In a 2-D system we will consider dynamical systems that look something like

where *x* and *y* are our two variables of interests.. Examples might
include, rabbits and grass, hosts and parasites or maybe Romeo and
Juliet (...see Below). The most important concepts to understand
about 2-D systems (and general dynamical systems are)

- the phase plane
- Flow on the phase plane
- Phase portraits
- Fixed points
- Stability

The **Phase plane** is a graph where the axes are just our variables
*x* and *y*, so instead of plotting rabbits against time and grass
against time, we want to look at the behavior of rabbits against
grass. If we had three variables, the volume that they span is known
as **phase space** (beyond three variables, life gets tricky).
**Flow on the phase plane** is exactly the same idea we used in
constructing direction sets (Figure 1), that is,
Equations (6) say that for every point *x*,*y* on the phase
plane, and give rules for how the point will change in time. If
is positive *x* will increase, if negative *x* will decrease.
The same is true for *y*, so at every point there will be a little
arrow saying where the system will go in a short time step. This is
easier to show than say. Once we know what the flow
field looks like, individual solutions simply trace out
**trajectories** in phase space.

Now in general, where the change functions are not zero, the system
will evolve in time along various trajectories. Things get more
interesting however, around the few
**fixed** points where things don't change. At a fixed point
both and are zero so if we start *exactly* at a fixed point we will
never move away from it. The more interesting question is what
happens if we start close to a fixed point. Like in the 1-D problem
we can have stable *attractors* and unstable *repellers* but
in two dimensions we can do some other strange things as well. The
next problem will illustrate what can happen. Nevertheless, the basic
rule for analysis of a 2-D system is

- formulate an interesting 2-D problem
- find the fixed points and categorize their stability
- sketch out a phase portrait
- Use stella to solve for a few crucial trajectories

Mon Sep 22 21:30:22 EDT 1997