[Math] Plot phase plane for system of differential equations

MATLABordinary differential equationssystems of equations

I'm in need of some help with matlab code. I'm working on a problem which gives the following system:

$$x'=x^2 – x – y$$

$$y'=x-y$$

We are asked to solve the system numerically starting with $(x(0), y(0))=(-0.3,-0.3)$ for $t \in [0,10]$. Additionally, we are asked to plot the solution in a phase plane and also as a function of time.

My initial reaction is to try and use the ode45 function, then plot the $x$ and $y$ components as functions of time. The thing that I'm really having trouble with is plotting the phase plane…

Any help/links/advice is greatly appreciated!

Best Answer

The phase portrait of a system of two first-order ODEs can be obtained in a similar manner as described in this post, e.g. using Matlab's quiver function. Otherwise, one can plot several trajectories $(x (t), y (t))$ obtained by numerical integration (here with ode45) and having different initial conditions.

Related Solutions

[Math] Phase plane for a system of differential equations.

Yes, but your graph did not designate any point as an "initial point". Nor did you say that "A" was an equilibrium point.

Does the phase-space plot of Lotka-Volterra equations grows chaotically

Your image is the result of drawing many large secants inside the convex closed solution curve. To see it happening, I ran the integation on some smaller time intervals, still using the time step 1

If you go further to T=4000 as you did, the shift of the secant positions will have covered the full solution curve and result in an image just as you got.

Letting the solver return the internally generated sampling nodes and values instead of prescribing a step size 1 results in the image

This was produced with python's scipy.integrate.solve_ivp Using matlab with

[T, Y] = ode45(@LV, [0,4000], Y0);

will do essentially the same, but insert 3 interpolated points in each segment, so that the plot will follow the curve much more closely.

Another measure is to reduce the step size to 0.1 or smaller. Adding some secants for time steps 1,2,4 to add some content gives