Sequences -- Classifying Equilibrium Points -- Discovery

You should use one of the computer algebra systems below with this module. Click on the appropriate icon for your preferred CAS and then arrange your screen so that you can easily move back-and-forth between this window and your CAS window. Click on the appropriate help button for help.

Before beginning this module you may want to review our work with

The purpose of this modules is for you to discover how to determine if an equilibrium point of a dynamical system

p(n + 1) = f(p(n))

is attracting.

We begin by looking at the simplest examples -- linear dynamical systems in which the function f(p) is linear -- that is, dynamical systems of the form

p(n + 1) = m p(n) + b

These models are particularly easy to work with because they have only one equilibrium point, found by solving the equation

Missing Equation

Actually there is one exception. If m = 1 then there are either no equilibrium points (if b is not zero) or every point is an equilibrium point (if b = 0.)

By studying examples like the following

p(n + 1) = 0.5 p(n) + b.
p(n + 1) = 0.8 p(n) + b.
p(n + 1) = 1.2 p(n) + b.
p(n + 1) = 1.5 p(n) + b.
p(n + 1) = -0.5 p(n) + b.
p(n + 1) = -0.8 p(n) + b.
p(n + 1) = -1.0 p(n) + b.
p(n + 1) = -1.2 p(n) + b.

try to find out the answers to the following two questions

When is each equilibrium point b/(1 - m) of the dynamical system
p(n + 1) = m p(n) + b
attracting?
Under what circumstances is

In particular, does the initial condition make any difference?

Try to answer the questions above using hand-drawn cobweb diagrams to look at the list of models above. You may also want to use your CAS window to check your hand-drawn cobweb diagrams.

Nonlinear dynamical systems of the form

p(n + 1) = f(p(n))

are more difficult to work with. One reason they are more difficult to work with is that they often have more than one equilibrium point. For example, the logistic models

p(n + 1) = a [1 - b p(n)] p(n)

have two equilibrium points, found by solving the equation

Missing Graphic

and pack-hunter models usually have three equilibrium points.

By using your CAS window to study all the equilibrium points of several different logistic models and several different pack-hunter models try to find out the answers to the following two questions

When is each equilibrium point attracting?
Under what circumstances is

In particular, does the initial condition make any difference?

Copyright c 1995 by PWS Publishing Company, a division of International Thomson Publishing Inc. Comments to Frank Wattenberg, Department of Mathematics, Carroll College, Helena, MT 59625.