In the last section, we used a coordinate system defined by the eigenvectors of a matrix to express matrix multiplication in a simpler form. For instance, if there is a basis of \(\real^n\) consisting of eigenvectors of \(A\text{,}\) we saw that multiplying a vector by \(A\text{,}\) when expressed in the coordinates defined by the basis of eigenvectors, was equivalent to multiplying by a diagonal matrix.

In this section, we will put these ideas to use as we explore discrete dynamical systems. Our goal here is to describe the types of phenomena that can occur and to develop a means of detecting these phenomena and then extracting essential information from them.

Preview Activity4.4.1

Suppose that we have a diagonalizable matrix \(A=PDP^{-1}\) where

Find the eigenvalues of \(A\) and find a basis for the associated eigenspaces.

Form a basis \(\bcal\) of \(\real^2\) consisting of eigenvectors of \(A\) and write the vector \(\xvec = \twovec{1}{4}\) as a linear combination of basis vectors.

Write \(A\xvec\) as a linear combination of basis vectors.

What is \(\coords{\xvec}{\bcal}\text{,}\) the representation of \(\xvec\) in the coordinate system defined by \(\bcal\text{?}\)

What is \(\coords{A\xvec}{\bcal}\text{,}\) the representation of \(A\xvec\) in the coordinate system defined by \(\bcal\text{?}\)

What is \(\coords{A^4\xvec}{\bcal}\text{,}\) the representation of \(A^4\xvec\) in the coordinate system defined by \(\bcal\text{?}\)

Subsection4.4.1A first example

We will begin our study of dynamical systems with an example that illustrates how eigenvalues and eigenvectors may be used to understand their behavior.

Activity4.4.2

Suppose we have two species \(R\) and \(S\) that interact with one another and that we record the change in their populations from year to year. When we begin our study, the populations, measured in thousands, are \(R_0\) and \(S_0\text{;}\) after \(k\) years, the populations are \(R_k\) and \(S_k\text{.}\)

If we know the populations in one year, they are determined in the following year by the expressions

We will combine the populations in a vectors \(\xvec_k = \twovec{R_k}{S_k}\) and note that \(\xvec_{k+1} = A\xvec_k\) where \(A = \left[\begin{array}{rr}
0.9 \amp 0.8 \\
0.2 \amp 0.9 \\
\end{array}\right]
\text{.}\)

are eigenvectors of \(A\) and find their respective eigenvalues.

Suppose that initially \(\xvec_0 = \twovec{2}{3}\text{.}\) Write \(\xvec_0\) as a linear combination of the eigenvectors \(\vvec_1\) and \(\vvec_2\text{.}\)

Write the vectors \(\xvec_1\text{,}\) \(\xvec_2\text{,}\) and \(\xvec_3\) as a linear combination of eigenvectors \(\vvec_1\) and \(\vvec_2\text{.}\)

When \(k\) becomes very large, what happens to the ratio of the populations \(R_k/S_k\text{?}\)

If we begin instead with \(\xvec_0 =
\twovec{4}{4}\text{,}\) what eventually happens to the ratio \(R_k/S_k\) as \(k\) becomes very large?

Explain what happens to the ratio \(R_k/S_k\) as \(k\) becomes very large no matter what the initial populations are.

After a very long time, by approximately what factor does the population of \(R\) grow every year? By approximately what factor does the population of \(S\) grow every year?

This activity demonstrates the type of questions we will be considering. In particular, we will assume that we have an initial vector \(\xvec_0\) and a matrix \(A\) and define \(\xvec_{k+1} = A\xvec_k\text{.}\) The eigenvalues and eigenvectors of \(A\) provide the key that helps us understand how the vectors \(\xvec_k\) evolve and enables us to make long-range predictions.

