Chapter 20: Discrete Dynamical Systems - CD
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Chapter Summary
Dynamical systems are mathematical models of "how things move." The motion of the bodies in the solar system comes to mind as a physical dynamical system. The moons and planets move in complicated ways around the sun and one another, each exerting forces on the others. Mathematical dynamical systems are broadly applicable to other kinds of "dynamics." We study discrete dynamical systems in this chapter. It appears only on the CD.

This introduction tries to answer the questions: What is a practical mathematical meaning of "dynamical system?" What does "discrete" or "continuous" mean?

In the first section, we will study two economic models of price adjustment in which the price is determined in discrete steps, p[1], p[2], p[3], .... The price at day t+1 is computed from the price at day t by an equation for the change in price. Prices do not change during the day. Time t moves in discrete steps, t=0,1,2,.... The change in the price is p[t+1]-p[t], and what drives this change in our first model is the "excess demand." Change in price is proportional to the amount by which demand exceeds supply:


Once we know an initial price p[0], we can rewrite the change equation in the form of a recursive definition with the unknown on one side and known values on the other,


The phrase "recursive definition" means that in order to compute p[4], you first must compute p[1], p[2], and p[3]. An equation like the ones above of the form

is called an autonomous difference equation of first order, but discrete dynamical systems are more than just the difference equation. "Autonomous" refers to the fact that the function f[p] depends only on the price at time t and not the time itself. "First order" refers to the fact that one time determines the next. (Whales take 9 years to produce babies, so the model of a whale population given in the Scientific Projects has order 9.)

The "dynamics" in our economic models comes down to the question: Do prices tend to a limiting value, or does the economy oscillate or "blow up?" This really entails two mathematical questions. First, how does the behavior of the sequence of prices depend on the initial price? Second, how does the behavior - limiting or oscillating - depend on the parameters in the difference equation?

A practical working definition of "discrete dynamical systems" then is the study of the behavior of solutions to a difference equation as a function of the initial condition p[0] and the equation itself. A single system (of first order) is the function that has as input the initial condition p[0] and as output the whole solution sequence


We seek ways to summarize this infinite output, such as by saying prices tend to an equilibrium value.

A continuous (first order, autonomous) dynamical system is the function that assigns a solution function y[t] to an initial value by means of an equation that says how the quantity changes in continuous time. The rate of change of y is the derivative, so the analog of the difference equation is a formula for the rate of change


or for the differential (hence, the name "differential equation")

The whole continuous dynamical system becomes the function that assigns the solution function y[t] for to the initial condition through the differential equation


Since the change in a differentiable function in a small time step is


with , or

a continuous dynamical system can be approximated on the computer by a discrete dynamical system that moves in steps of size

This is called "Euler's Method" of approximating solutions of differential equations. We used this discrete approximation to the continuous system in our study of S-I-R epidemics in Chapter 2, for the Cool Canary in Chapter 4, and in the theory of natural logs and exponentials in Chapter 8. Now, we study the discrete systems themselves more systematically.

20.1 Two Models for Price Adjustment

Section Summary
In Section CD-11.5, we studied the maximum profit of a monopoly producer, but now we want to open up the production side of our model. The quantity of a product that producers are willing to make and the quantity that consumers are willing to buy both depend on the price of the product. The intuition is that producers will make more when the price of their product is high, but that consumers will buy less at a higher price.

The quantity consumers demand q=D[p] is a decreasing function of price, and the quantity producers are willing to supply q=S[p] is an increasing function of price. We begin with linear supply and demand:


The Figure CD-20.1 shows supply and demand when

In this example, the supply and demand curves cross when p=0.93. This is the solution of the equation that says "supply equals demand."


Figure CD-20.1: Linear supply and demand

20.1.1 Price Adjustment

What happens if supply is not equal to demand? There is more than one approach to this question, so we need more economic assumptions to formulate an answer. Suppose that we are considering a fast but not continuous adjustment. One of the Scientific Project proposals asks you to explore a model of a baking economy. Each morning, bakers make "Byties" and sell them at the prevailing price. If the demand is very high, consumers buy them all up early in the day and suppliers respond the next day with more Byties - and a higher price. (Since we assume that the aggregate of producers supplies strictly in accordance with the price.) If today's Byties are selling at too high a price for consumer demand, some will go unbought, spoil, and be thrown out. Producers will respond tomorrow by baking fewer Byties - and selling at a lower price. If things work nicely, prices tend to a place at which producers and consumers agree. However, it is conceivable that daily adjustments are extreme. One day Byties are left over, so supplies and prices drop a lot, and the next day everything is sold out by 8:00 am. Producers raise production and prices too much, and consumers respond by not buying very many. Next, prices drop too far again... How can we capture the economic dynamics of this situation and develop mathematical criteria for "stability" of our economy? We can make the simplest possible assumption about how prices change - namely that the change in price is proportional to the amount by which demand exceeds supply,


