3502 The Phase Plane

The sort of system for which we will be trying to sketch solutions can be written in the form $$\begin{aligned} x'=ax+by\\ y'=cx+dy \end{aligned} \lrArr \boldsymbol{x'}=A\boldsymbol{x}, \text{ where } A=\begin{pmatrix} a&b\\c&d \end{pmatrix}\tag{1}$$ where $a, b, c, d$ are constants.
A solution of this system has the form (we write it two ways) $$\begin{aligned} \boldsymbol{x}(t)=\begin{pmatrix} x(t)\\y(t) \end{pmatrix} \end{aligned}, \begin{aligned} x=x(t)\\y=y(t) \end{aligned}\tag{2}$$ It is a vector function of $t$ whose components satisfy the system $(1)$ when they are substituted in for $x$ and $y$. In general, you learned in 18.02 and physics that such a vector function describes a motion in the $xy$-plane; the equations in $(2)$ tell how the point $(x, y)$ moves in the $xy$-plane as the time $t$ varies. The moving point traces out a curve called the trajectory of the solution $(2)4. The$xy$-plane itself is called the **phase plane** for the system$(1)$. We show a sketch of a trajectory at right. Notice the arrow is used to indicate the direction of increasing time.  We use the term **phase portrait** to mean the graphs of enough trajectories to give a good sense of all the solutions to the system$(1)$.

Critical Points

Definition. A critical point is a point where the derivatives are 0. Therefore a point $(x_0, y_0)$ is a critical point of the system $(1)$ if $$\begin{pmatrix} x'\\y' \end{pmatrix}=A\begin{pmatrix} x_0\\y_0 \end{pmatrix}=\begin{pmatrix} 0\\0 \end{pmatrix}$$ The equations of the system $(1)$ show this is equivalent to $$x = x_0, y = y_0 \text{ is a (constant) solution to (1).}$$ Critical points are the key to our qualitative view of systems. We classify the linear systems by their behavior near critical points.
For the linear system constant coefficient system $(1)$ there is always a critical point at $(0, 0)$. If the matrix $A$ is invertible then this is the only critical point.

Sketching Principle

When sketching integral curves for direction fields we saw that integral curves did not cross. For the system $(1)$ we have a similar principle.

Sketching Principle. Two trajectories of $(1)$ cannot intersect.