Is there a way to plot a function f with equations (from a ODE system) inside of it?

Question

I am trying to solve this assignment for my Differential Equations class and I am encountering some problems.

Given this system of ODEs:

dy(1) = y(2) − y(3);
dy(2) = −3*y(2) − 5*sin(ω*y(1));
dy(3) = y(1)*y(2);

with omega=3. Initial values: y(1)=0; y(2)=4; y(3)=1;

Plot f between t=0 and t=20:

f(t) = |2*cos(y1(t)) + y2(t)|

I have first tried simulating the equation system with ODE45 with this code:

On editor:

function dy=modelo10(t,y)
    global omega

    dy = zeros(3,1);
    dy(1)= y(2) - y(3);
    dy(2)= -3*y(2) - 5*sin(omega*y(1));
    dy(3)= y(1)*y(2);

end

On command tab:

>>global omega;
>>omega = 3;
>>[T,Y] = ode45(@modelo10, [0,20], [0,4,1]); 
%% I assign the function to variable f
>>f = @(t) 2.*cos(y(:,1)) + y(:,2);
%% And finally plot it with the values t
>>fplot(f, [0,20]);

I get a blank graph with the following error:

Warning: Function behaves unexpectedly on array inputs. To improve 
performance, properly
vectorize your function to return an output with the same size and 
shape as the input
arguments. 
  In matlab.graphics.function.FunctionLine>getFunction
  In matlab.graphics.function.FunctionLine/updateFunction
  In matlab.graphics.function.FunctionLine/set.Function_I
  In matlab.graphics.function.FunctionLine/set.Function
  In matlab.graphics.function.FunctionLine
  In fplot>singleFplot (line 241)
  In fplot>@(f)singleFplot(cax,{f},limits,extraOpts,args) (line 196)
  In fplot>vectorizeFplot (line 196)
  In fplot (line 166) 
Warning: Error updating FunctionLine.

The following error was reported evaluating the function in 
FunctionLine update:
Non-scalar in Uniform output, at index 1, output 1.
Set 'UniformOutput' to false.

Warning: Error updating FunctionLine.

The following error was reported evaluating the function in 
FunctionLine update:
Non-scalar in Uniform output, at index 1, output 1.
Set 'UniformOutput' to false

So my question is what are the steps (or commands) to plot f?

I am using MATLAB R2019a.

Cris Luengo Cris Luengo · Accepted Answer · 2019-05-06T17:06:30

Your first problem is in defining f:

[T,Y] = ode45(@modelo10, [0,20], [0,4,1]); 
f = @(t) 2.*cos(y(:,1)) + y(:,2);

Note that the first line defines a variable Y, but in the second one you use y. These are different variables. This likely works for you because you have a variable y defined in your workspace, but it is doing the wrong thing.

Next, your f is a function f(t), but it is defined not using the input t. It always outputs the same thing. fplot expects a function that outputs an array of the same size as its input, this is the error message you are getting.

However, you don't need to define a function here, you can directly compute all values for f and plot them using plot:

[t,y] = ode45(@modelo10, [0,20], [0,4,1]);
f = 2.*cos(y(:,1)) + y(:,2);
plot(t,f)

Additionally, I want to recommend that you don't use global. You don't need it here, you can define omega as an input argument to modelo10:

function dy = modelo10(t,y,omega)
   dy = zeros(3,1);
   dy(1) = y(2) - y(3);
   dy(2) = -3*y(2) - 5*sin(omega*y(1));
   dy(3) = y(1)*y(2);
end

and then call ode45 as follows:

[t,y] = ode45(@(t,y)modelo10(t,y,omega), [0,20], [0,4,1]);

Here, @(t,y)modelo10(t,y,omega) defines an anonymous function that carries the value of omega with it. This anonymous function has two input arguments (t and y), as required by ode45.

Finally, you can simplify your code by defining modelo10 on a single line:

function dy = modelo10(t,y,omega)
   dy = [ y(2) - y(3); -3*y(2) - 5*sin(omega*y(1)); y(1)*y(2) ];
end

You can now do:

modelo10 = @(t,y) [ y(2) - y(3); -3*y(2) - 5*sin(omega*y(1)); y(1)*y(2) ];
[t,y] = ode45(modelo10, [0,20], [0,4,1]);

Is there a way to plot a function f with equations (from a ODE system) inside of it?

2 Answers