Notice that when demand exceeds supply D[p]-S[p] is positive, and the new price is higher than the old one (assuming k is positive). If supply exceeds demand, D[p]-S[p] is negative, and the new price is lower than the old one. In our specific linear supply and demand model, this gives us the dynamic model

or simply

with and Using the values above for the economic parameters aD, aS, bD, bS, and k=0.0001, beginning at a price of p[0]=0.20, we obtain the solution shown in Figure CD-20.2.


Figure CD-20.2: Prices tend to equilibrium from below

With an initial price of p=1.20, the solution looks like Figure CD-20.3.


Figure CD-20.3: Prices tend to equilibrium from above

The dynamics in this case are that the economy adjusts production so that the price tends to equilibrium.

Suppose we want to adjust prices faster by making k=0.00135? The initial price p=0.20 produces the solution in Figure CD-20.4.


Figure CD-20.4: Bytie prices go ape

Stability of our price adjustment model apparently depends on the price adjustment constant k or the mathematical parameters and . We want you to explore this in an exercise below.

20.1.2 Supply Adjustment

Another possible adjustment mechanism is suggested by the following different kind of economy. Farmers decide in the spring how much corn they will plant. Of course, each farmer does this separately, but the aggregate response is roughly an increasing supply for larger prices. In the fall, after harvest (and all of the uncertainties of farming that our model neglects), farmers sell in a market that adjusts prices very rapidly. For our purposes, suppose that they sell at the equilibrium price at which supply equals demand. They are stuck with the price, but, in spring, they can adjust supply by planting more or less. Again, for simplicity, we suppose that supply and demand are linear


A simple equilibrium is the price p such that S[p]=D[p]

but the dynamics of this model comes from the different times when decisions are made. Producers set supply by last year's prices

whereas consumers decide on purchases by this year's prices

The equilibrium criterion becomes


Notice that our basic difference equation is of the same mathematical form as in the first model but that the economics is different, so your experiments will lead to different conjectures economically. In both models, the next price is an affine linear function of the current price , where in the first economy and in the second. Some general mathematical results about the stability of affine linear dynamical systems


would apply to both of our economies (and several other applications). For example, the general equation is stable when as well as other cases. This case means that the (mythical) Bytie economy is stable when or k=0.001 and the (simplified) corn economy is stable in the case or . We would really like some general rules to tell us when we have stability in terms of the parameters of our various models. A mathematical simplification results if we first understand just the role of two parameters and .

Exercise set CD-20.1

  1. (Computer Exercise) Stability of Experiment with the FstDscrDySy or CobWeb program to formulate a conjecture about the role of and in determining both the stability and limiting value of the general affine linear dynamical system given above. Consider the following cases:
    • 1. ,    ,   P0=100,500,1000
    • 2. ,    ,   P0=100,500,1000
    • 3. ,    ,   P0=100,500,1000
    • 4. ,    ,   P0=100,500,1000
    • 5. ,    ,   P0=100,500,1000
    • 6. ,    ,   P0=100,500,1000
    • 7. ,    ,   P0=100,500,1000
    • 8. ,    ,   P0=100,500,1000
    • 9. ,    ,   P0=100,500,1000

      Apply the results of your conjectures to make predictions about the Bytie and corn economy models above; that is, re-formulate your conjectures in terms of the parameters k, aD, aS, bD, bS, in both cases.

    • Compute 5 steps of the nonlinear system

      with P0=25. Check your work with the program FstDscrDySy.

20.2 Cobwebs

Section Summary
There is a helpful graphical device to use in computing iterates of a discrete dynamical system. The resulting figures look something like cobwebs, although they have nothing to do with real spiders.

We program the computer to compute the sequences of a discrete dynamical system by iteration of a function. The general dynamical system


recursively generates the sequence

but, written in terms of p[0], is

This looks pretty messy but is just what we get by plugging the previous value into the equation for change. This has a simpler expression. Let

Then, the recursion is

and the sequence of values is

the computer has a command to perform this repeated function iteration and we will use it for both discrete and continuous dynamical system computations.