Let's look at the specific example in the preview activity more carefully. We see that we have

and that the matrix \(A\) has eigenvectors \(\vvec_1 =
\twovec{2}{1}\) and \(\vvec_2=\twovec{-2}{1}\) with associated eigenvalues \(\lambda_1=1.3\) and \(\lambda_2=0.5\text{.}\)

Notice that the eigenvectors \(\vvec_1\) and \(\vvec_2\) form a basis \(\bcal\) of \(\real^2\text{.}\) This means that \(A\) is diagonalizable so we can write \(A=PDP^{-1}\) where

With intial populations \(\xvec_0=\twovec{2}{3}\text{,}\) we have \(\xvec_0 = 2\vvec_1+\vvec_2\text{,}\) which means that \(\coords{\xvec_0}{\bcal} = \twovec{2}{1}\text{.}\) Therefore,

Thinking about this geometrically, we begin with the vector \(\coords{\xvec_0}{\bcal}=\ctwovec{2}{1}\text{.}\) Subsequent vectors \(\coords{\xvec_k}{\bcal}\) are obtained by scaling horizontally by a factor of \(1.3\) and scaling vertically by a factor \(0.5\text{.}\) Notice how the points move along a curve away from the origin becoming ever closer to the horizontal axis.

After a very long time, \(\coords{\xvec_k}{\bcal} \approx
\ctwovec{1.3^k\cdot2}{0}\text{,}\) which says that \(\coords{\xvec_{k+1}}{\bcal} \approx
1.3\coords{\xvec_k}{\bcal}\text{.}\) That is, the vector \(\coords{\xvec_k}{\bcal}\) grows by a factor of \(1.3\) every year.

To recover the behavior of the sequence \(\xvec_0, \xvec_1,
\xvec_2, \ldots\text{,}\) we change coordinate systems using the basis defined by \(\vvec_1\) and \(\vvec_2\text{.}\) Here, the points move along a curve away from the origin becoming ever closer to the line defined by \(\vvec_1\text{.}\)

Eventually, the vectors become practically indistinguishable from a scalar multiple of \(\vvec_1 = \twovec{2}{1}\text{;}\) that is, \(\xvec_k\approx s\vvec_1\text{.}\) This means that

so that \(R_k/S_k \approx 2\text{.}\) In addition, \(\xvec_{k+1}
\approx 1.3\xvec_k\) so that \(R_{k+1}\approx 1.3 R_k\) and \(S_{k+1}\approx 1.3 S_k\text{.}\) We conclude that, after a very long time, the ratio of the populations \(R_k\) to \(S_k\) is very close to 2 to 1. We also see that each population is multiplied by 1.3 every year meaning the annual growth rate for both populations is about 30%.

In the same way, we can consider other possible initial populations \(\xvec_0\) as shown in Figure 1. Regardless of \(\xvec_0\text{,}\) the population vectors, in the coordinates defined by \(\bcal\text{,}\) are scaled horizontally by a factor of \(1.3\) and vertically by a factor of \(0.5\text{.}\) The sequence of points \(\coords{\xvec_k}{\bcal}\text{,}\) called trajectories, move along the curves, as shown on the left. In the standard coordinate system, we see that the trajectories converge to the eigenspace \(E_{1.3}\text{.}\)

We conclude that, regardless of the initial populations, the ratio of the populations \(R_k/S_k\) will approach 2 to 1 and that the growth rate for both populations approaches 30%. This example demonstrates the power of using eigenvalues and eigenvectors to rewrite the problem in terms of a new coordinate system. By doing so, we are able to predict the long-term behavior of the populations independently of the initial populations.

Subsection4.4.2Classifying dynamical systems

In the previous example, we were able to make predictions about the behavior of trajectories \(\xvec_k=A^k\xvec_0\) by considering the eigenvalues and eigenvectors of the matrix \(A\text{.}\) The next activity looks at a collection of matrices that demonstrate the types of behavior a \(2\times2\) dynamical system can exhibit.

Activity4.4.3

We will now look at several more examples of dynamical systems. If \(P = \left[\begin{array}{rr}
1 \amp -1 \\
1 \amp 1 \\
\end{array}\right]
\text{,}\) we note that the columns of \(P\) form a basis \(\bcal\) of \(\real^2\text{.}\) Given below are several matrices \(A\) written in the form \(A=PEP^{-1}\) for some matrix \(E\text{.}\) For each matrix, state the eigenvalues of \(A\) and create a sketch similar to that of Figure 1; that is, on the left, sketch the trajectories \(\coords{A^k\xvec_0}{\bcal}\) for some initial vectors \(\xvec_0\) and on the right, sketch \(A^k\xvec_0\text{.}\) Describe the behavior of \(A^k\xvec_0\) as \(k\) becomes very large for a typical initial vector \(\xvec_0\text{.}\)

