i),x)';          % standard Euler scheme/method

Look here

dx(2)=-(OmegaR.^2/4).*sin(4*x(1))+2*Omegae.*gamma.*(Hy.*cos(x(1))-Hx.*sin(x(1)))-...
     2*Omegae.*alpha.*x(2);
% And look this
x(2) = [1 2 3]              % wring! - doesn't work

ode45 works as loop. So if you pass some parameters (Hx,Hy) they can't be arrays

Best Answer

Related Solutions

MATLAB: Solve a second-order differential equation with constant parameters changing through an external parameter

MATLAB: Ode45 inside a nested loop

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