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plotting a phase plane with a system of linear diff equations.

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Bryan Proa
Bryan Proa on 27 Jul 2021
Commented: Star Strider on 28 Jul 2021
Hi,
I have this system of differential equations(at the very bottom), and I'm not sure how to go about plotting a phase plane. Something that looks like the graph below.(This is just an example for another set of equations.)
syms u(t) v(t)
ode1 = diff(u) == ((1/40)*v)-(u*(9/100))
ode2 = diff(v) == (u*(9/100))-(v*(9/200))

Accepted Answer

Star Strider
Star Strider on 28 Jul 2021
Using the numeric ODE solvers, likely the easiest way to do that is to use odeset to set options, and then choose 'OutputFcn', @odephas2 as described in the documentation section on Solver Output .
syms u(t) v(t) t Y
ode1 = diff(u) == ((1/40)*v)-(u*(9/100))
ode1(t) = 
ode2 = diff(v) == (u*(9/100))-(v*(9/200))
ode2(t) = 
[VF,Sbs] = odeToVectorField(ode1,ode2)
VF = 
Sbs = 
ode12fcn = matlabFunction(VF, 'Vars',{t,Y})
ode12fcn = function_handle with value:
@(t,Y)[Y(1).*(-9.0./2.0e+2)+Y(2).*(9.0./1.0e+2);Y(1)./4.0e+1-Y(2).*(9.0./1.0e+2)]
% ic = randn(2,1)
ic = [-0.4 0.2]
ic = 1×2
-0.4000 0.2000
[t,y] = ode45(ode12fcn,[0 100], ic);
for k = 1:numel(t)
dy(:,k) = ode12fcn(t(k),y(k,:));
end
figure
plot(y(:,1), y(:,2))
grid
xlabel('y_1')
ylabel('y_2')
figure
quiver(y(:,1), y(:,2), dy(1,:).',dy(2,:).')
grid
xlabel('y_1')
ylabel('y_2')
opts = odeset('OutputFcn', @odephas2);
figure
[t,y] = ode45(ode12fcn,[0 100], ic, opts);
Error using uicontrol
This functionality is not available on remote platforms.

Error in odephas2 (line 94)
uicontrol( ...

Error in ode45 (line 269)
feval(outputFcn,[t tfinal],y(outputs),'init',outputArgs{:});
figure
plot(t, y)
grid
xlabel('t')
ylabel('y')
legend(string(Sbs))
That is the essential approach.
I will defer to you for the rest.
.

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