How to draw phase portrait for 2x2 nonlinear system of DE?

  • #1
zenterix
617
81
Homework Statement
Consider the system of nonlinear differential equations
Relevant Equations
##x'=-x+xy##

##y'=-2y+xy##
The critical points are ##(0,0)## and ##(2,1)##.

The linearization of these equations is

$$\begin{bmatrix}x'\\y'\end{bmatrix}=\begin{bmatrix}-1+y_0&x_0\\y_0&x_0-2\end{bmatrix}\begin{bmatrix}x-x_0\\y-y_0\end{bmatrix}$$

At ##(0,0)## we have

$$\begin{bmatrix}x'\\ y'\end{bmatrix}=\begin{bmatrix}-1&0\\0&-2\end{bmatrix}\begin{bmatrix}x\\ y\end{bmatrix}$$

The eigenvalues are ##-1## and ##-2## and associated eigenvectors are ##\hat{i}## and ##\hat{j}##.

The general solution is

$$\vec{x}=c_1e^{-t}\hat{i}+c_2e^{-2t}\hat{j}$$

and here is a phase portrait

1716343856115.png


At ##(2,1)## we have

$$\begin{bmatrix}x'\\ y'\end{bmatrix}=\begin{bmatrix}0&2\\1&0\end{bmatrix}\begin{bmatrix}x-2\\ y-1\end{bmatrix}$$

Eigenvalues are ##\pm\sqrt{2}## and associated eigenvectors are ##\langle 1,\pm\sqrt{2}\rangle##.

The general solution is

$$\begin{bmatrix}x-2\\ y-1\end{bmatrix}=c_1e^{\sqrt{2}t}\begin{bmatrix}1\\\sqrt{2}\end{bmatrix}+c_2e^{-\sqrt{2}t}\begin{bmatrix} 1\\ -\sqrt{2}\end{bmatrix}$$

Here is a phase portrait

1716344046086.png


My question is how to draw the phase portrait of the original system by combining the two phase portraits I drew.

Note that the phase portraits represent linearized behavior of the system about the critical points. As far as I know, though I drew the phase portraits showing how linear solutions behave for all $t$, we are only using these linear solutions to get an idea for the behavior near the critical points.

There is an accompanying example in some notes I am following and they draw the phase portrait as

1716344422259.png


It is not clear to me how some of these solutions were obtained.

For example, the one in the bottom right.

So far I have the following

1716347185120.png


I guess my doubt is about what happens, for example, around region A above.

But I realize now that the behavior far away from the critical points is not important, or at least not confidently characterizable by these linear approximations, which are maybe accurate only near the critical points.
 
Last edited:
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  • #2
I am asking about general techniques and tips on how to do this.

Here is another, more complicated example

$$x'=14x-\frac{x^2}{2}-xy$$

$$y'=16y-\frac{y^2}{2}-xy$$

Since it will take too long to write out the calculations, here they are by hand

1716346369648.png


Above, initially I show four critical points and the Jacobian matrix for linearization.

Then I evaluate the Jacobian at each of the critical points and compute eigenvalues, eigenvectors and draw a small rough phase portrait (note that I am centering at the origin but actually the solutions should be at the critical points).

The final question is how to combine all four phase portraits into a phase portrait of the initial system.

Is this more like an art?

1716347015635.png


1716347030946.png


I think one important thing is to consider whether a critical point is stable or not.
 
Last edited:

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