4

If we're given:

$$\dot{x}=-x+y$$ $$\dot{y}=xy-1$$

How do I draw a phase portrait of this system? I don't understand which direction the arrows are supposed to point.

This is what I got so far:

I found the nullclines:

$$\dot{x}=0$$ $$x=y$$

and

$$\dot{y}=0$$ $$xy=1$$

Then I drew the lines $x=y$ and $xy=1$.

Don't know what to do from here.

sucksatmath
  • 877

2 Answers2

5

This three step process is a summary from the excellent book series "Differential Equations: A Dynamical Systems Approach, Higher-Dimensional Systems" by Hubbard and West.

  • $(1)$ Write the equations as: $$\dfrac{dy}{dx} = \dfrac{\frac{dy}{dt}}{\frac{dx}{dt}} = \dfrac{xy - 1}{y - x}$$

and sketch the isoclines of $(a)$ Horizontal Slope (where $y' = 0$) and $(b)$ Vertical Slope (where $x' = 0$).

  • $(2)$ In each region determined by these isoclines, put together the horizontal and vertical arrows and then sketch the resultant direction field.

Here you are using the above equation, choosing sample $(x, y)$ pairs and drawing the arrows that have direction and magnitude based on the slope.

  • $(3)$ Trace sample trajectories through the direction field.

Some other things that I find helpful are to determine the type of critical points which you see at the intersection of the nullclines, that is $(-1, -1)$ and $(1, 1)$ in this example. One is a stable spiral at $(-1, -1)$ and the other is an unstable saddle point $(1, 1)$. Additionally, you can look at the eigenvectors.

Putting all of these things together, we arrive at the phase portrait:

enter image description here

Lastly, it is worth noting, that these are not hard rules. Practice makes perfect and you'll develop your own approach in a way that is easy for you.

Moo
  • 11,311
  • 1
    How is the part "Write the equations as $\frac{dy}{dx}=\ldots$" useful in this program? – Did Oct 13 '16 at 05:39
  • I think the OP wants to do it by hand and the PP is only meant to compare the hand sketch, which you likely already know. – Moo Oct 13 '16 at 05:41
  • 1
    Sorry but I fail to see the use of this step. On the contrary, the remaining part of (1) ("sketch the isoclines") is of course crucial. – Did Oct 13 '16 at 05:44
  • The authors use $\dfrac{dy}{dx}$ by choosing a bunch of points in each quadrant, calculating that value, which they use for sketching a bunch of slope-direction arrows. These arrow also encode the magnitude of the slope along with the slope-direction. Having a bunch of these helps immensely when doing step $3$. – Moo Oct 13 '16 at 05:47
  • 1
    Not really, often it is only the quadrant where $(\dot x,\dot y)$ is pointing at that is useful, thus, the signs of $\dot x$ and $\dot y$. – Did Oct 13 '16 at 05:50
  • I don't disagree with your statement, but I also find nothing wrong with either approach when trying to do this by hand. After some practice, not all of those steps are necessary, but when first starting to draw these, I found that step helpful. – Moo Oct 13 '16 at 05:55
  • 1
    FWIW, I do not endorse the epistemic relativism currently expressed at the end of this post. – Did Oct 18 '16 at 06:31
-2

Note that your ODE is of the form $(\dot{x},\dot{y})=(F_1(x,y),F_2(x,y))$ and so you can plot the vector field $$(x,y)\mapsto(F_1(x,y),F_2(x,y)).$$

yngabl
  • 1,024