This activity demonstrates six possible types of dynamical systems, which are determined by the eigenvalues of \(A\text{.}\)

Suppose that \(A\) has two real eigenvalues \(\lambda_1\) and \(\lambda_2\) and that both \(|\lambda_1|, |\lambda_2| \gt
1\text{.}\) In this case, any nonzero vector \(\xvec_0\) forms a trajectory that moves away from the origin so we say that the origin is a repellor. This is illustrated in Figure 2.

Suppose that \(A\) has two real eigenvalues \(\lambda_1\) and \(\lambda_2\) and that \(|\lambda_1| \gt 1 \gt |\lambda_2|
\text{.}\) In this case, most nonzero vectors \(\xvec_0\) form trajectories that converge to the eigenspace \(E_{\lambda_1}\text{.}\) In this case, we say that the origin is a saddle as illustrated in Figure 3.

Suppose that \(A\) has two real eigenvalues \(\lambda_1\) and \(\lambda_2\) and that both \(|\lambda_1|, |\lambda_2| \lt
1\text{.}\) In this case, any nonzero vector \(\xvec_0\) forms a trajectory that moves into the origin so we say that the origin is an attractor. This is illustrated in Figure 4.

Suppose that \(A\) has a complex eigenvalue \(\lambda = a+bi\) where \(|\lambda| \gt 1\text{.}\) In this case, a nonzero vector \(\xvec_0\) forms a trajectory that spirals away from the origin. We say that the origin is a spiral repellor, as illustrated in Figure 5.

Suppose that \(A\) has a complex eigenvalue \(\lambda = a+bi\) where \(|\lambda| = 1\text{.}\) In this case, a nonzero vector \(\xvec_0\) forms a trajectory that moves on a closed curve around the origin. We say that the origin is a center, as illustrated in Figure 6.

Suppose that \(A\) has a complex eigenvalue \(\lambda = a+bi\) where \(|\lambda| \lt 1\text{.}\) In this case, a nonzero vector \(\xvec_0\) forms a trajectory that spirals into the origin. We say that the origin is a spiral attractor, as illustrated in Figure 7.

Activity4.4.4

In this activity, we will consider several ways in which two species might interact with one another. Throughout, we will consider two species \(R\) and \(S\) whose populations in year \(k\) form a vector \(\xvec_k=\twovec{R_k}{S_k}\) and which evolve according to the rule

Explain why the species do not interact with one another. Which of the six types of dynamical systems do we have? What happens to both species after a long time?

Suppose now that \(A = \left[\begin{array}{rr}
0.7 \amp 0.3 \\
0 \amp 1.6 \\
\end{array}\right]
\text{.}\)

Explain why \(S\) is a beneficial species for \(S\text{.}\) Which of the six types of dynamical systems do we have? What happens to both species after a long time?

Suppose now that \(A = \left[\begin{array}{rr}
0.7 \amp 0.5 \\
-0.4 \amp 1.6 \\
\end{array}\right]
\text{.}\)

Explain why this describes a predator-prey system. Which of the species is the predator and which is the prey? Which of the six types of dynamical systems do we have? What happens to both species after a long time?

Suppose now that \(A = \left[\begin{array}{rr}
0.5 \amp 0.2 \\
-0.4 \amp 1.1 \\
\end{array}\right]
\text{.}\)

Compare this predator-prey system to the one in the previous part. Which of the six types of dynamical systems do we have? What happens to both species after a long time?

