How to draw a phase portrait of a two-dimensional ODE?

Question

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.

Answer 1

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:

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.

Answer 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 $$(x,y)\mapsto(F_1(x,y),F_2(x,y)).$$