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## Introduction to 2-D systems: Basic concepts

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

1. formulate an interesting 2-D problem
2. find the fixed points and categorize their stability
3. sketch out a phase portrait
4. Use stella to solve for a few crucial trajectories
When you are done, you will have a lovely picture that tells you exactly how the entire system will evolve in time. Many times you can guess what will happen without even solving the equations. The next example shows how.

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marc spiegelman
Mon Sep 22 21:30:22 EDT 1997