Subsection4.4.3A \(3\times3\) system

Up to this point, we have focused on \(2\times2\) systems. In fact, the general case is quite similar. As an example, consider a \(3\times3\) system \(\xvec_{k+1}=A\xvec_k\) where the matrix \(A\) has eigenvalues \(\lambda_1 = 0.6\text{,}\) \(\lambda_2 = 0.8\text{,}\) and \(\lambda_3=1.1\text{.}\) The matrix \(A\) therefore has a basis \(\bcal\) consisting of eigenvectors so we can look at the trajectories \(\coords{\xvec_k}{\bcal}\) in the coordinate system defined by \(\bcal\text{.}\) Figure 8 shows how the trajectories will evolve. We see that all the trajectories will converge into the eigenspace \(E_{1.1}\text{.}\)

In the same way, suppose we have a \(3\times3\) system with complex eigenvalues \(\lambda=0.8 \pm 0.5i\) and \(\lambda_3=1.1\text{.}\) Since the complex eigenvalues satisfy \(|\lambda| \lt 1\text{,}\) there is a two-dimensional subspace in which the trajectories spiral in toward the origin. Figure 9 shows some of the trajectories. Once again, we see that all the trajectories converge into the eigenspace \(E_{1.1}\text{.}\)

Activity4.4.5

The following type of analysis has been used to study the population of a bison herd. We will divide the population of female bison into three groups: juveniles who are less than one year old; yearlings between one and two years old; and adults who are older than two years.

Each year,

80% of the juveniles survive to become yearlings.

90% of the yearlings survive to become adults.

80% of the adults survive.

40% of the adults give birth to a juvenile.

By \(J_k\text{,}\) \(Y_k\text{,}\) and \(A_k\text{,}\) we denote the number of juveniles, yearlings, and adults in year \(k\text{.}\) We have

Make a prediction about the long-term behavior of \(\xvec_k\text{.}\) For instance, at what rate does it grow? For every 100 adults, how many juveniles, and yearlings are there?

Suppose that the birth rate decreases so that only 30% of adults give birth to a juvenile. How does this affect the long-term growth rate of the herd?

Suppose that the birth rate decreases further so that only 20% of adults give birth to a juvenile. How does this affect the long-term growth rate of the herd?

Find the smallest birth rate that supports a stable population.

Subsection4.4.4Summary

We have been exploring discrete dynamical systems, which have the form \(\xvec_{k+1}=A\xvec_k\text{,}\) by looking at the eigenvalues and eigenvectors of \(A\text{.}\) In the \(2\times2\) case, we saw that

\(|\lambda_1|, |\lambda_2| \lt 1\) produces an attractor so that trajectories are pulled in toward the origin.

\(|\lambda_1| \gt 1\) and \(|\lambda_2| \lt 1\) produces a saddle in which most trajectories are pushed away from the origin and in the direction of \(E_{\lambda_1}\text{.}\)

\(|\lambda_1|, |\lambda_2| \gt 1\) produces a repellor in which trajectories are pushed away from the origin.

The same kind of reasoning allows us to analyze \(n\times n\) systems as well.

Subsection4.4.5Exercises

1

For each of the \(2\times2\) matrices below, determine which of the six types the dynamical system \(\xvec_{k+1}=A\xvec_k\) belongs to and describe the behavior of typical trajectories as \(k\) becomes large.

Find the eigenvalues of \(A\text{.}\) To which of the six types does the system \(\xvec_{k+1}=A\xvec_{k}\) belong?

Using the eigenvalues of \(A\text{,}\) we can write \(A=PEP^{-1}\) for some matrices \(E\) and \(P\text{.}\) What is the matrix \(E\) and what geometric effect does multiplication by \(E\) have on vectors in the plane?

If we remember that \(A^k = PE^kP^{-1}\text{,}\) determine the smallest value of \(k\) for which \(A^k=I\text{?}\)

Find the eigenvalues of \(B\text{.}\)

Then find a matrix \(E\) such that \(B =
PEP^{-1}\) for some matrix \(P\text{.}\) What geometric effect does multiplication by \(E\) have on vectors in the plane?

Determine the smallest value of \(k\) for which \(A^k=I\text{.}\)

4

Suppose we have the female population of a species is divided into juveniles, yearlings, and adults and that each year

90% of the juveniles live to be yearlings.

80% of the yearlings live to be adults.

60% of the adults survive to the next year.

50% of the adults give birth to a juvenile.

Set up a system of the form \(\xvec_{k+1}=A\xvec_k\) that describes this situation.

Find the eigenvalues of the matrix \(A\text{.}\)

What prediction can you make about these populations after a very long time?

If the birth rate goes up to 60%, what prediction can you make about these populations after a very long time? For every 100 adults, how many juveniles, and yearlings are there?