We have not used the basic formulation


for our dynamical systems because we want to emphasize the meaning of the equation as a prescription for change in the form

The connection is simply g[p]=p+f[p], but we want to formulate our results in terms of the function f[p] in order to make the clearest connection between discrete and continuous systems.

20.2.1 Equilibria

The first example of the difference between using f[p] and using g[p] is in saying what an equilibrium value is dynamically. If Pe is an equilibrium value, then when you start with p[0]=Pe or get to Pe somehow, the sequence does not change any more.


in other words, the change f[Pe]=0.

Definition: Equilibrium Point of a Discrete Dynamical System The number Pe is an equilibrium point for

if f[Pe]=0. In this case, the constant sequence is a solution of the difference equation.

It might not always be most convenient to write a difference equation in this change form, but we can always do it by adding and subtracting p[t]. For example,


We prefer this form because of the relation to continuous systems. In both cases, the change is zero at an equilibrium. In this case, this means

so either Pe=0 or Pe=60.

20.2.2 Iteration

Here is a way to draw the function iteration


Graph q=g[p] and q=p on the same axes. Begin at the point q=P0 on the line q=p. Move vertically to the graph q=g[p] over p=P0. The value q=g(P0)=p[1] by the form of the dynamical system p[t+1]=g[p[t]].


Figure CD-20.5: First iterate

Now, move horizontally from the point q=p[1] on the graph of q=g[p] over to the line q=p. This point is still p[1], but next we view it as input to p[2]=g[p[1]] by moving vertically to the graph q=g[p] along p=p[1]. This gives us q=g[p[1]]=p[2].


Figure CD-20.6: Two iterates

Continue the process, alternately moving vertically from q=p to q=g[p] and horizontally back to q=p. This is simply a graphical representation of




Figure CD-20.7: Five iterates

A stable equilibrium point looks like a spider working inward toward a place at which the line q=p crosses q=g[p], and an unstable system looks like a spider working outward.

Exercise set CD-20.2

  1. Show that the value Pe is an equilibrium of p[t+1]=p[t]+f[p[t]] if g[Pe]=Pe where g[p]=p+f[p].

  2. Could the constant sequence be a solution to p[t+1]=p[t]+f[p[t]] if ?

    • 1. Find the equilibria of the system

    • 2. Write the difference equation

      in the change form

      What is the function f[p]=? What are the equilibrium points of this system?
    • 3. Find the equilibrium point of the general affine linear discrete dynamical system

      How many equilibria can such a system have? What happens if ? How does this result compare with the conjecture you made in Exercise CD-20.1.1?

    • Sketch the first three iterates of the cobweb for the dynamical system:
      • 1. First initial condition:

      • 2. Second initial condition:

        The computer program FstDscrDySy draws cobwebs as well as conventional graphs of solutions to dynamical systems. Check your solutions to the previous exercise with it.


        Figure CD-20.8: Six Iterates

      • The line q=p crosses q=g[p] when Pe=g[Pe]. Show that this is the same condition as f[Pe]=0.

20.3 The Linear System

Section Summary
The simplest (nontrivial) dynamical system we can study is

or

It is not hard to find a formula for the solution of


Plug in and do some algebra:

The stability of the system


is simply a question of

and this has a simple answer in five cases, which are given in Exercise CD-20.3.2. The case proves a basic stability result:

Theorem CD-20.1 The Linear Stability Theorem If , the linear dynamical system

has the unique equilibrium point Pe=0. If , then 0 is a globally stable attractor; that is, every solution satisfies

20.3.1 The Affine Linear Equation
Now, we want to extend our stability result to the dynamical systems of the form,

We make a change of variables to convert our affine system into a truly linear one. This is a very useful trick in many mathematical problems. We are introducing a "local variable" somewhat like the one in Chapter 1. Our "local variable" is q=p-Pe, where . The value Pe is an equilibrium because and f[Pe]=0. We rewrite the system in terms of the local variable, q[t]=p[t]-Pe:


The dynamical system for q[t] is completely equivalent to the one for p[t], except that we must add or subtract Pe. The change of variables also proves

Theorem CD-20.2 The Affine Stability Theorem If , the affine dynamical system

has the unique equilibrium point . If , then Pe is a globally stable attractor; that is, every solution satisfies

PROOF:

Apply the linear stability to q[t] and unchange the variables.

