This chapter covers linear systems (mostly two-dimensional systems) as well as homogeneous, constant-coefficient, second-order linear equations. Linear algebra is not a prerequisite for our course, and we introduce the required concepts and notation as needed. We find that even those students who have had some linear algebra generally do not have a good geometric understanding of eigenvalues and eigenvectors, so most students learn some linear algebra while studying this chapter.

There are two examples that appear frequently throughout the chapter. They are the harmonic oscillator and Paul's and Bob's Cafes. The purpose of the Cafe model is to provide an example for understanding the meaning of the coefficients of a linear system and for practicing interpretation of solutions in everyday language. Hopefully the readers will not be annoyed by our poking fun at each other.

Section 3.1 covers matrix notation and the basic properties of linear systems. In Sections 3.2 and 3.3, we study linear systems with two real, distinct, nonzero eigenvalues. Section 3.4 generalizes the discussion to the case of nonreal eigenvalues, and Section 3.5 covers certain special cases---repeated eigenvalues and zero eigenvalues. Section 3.6 is a summary of everything presented in Chapter 3 as it applies to second-order, linear equations. It completes the discussion started in Section 2.3. In Section 3.7 we summarize the results of the chapter in terms of the trace-determinant plane. Finally, in preparation for a more detailed discussion of the Lorenz equations, we briefly discuss three-dimensional linear systems in Section 3.8. Sections 3.1-3.6 are fundamental and should be covered completely. Sections 3.7 and 3.8 are less central and may be safely skipped.

DETools

We use HPGSystemSolver constantly throughout the chapter along with the special purpose tools mentioned below.

3.1 Properties of Linear Systems and The Linearity Principle

In this section we introduce matrix notation for linear systems. Most of our students have seen 2x2 matrices previously, often in high school. The determinant of a 2x2 matrix arises naturally when checking for equilibrium points of a linear system. We also mention that that a linear system of algebraic equations has nontrivial solutions if and only if the determinant of the matrix vanishes. This property is used in Section 3.2.

The main goal of the section is to present the Linearity Principle (or Principle of Superposition) and the idea that the general solution of a two-dimensional, linear system can be obtained from two independent solutions. Linearly independent solutions are defined as those with linearly independent initial conditions. The Wronskian appears in Exercise 35 in this section.

Comments on selected exercises

Exercises 1-4 consider the meaning of the parameters in the CD store model. Students are asked to describe the underlying assumptions built into these choices of coefficients.

Exercises 5-9 are straightforward practice with matrix notation for systems.

Exercises 10-13 use the geometric methods of Chapter 2 to view the direction field and the x(t)- and y(t)-graphs. Students should use technology to help with the direction field, but then they should be able to produce rough sketches of the phase portrait and the x(t)- and y(t)-graphs without technology.

Exercise 14 deals with the equilibrium points of a matrix whose determinant is nonzero (completing the discussion in the text). Exercise 15 discusses the case where det A =0.

There are a number of exercises throughout the chapter on second-order equations, and it is important that the students have a lot of practice with the relationship between second-order equations and first-order systems. Exercises 16-19 are their first opportunity for this practice.

Exercises 20-23 consider a simple model of a housing market. These questions are similar to those of Exercises 1-4 (the CD stores).

Exercises 24-30, 33, and 34 concern the Linearity Principle. Exercises 26 and 27 involve the same initial-value problem, so the answers to part (c) are indentical. Making this observation in class after the students have done both exercises can lead to a valuable discussion.

Exercises 31, 32, and 35 are theoretical. They concern linear independence of vectors and solutions as well as the Wronskian.

3.2 Straight-Line Solutions

Sections 3.2 and 3.3 are closely related. In Section 3.2, we find solutions using eigenvalues and eigenvectors. In Section 3.3, we show how this technique leads to an understanding of the phase portrait if the eigenvalues of the linear system are nonzero, real, and distinct.

In Section 3.2, we begin with the geometric observation that certain solution curves for linear systems form straight lines (actually rays) in the phase plane. Using this geometric observation, we derive algebraic tools for finding solutions. Eigenvectors and eigenvalues are introduced geometrically ("Where does the vector field point directly toward or directly away from the origin?"). Then this geometric condition is converted into an algebraic condition. In turn, the algebra gives rise to the characteristic polynomial. Once eigenvalues and eigenvectors have been discussed, the formula for the corresponding particular solution is obtained and justified using a guess-and-test technique.