5

Determine whether the following statements are true or false and provide a justification for your response. In each case, we are considering a dynamical system of the form \(\xvec_{k+1} =
A\xvec_k\text{.}\)

If the \(2\times2\) matrix \(A\) has a complex eigenvalue, we cannot make a prediction about the behavior of the trajectories.

If \(A\) has eigenvalues whose absolute value is smaller than 1, then all the trajectories are pulled in toward the origin.

If the origin is a repellor, then it is an attractor for the system \(\xvec_{k+1} = A^{-1}\xvec_k\text{.}\)

If a \(4\times4\) matrix has complex eigenvalues \(\lambda_1\text{,}\) \(\lambda_2\text{,}\) \(\lambda_3\text{,}\) and \(\lambda_4\text{,}\) all of which satisfy \(|\lambda_j| \gt
1\text{,}\) then all the trajectories are pushed away from the origin.

If the origin is a saddle, then all the trajectories are pushed away from the origin.

6

The Fibonacci numbers form the sequence of numbers that begins \(0, 1, 1, 2, 3, 5, 8, 13, \ldots\text{.}\) If we let \(F_n\) denote the \(n^{th}\) Fibonacci number, then

Use this relationship to compute \(F_{20}\text{.}\)

Explain why \(F_{n+1}/F_{n}\approx \lambda_1\) when \(n\) is very large.

The number \(\lambda_1=\frac{1+\sqrt{5}}{2} = \phi\) is called the golden ratio and is one of mathematics' special numbers.

7

This exercise is a continuation of the previous one.

The Lucas numbers \(L_n\) are defined by the same relationship as the Fibonacci numbers: \(L_{n+2}=L_{n+1}+L_n\text{.}\) However, we begin with \(L_0=2\) and \(L_1=1\text{,}\) which leads to the sequence \(2,1,3,4,7,11,\ldots\text{.}\)

As before, form the vector \(\xvec_n=\twovec{L_{n+1}}{L_n}\) so that \(\xvec_{n+1}=A\xvec_n\text{.}\) Express \(\xvec_0\) as a linear combination of \(\vvec_1\) and \(\vvec_2\text{,}\) eigenvectors of \(A\text{.}\)

Explain why \(L_n\) is the closest integer to \(\phi^n\) when \(n\) is large.

Use this observation to find \(L_{20}\text{.}\)

8

Strang defines the Gibonacci numbers \(G_n\) as follows. We begin with \(G_0 = 0\) and \(G_1=1\text{.}\) A subsequent Gibonacci number is the average of the two previous; that is, \(G_{n+2} = \frac12(G_{n}+G_{n+1})\text{.}\) We then have

If \(\xvec_n=\twovec{G_{n+1}}{G_n}\text{,}\) find the matrix \(A\) such that \(\xvec_{n+1} =
A\xvec_n\text{.}\)

Find the eigenvalues and associated eigenvectors of \(A\text{.}\)

Explain why this dynamical system does not neatly fit into one of the six types that we saw in this section.

Write \(\xvec_{0}\) as a linear combination of eigenvectors of \(A\text{.}\)

Write \(\xvec_n\) as a linear combination of eigenvectors of \(A\text{.}\)

What happens to \(G_n\) as \(n\) becomes very large?

9

Consider a small rodent that lives for three years. Once again, we can separate a population of females into juveniles, yearlings, and adults. Suppose that, each year,

Half of the juveniles live to be yearlings.

One quarter of the juveniles live to be adults.

Adult females produce eight female offspring.

Writing the populations of juveniles, yearlings, and adults in year \(k\) using the vector \(\xvec_k=\threevec{J_k}{Y_k}{A_k}\text{,}\) find the matrix \(A\) such that \(\xvec_{k+1} = A\xvec_k\text{.}\)

Show that \(A^3=I\text{.}\)

What are the eigenvalues of \(A^3\text{?}\) What does this say about the eigenvalues of \(A\text{?}\)

Verify your observation by finding the eigenvalues of \(A\text{.}\)

What can you say about the trajectories of this dynamical system?

What does this mean about the population of rodents?

Find a population vector \(\xvec_0\) that is unchanged from year to year.