In the module on Cobweb Diagrams we looked at linear discrete dynamical systems -- that is, models of the form

where the function f(p) on the right side of the equation is a linear function

We observed that if m is not one then there is a unique equilibrium point

and that sometimes this equilibrium point is attracting -- that is, over the long term the terms p_n approach p_* -- and sometimes it is repelling -- that is, the terms go off to plus or minus infinity.

In that module you experimented with this phenomenon and were asked to find and prove a theorem that would determine when the equilibrium point was attracting and when it was repelling. We also looked at some nonlinear models. Now, in this module, we formulate and prove two theorems that determine the character of equilibrium points for linear and nonlinear models.

The Classification Theorem for Linear Discrete Dynamical Systems

Consider the linear dynamical system

and if m is not one let

denote the unique equilibrium point. Then

• If |m| < 1 then p_* is attracting in the sense that for any initial condition, p₁,
  

• If |m| > 1 then p_* is repelling and unless p₁ = p_*,
  

Proofs

We present three different proofs. This theorem is very accessible to high school students. The different proofs can be used at different times in the high school curriculum. The first proof uses geometric series.

A proof using geometric series

This proof applies only to the first clause of the conclusion. We begin by computing p₂,

then p₃,

and then p₄.

Now we can see a pattern emerging (the proof is a nice exercise in mathematical induction).

Thus, we see that

If |m| < 1 the first term on the right side of the equation above goes to zero as n goes to infinity and the last term on the right side of the equation above involves a geometric series whose sum is given by

Thus, we see that

The next two proofs are both based on the same idea. We compare the distance

from the n-th term to the equilibrium point to the distance

from the preceding term to the equilibrium point. That is, we want to compute

If this fraction is less than one than the terms are getting closer to the equilibrium point and the equilibrium point is attracting -- that is,

but, if the fraction is bigger than one the terms are going away from the equilibrium point and the equilibrium point is repelling.

A geometric proof

Looking at the cobweb diagram

we see that

is just the absolute value of the slope of the line f(p) = m p + b, or m. Thus, if |m| < 1 the equilibrium point is attracting and if |m| > 1 th equilibrium point is repelling.

An algebraic proof

Our algebraic proof is based on the following computation

So, once again, we see that if |m| < 1 the terms of the sequence are approaching p_* and if |m| > 1 they are getting further away.

Nonlinear Discrete Dynamical Systems

For nonlinear discrete dynamical systems -- that is, systems of the form

where the function f(p) is nonlinear -- the situation is different. The first thing to notice is that a nonlinear discrete dynamical system can have many equlibrium points. The graph below shows one example of a nonlinear dynamical system. The beginning of a cobweb diagram is shown on the green graph paper and the beginning of a graph of the sequence is shown on the blue graph paper. This figure is a live Java applet and is very similar to the Java applets in the preceding two modules. You can change the initial condition by clicking along the bottom edge of the green graph and you can graph the resulting sequence and cobweb diagram one term at a time by clicking repeatedly in the upper two-thirds of the applet.

• Notice first that this nonlinear dynamical systems has five equilibrium points -- the points on the green graph where the red curve f(p) crosses the diagional line.

• Experiment with various different initial conditions. Notice that two of the equilibrium points are attracting and three are repelling. Which ones are attracting and which ones are repelling?

• Are there any other kinds of equilibrium points?

• Can adjacent equilibrium points both be attracting? Or, can they both be repelling?

The phenomena that we observe in the example above can be used in two different ways in the high school curriculum.

• In a high school algebra class this situation is wonderful motivation for calculus and, in particular, for motivating the need for the derivative. For linear dynamical systems the character of the unique equilibrium point is determined by the slope of the linear function
  f(p) = m p + b

  In the nonlinear case the character of each equilibrium point, p_*, is determined by the slope of the curve f(p) at the point p_*. Thus, we need to know how to find this slope mathematically.

• In a calculus class this is a wonderful application of the derivative. In particular, it highlights two important facts about the derivative.
  • The derivative does some of the same jobs for a nonlinear function that the slope does for a linear function.

  • The derivative at a point gives us information about a function close to the point. For linear dynamical systems if the absolute value of the slope is less than one then the equilibrium point is attracting no matter what the initial condition is. For nonlinear dynamical systems if the derivative at an equilibrium point has absolute value less than one then the equilibrium is attracting in the sense that if the initial condition is close to the equilibrium point then the sequence is pulled in to that equilibrium point.

The Classification Theorem for Nonlinear Discrete Dynamical Systems

Consider the dynamical system

and let p_* be an equilibrium point -- that is, suppose

Then, if |f'(p_*)| < 1, the equilibrium point is attracting in the sense that if the initial condition p₁ is sufficiently close to p_* then

We will not prove this theorem here.

An equilibrium point that is attracting in the sense described above is sometimes called stable because if a system moves slightly away from the equilibrium because of some transient problem then the system will return to the equilibrium.

Consider the logistic models

where

In the module on logistic models we experimented with these models and looked at how the constant a affected their long term behavior. Now, using the Classification Theorem for Nonlinear Discrete Dynamical Systems, we can learn more about these logistic models.

• Find all the equilibrium points for these models. We are interested in the nonzero equilirium point. For which values of a is the nonzero equilibrium point positive?

• For which values of a is the nonzero equilibrium point stable?

• Verify your answer to the last question experimentally.