Although these solutions are exponentials, we call them "straight-line solutions" because they manifest themselves as lines in the phase plane. It is important to emphasize that these lines really correspond to functions x(t) and y(t) that are exponentials.

DETools

The MatrixFields tool helps with the geometric interpretation of eigenvectors. The vector AY is attached to the end of the position vector Y. Start the tool with both the direction field and eigenvalue plane turned off. After playing with the tool for a few minutes, turn on the direction field. Ultimately, you should turn on the eigenvalue plane to illustrate the eigendirections. The discussion of why the eigenvalues are plotted in the complex plane is delayed until Section 3.4.

Comments on selected exercises

Exercises 1-10 provide routine practice with the computation of eigenvalues, eigenvectors, straight-line solutions, and general solutions. The results of Exercises 1-3 and 6-10 can be used in Exercises 1-8 in Section 3.3.

Exercises 11-14 use eigenvalues and eigenvectors to solve initial-value problems. These initial-value problems reappear in the Exercises 9-12 of Section 3.3.

Exercises 15-18 are more theoretical. The students are asked to make general observations regarding eigenvalues of matrices of certain special forms.

Exercises 19-25 apply the techniques developed in this section to the important special case of second-order, homogeneous linear equations. Exercises 21-24 involve the same equations as those in Exercises 1-4 in Section 2.3. They also reappear in Exercises 13-16 in Section 3.3.

3.3 Phase Planes for Linear Systems with Real Eigenvalues

The qualitative analysis of linear systems with real, distinct, nonzero eigenvalues is completed in this section. We emphasize that a good qualitative picture of the system can be obtained from the eigenvalues and eigenvectors alone and that one can often get a good idea of the behavior of the solutions without having to compute them in complete detail.

DETools

In addition to HPGSystemSolver, one can use LinearPhasePortraits to illustrate the various types of phase portraits. We recommend keeping the trace-determinant plane hidden until Section 3.7.

Comments on selected exercises

Exercises 1-8 involve phase portraits corresponding to some of the systems that arose in the exercise set for Section 3.2. Similarly, Exercises 9-12 and 13-16 are related to Exercises 11-14 and 21-24 in Section 3.2, respectively.

The model of Paul's and Bob's cafe returns in Exercises 17 and 18. Now the students have more tools at their disposal.

Exercises 19 and 20 have been very successful examination questions to test to see if students understand the relationship between the phase portrait and x(t)- and y(t)-graphs. Although one is tempted to use technology to help with these problems, they are most instructive if they are done without the use of technology.

Exercises 21 and 22 apply the techniques of this section to a particular harmonic oscillators.

Exercises 23-26 consider simple linear models of a pond inhabited by two species of fish. We ask the students to make predictions based on the phase portraits. These are good examples to revisit during the discussion of linearization in Section 5.1.

3.4 Complex Eigenvalues

The careful student will have already asked if all linear systems have straight-line solutions. In this section we deal with the case of complex eigenvalues. The idea is to use the algebra developed in the previous two sections to obtain a formula for the general solution and then to use the formula to obtain the qualitative analysis. Of particular importance is the idea of a natural period for a system. This important quantity is qualitative information that cannot be seen in the phase plane.

Most students have seen complex numbers before but need a little review in the basics, i.e., multiplication and division of complex numbers. Also, a review of (or an introduction to) Euler's formula is necessary. We have provided a very brief summary of some of this material in Appendix C.

Again, it is stressed that a great deal of qualitative (and quantitative) information can be obtained from the eigenvalues alone. When sketching phase portraits, the easiest way to determine whether solutions spiral clockwise or counter-clockwise is by looking at any nonzero vector in the vector field.

DETools

Both MatrixFields and LinearPhasePortraits illustrate the geometric significance of complex eigenvalues. When you use these tools, you should turn on the eigenvalue plane, but we still recommend that you keep the trace-determinant plane turned off in LinearPhasePortraits until Section 3.7.

Comments on selected exercises

Exercises 1 and 2 ask the student to produce the general solution from one complex eigenvalue and one eigenvector.