Exercise set CD-20.3

  1. Show that the only equilibrium of the system is Pe=0, draw the cobwebs, and write the closed formula solutions in the cases
    • 1.
    • 2.
    • 3.
    • 4.

    • Show the following:

      (HINT: In the positive cases, you can take .)

    • State and prove two kinds of Linear Instability Theorems, one in the case and the other in the case .

    • Show that the above system has as its unique equilibrium point if .

    • Show that a closed formula for the solution of

      in terms of t, , , and P0 is

      Note that the solution to

      is . (HINT: Resubstitute p[t]-Pe=q[t], . Q0=? Pe=?)

    • Give the equilibria and their stability for the dynamical systems in Exercise CD-20.1.1.

    • Stable Supply Adjustment
      Consider the supply adjusting economy

      Show that prices in this economy stably approach exactly when farmers are less excitable than consumers, aS<aD. Give an economic interpretation of the expression .

    • Stability of Excess Demand Adjustment
      Consider the price adjustment economy driven by a multiple of excess demand:

      Show that prices stably approach under the condition that . Can you interpret this condition in terms of average sensitivity to price?

Problem CD-20.1 A Geometric Statement of Affine Stability

Rephrase the Linear Stability Theorem in terms of the cobweb diagram.
"If the dynamical system line crosses the line q=p at slopes between ? and ? (or angle between ? and ?), then the linear system is stable..."

Use the CobWeb program to illustrate some examples.

20.4 Logistic Growth

Section Summary
This section extends our study of equilibria to a simple nonlinear case.

The population of species with distinct generations can be modeled by discrete dynamical systems. For example, p[t] might measure the peak population of May flies in year t. May flies hatch, live for a short period to lay eggs, and die. The first model we consider neglects environmental limitations, such as space or food, and only concentrates on basic fertility. The model is realistic only for the early growth of "small" populations. Review the continuous model of algae growth in Exercise CD-28.2 and the the computer program ExpGth for comparison.

Our model is simply

Example CD-20.1 The Basic Fertility Model

where represents the per capita fertility (or "percent" fertility as a fraction). If , each bug just manages to replace him or herself. This is an average rate, so there may be many more females than males. If , each bug replaces itself and also produces a new bug. The next year there are twice as many bugs. The second year, there are four times as many, since the doubled population doubles, and so on. The solution to this model is


This kind of prolific growth cannot continue for very long. The computer program ExpGth (of Chapter 28) shows that algae doubling every 3 hours would exceed the mass of all of Lake Michigan in a short period. Whenever , this model eventually grows very fast.

Notice that we can express this in terms of instability of the equilibrium point Pe=0. When , we know that


If , there is an annual deficit in fertility. For example, if , then the next year the population is only of this year's, the second only or 90.25% of the starting population. As time progresses, the population goes to zero. Mathematically,


In the mathematically stable cases where , the species tends to extinction.

We want to put per capita fertility at small populations together with a limited ability of the environment to support large populations. The next model has these features. The parameter C is a positive constant, and we are interested biologically in the case when .

Example CD-20.2 The Logistic Growth Model

When p[t] is small, the term is near 1, so the model behaves like our Basic Fertility Model; but, as p[t] increases, the term tends to zero and stops growth. In fact, if the population ever exceeds this value p[t]>C, the term is negative , so p[t] "grows" negatively; in other words, p[t] declines. Biologically speaking, we want the population to persist - that is - and we also seek stability - the population in harmony with the environment rather than undergoing wild oscillations in population.

Exercise set CD-20.4

  1. A population governed by linear growth could survive with if there were migration. Imagine bugs on a desert island where survival is difficult but where new bugs come in on birds that visit.

    • 1. Suppose in the migration model above. If 100 new bugs migrate in each year, what is the equilibrium population?
    • 2. Is the equilibrium value the dynamic limit of the system if ?
    • 3. What is the answer to this question in terms of more general parameter values, and ?
    • 4. What is the closed formula solution to the above system?

      Bugs survive in the migration model without filling the entire universe, but the biological assumptions are quite restricted. Desert islands visited by bug-carrying birds is a narrow scope.

