Part II: Two-dimensional systems: flows in the plane - Chapter 2

Phase portraits

The phase portrait of a given equation $\dot{\boldsymbol{x}}=\boldsymbol{f}(\boldsymbol{x})$ is a plot of typical trajectories in the $xy$-plane. For most interesting examples arising in applications, there is no hope of finding the trajectories analytically. And even when explicit formulas are available (which occurs very seldom), they are often two complicated to provide much insight. Instead we will try to determine the qualitative behavior of the solutions. Digital experiments will allow us to explore and observe some phenomena that we will analyse afterwards, more or less rigorously, depending on their level of difficulty.

Sketching the phase portrait

The two basic features of any phase portrait are:

  • the fixed points that are the points $\boldsymbol{x}$ where $\boldsymbol{f} (\boldsymbol{x})=\boldsymbol{0}$;
  • special isoclines, namely the nullclines. They are defined as the curves where either $\dot{x}=0$ or $\dot{y}=0$. The nullclines indicate where the vector field is purely horizontal or vertical. By definition, fixed points lie at the intersection of the nullclines. They also partition the plane into regions where $\dot{x}$ and $\dot{y}$ have a fixed sign.

We will see how we can approximate the phase portrait near a fixed point by that of a corresponding linear system. This will extend what we developed earlier for one-dimensional systems.

Reversing time is so easy!

Consider an equation $\dot{\boldsymbol{x}}=\boldsymbol{f}(\boldsymbol{x})$ that we write as

$$ \frac{\text{d}\boldsymbol{x}}{\text{d}t}=\boldsymbol{f}(\boldsymbol{x}). $$

Remember that

$$ \dot{\boldsymbol{x}}=\frac{\text{d}\boldsymbol{x}}{\text{d}t}. $$

We want to reverse time, which means that we change $t$ into $-t$. We see that it is equivalent to switch the direction of the vector field because

$$ \frac{\text{d}\boldsymbol{x}}{\text{d}(-t)}=-\frac{\text{d}\boldsymbol{x}}{\text{d}t}=-\,\boldsymbol{f}(\boldsymbol{x}). $$

This shows in particular that it is enough to look forward in time, since to know what happens backward in time we just have to change consider the vector field $-\boldsymbol{f}$.

Flowing blobs of initial conditions

So far we have been emphasizing solutions of differential equations as functions of time. We now emphasize the dependence on initial conditions. Let $\dot{\boldsymbol{x}}=\boldsymbol{f}(\boldsymbol{x})$ be a differential equation and imagine drawing some shape in the $xy$-plane (a rectangle, for instance) and solving the equation, starting at each point of this shape. Using our image of an imaginary fluid whose speed at point $\boldsymbol{x}$ is given by $\boldsymbol{f}(\boldsymbol{x})$, we can imagine that we drop a bunch of particles and look how they flow. The shape will move, and likely become distorted, as the following examples illustrate.
You can draw a blob of any shape with your mouse and observe how it evolves. You can also reverse time by changing $t$ into $-t$.