Exercises 3-8 and 9-14 are parallel groups of problems. In the first group, the students are asked to determine qualitative information about the solutions without actually finding closed-form expressions for them. In the second group, the students are asked to compute closed-form expressions for solutions and compare their results to those they obtained in the first group. At this point, the arithmetic of finding solutions starts to become fairly involved, and it is easy to assign more problems than the students can reasonably do.

Exercises 15 and 24 give the students practice working with the graphs that arise if the eigenvalues are complex.

Exercises 16-20 consider certain theoretical issues that arise in the case of complex eigenvalues.

Exercises 21 and 22 connect our approach with the approach that is more more common in core engineering courses (amplitude, phase,...). We expand on these connections in Sections 4.2-4.4.

Exercise 23 is a theoretical exercise that relates the material of this section to the behavior of harmonic oscillators.

Exercise 25 is an essay on why there are no spiral saddles in two dimensions.

Exercise 26 involves an elliptical center.

3.5 The Special Cases: Repeated and Zero Eigenvalues

In this section we study the atypical cases. The amount of time spent on this section varies widely depending on how much detail you want the students to master. If you are so inclined, the details of how to find the general solution can be safely skipped. Lab 3.1 can be used as a way for students to discover much of the qualitative material from this section on their own.

We derive the general solution for a 2x2 system with repeated eigenvalues by taking advantage of the fact that every vector is a generalized eigenvector. This approach leads to general solutions that are expressed in terms of their initial conditions, and it should be contrasted with the approach that was taken in Sections 3.2-3.4.

DETools

LinearPhasePortraits can be used to illustrate the fact that the repeated eigenvalue case separates the case of real and distinct eigenvalues from complex eigenvalues. This is a good point to illustrate here as well as when the trace-determinant plane is discussed.

Comments on selected exercises

Exercises 1-4 and 5-8 is another parallel grouping of exercises. Once again, qualitative information is determined initially, and closed-form solutions are obtained later.

Exercises 17-19 and 21-22 are also computational, involving double roots and zero eigenvalues. Exercises 21 and 22 are a pair similar to Exercises 1-8.

Exercises 9-16, 20, and 23 are theoretical. In particular, Exercises 15 and 16 verify two claims that are made in the section.

3.6 Second-Order Linear Equations

When we mentioned the harmonic oscillator as an example in Sections 3.1-3.5, the students were expected to write the second-order equation, convert it to a first-order system, and then complete the analysis described in this chapter. In this section, we classify the harmonic oscillators and explain why it is not necessary to go through all of these steps. In other words, we complete the connection between the methods of this chapter and the "guessing" method described in Section 2.3. It is important that the students completely understand this simplification since it saves a great deal of labor in Chapter 4.

DETools

This is a good place to bring back the MassSpring tool. RLCCircuits with A=0 is another tool that is appropriate for this section.

Comments on selected exercises

Exercises 1-12 are routine computational exercises. They ensure that the students have a basic understanding of the algebraic techniques described in this section.

Exercises 13-20 and 21-28 are parallel groups of exercises involving harmonic oscillators. By the end of these pairings, we hope that students have a good idea of what they can determine easily from the eigenvalues and eigenvectors and what they learn from calculating the closed-form solution.

Exercise 29 asks for a table of all possible harmonic oscillators. It is a precursor to the discussion included in Section 3.7.

Exercises 30 and 31 revisit the Linearity Principle for second-order equations.

Exercises 32 and 33 involve the relationship between the method described in this section and the eigenvalue/eigenvector methods of Sections 3.2-3.5.

Exercises 34 and 35 involve the behavior of solutions for a particular one-parameter family of second-order equations.

Exercises 36-40 describe applications that can be modeled by a second-order equation. We have always enjoyed discussing Exercise 22 with our students. Exercises 37-40 are designed to see if the students can make qualitative predictions if certain parameters are varied slightly. You may want to think about the answers to these exercises before the students appear at your office hours.

3.7 The Trace-Determinant Plane

In this section we present a classification of the behavior of linear systems using the trace-determinant plane. This section is essentially a compilation of everything that has been discussed earlier in the chapter, and therefore it can be omitted. However, the resulting diagram is quite appealing, and students seem to enjoy this method of classification (as opposed to the construction of a table as is suggested in Exercise 1 at the end of this section).