    • Nonlinear Stability Experiments
      • 1. Find the two equilibrium values of the Logistic Growth Model in terms of the parameters of the model.
      • 2. Formulate conjectures about the stability of these two equilibria by doing experiments with the FstDscrDySy program in various cases - for example:     C=50     P0=10,60,70     C=50     P0=10,60,70

20.5 Calculus and Nonlinearity

Section Summary
The calculus microscope lets us "see" a nonlinear system near equilibrium.

You probably found it rather difficult to formulate a general conjecture about even the Logistic Growth Model's stability. That model is about as simple as a nonlinear model can be. We need some additional tools.

The main idea of differential calculus is simply that smooth nonlinear functions "look" linear under a powerful microscope. What would we "see" if we focused our microscope at an equilibrium of a nonlinear cobweb diagram? A linear cobweb diagram as in Figure CD-20.9.



Figure CD-20.9: A microscopic view of a cobweb

We would expect the nonlinear cobweb to behave just like the linear magnification as long as we stayed close to the equilibrium. We know that the stability of the linear diagram is completely covered by the Linear Stability Theorem above. How can we support the intuitive idea of microscopic stability with symbolic computations? What is the intuitive idea in graphical terms?

Problem CD-20.2 A Geometric Local Stability

Use Problem 20.1 to make a conjecture about stability of a nonlinear system that you view in a microscope focused at a point of equilibrium. In the linear case, you showed the following.

If the line

inside the sector shown, then the linear system is stable.

Figure CD-20.10: The region of linear stability.

Our formulation of the stability theorem is the following symbolic result.

Theorem CD-20.3 Local Nonlinear Stability Suppose that f[p] is a differentiable function and Pe is an equilibrium value of the dynamical system

that is, f[Pe]=0. If -2<f'[Pe]<0, then when P0 is sufficiently close to Pe,

that is, Pe is a stable attractor for near enough initial conditions P0.

PROOF:

We use two ideas to give a symbolic proof of the geometrically obvious result of your work in Problem 20.2: The microscope or increment approximation


and the local variable trick that we used in the Affine Stability Theorem. The increment equation is used with p=Pe and , so it becomes

First, we look in the microscope, and then we show that we can look outside. If ,


Now, we localize our variable by subtracting Pe on both sides

where q=p-Pe. Any fixed number, such as m = Ave[ 1, |1+f'[Pe]|] = , satisfies

because we have assumed that the fixed number f'[Pe] lies between -2 and 0.

If we take absolute values of the estimate for the change in our dynamical system and further estimate the approximate term by m, then we see that for any ,


For p[t] sufficiently close to Pe, we still have

It follows by successively plugging in and estimating that

and

so . (Further details of the proof are contained in the Mathematical Background.)

Example CD-20.3 Local Stability for Logistic Systems Apply the theorem to the logistic systems


where .

Solution:

The equilibria are Pe=0 and Pe=C. The derivative


At the equilibrium Pe=0, , so 0 is a locally stable when .

At the equilibrium Pe=C, , so C is locally stable when .

Exercise set CD-20.5

  1. Classify the equilibria of the examples in Exercise CD-20.4.2.

    Now, put your knowledge together on some new examples.

  2. For the following difference equations, sketch the cobweb figure: q=p and q=g[p] on the same graph. You will need to find all equilibria. Where do these appear on your figures? Apply the Local Stability Theorem to all of the equilibria. Which ones are locally stable?

20.6 Projects

20.6.1 A Model Economy
A price adjustment model economy is explored in the Scientific Projects.

20.6.2 Sustained Harvest of Sei Whales
One of the Scientific Projects deals with a real scientific model of Sei Whales. The question there is, "How many whales per year can we harvest without sending the population to extinction?" Unfortunately for the whales, the answer depends on their not being overexploited initially. Mathematically, there is a positive local attracting equilibrium, but that is not the whole dynamic story. The computer program Whales offers you some help with the simulation because the dynamical system is order 9, corresponding to the number of years it takes to produce a baby whale. The mathematical model was derived by J.R. Beddington in a Report of the International Whaling Commission in 1978.

20.6.3 Computation of Inverse Functions
The inverse of a function y=f[x] is the function g[y] that "un-does" what f[x] does. That is, given a value of y=y1, the value of x=x1 that makes f[x1] equal the given y1 is x1=g[y1]. For example, gives the angle whose tangent equals y. A method for you to use yourself to compute the inverse of complicated functions such as y=xx is given in the project on inverse functions. The method is a discrete dynamical system such that, given a value of y1, the system converges to an equilibrium xe, with xexe=y1.

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