DETools

Now turn on the trace-determinant plane in the LinearPhasePortraits tool. Also, note the comment about Exercise 14 below.

Comments on selected exercises

A word of warning regarding the exercises for this section: A detailed analysis of the qualitative behavior of any of the one- and two-parameter families that are given in the exercises is not a task for the faint of heart. Do not assign many problems from this exercise set.

Exercise 14 is an "animated" exercise involving one-parameter families of phase portraits and paths in the trace-determinant plane.

3.8 Linear Systems in Three Dimensions

The algebra and geometry of three-dimensional linear systems is presented quickly and in a very matter-of-fact fashion in this section, and students without a background in linear algebra may find this section a bit terse. Our goals are to show that the ideas involved in the classification of two-dimensional linear systems generalize to higher dimensions and that, while the fundamental ideas remain the same in higher dimensions, the calculations become more complicated. This section is used only in the analysis of the Lorenz system in Section 5.5.

Comments on selected exercises

Exercises 1, 4-7, and 10-18 are computational. They take advantage of simple matrices or given information (either solutions or eigenvectors) to make the computations reasonable. However, even for these exercises, it is often difficult to visualize the solution curves in phase space.

Exercises 2 and 3 concern linear independence in three dimensions.

Exercises 8 and 9 recall some facts about cubic polynomials that help explain certain facts about eigenvalues.

Exercises 19-22 consider the CD store one last time (in three dimensions).

Review Exercises

Exercises 1-10 are "short answer" exercises. The answers are (usually) one or two sentences. Most (but not all) are relatively straightforward.

Exercises 11-18 are true/false problems. We always expect our students to justify their answers.

Exercises 19 and 21 are matching problems involving phase portraits and x(t)- and y(t)-graphs respectively.

Exercise 20 involves analyzing a one-parameter family of linear systems using the trace-determinant plane.

Exercise 22 involves systems that have a given line of equilibrium points and systems that have a given straight-line solution.

Exercises 23-26 are routine initial-value problems for second-order equations.

Exercises 27-32 involve general solutions, phase portraits, initial-value problems, and x(t)- and y(t)-graphs for an assorted collection of linear systems.

Comments on the Labs

Lab 3.1. Bifurcations in Linear Systems

Students find this lab difficult but very useful in understanding linear systems. Some students attempt to discover the diagram necessary for part 1 by using technology and trying (many) different values of a and b. Others realize immediately that this approach is hopelessly tedious and have no idea how to start. Compare this lab to some of the exercises in Section 3.7.

Lab 3.2. RLC Circuits

The equations modeling RLC circuits, which are frequently given as a main motivation why second-order equations must be studied prior to first-order systems, are actually first expressed as a first-order system in most circuit theory courses. The dependent variables are voltage across the capacitor and the current. In most circuit theory texts, this system is immediately converted into a second-order equation. This lab is meant to emphasize this point and to help our engineering students make the connection between our approach and what they are learning in their circuit theory class. This lab can be paired with Lab 4.2.

Lab 3.3. Measuring Mass in Space

A "harmonic-oscillator-like" device is actually used to measure the mass of astronauts during long space flights. ("How would you measure mass in space?" is a good question for discussion.)

Lab 3.4. Exploring a Parameter Space

This investigation of a three-parameter family of linear systems has resulted in some of our most spectacular lab "reports" ever. A number of our students who are artistically gifted have come up with especially creative ways of presenting their results.

Lab 3.5. Find Your Own Harmonic Oscillator

Part 1 of this lab can also be assigned as a "mini-lab" or homework essay. Some students have difficulty distinguishing external forces from restoring forces. Also, some students identify anything that oscillates as a harmonic oscillator. On the other hand, other students will produce some very clever models.

Lab 3.6. A Baby Bottle Harmonic Oscillator

Baby bottles are fairly foreign to the everyday life of most students in this course. The point of including the lab is that systems that can be modeled using the harmonic oscillator really do occur in unlikely places. (It also gave us something to think about when we were feeding Gib.) Students who have tried this lab have had difficulty obtaining useful data. If you assign this lab, it may be the only time that you ever hear "my baby ate my homework."