This chapter covers two-dimensional, linear systems and, as an application, linear, homogeneous, constant-coefficient, second-order equations. Linear algebra is not a prerequisite for our course, and we introduce the required concepts and notation as they are needed. Even those students who have had some linear algebra generally don't have a good geometric understanding of, for example, eigenvalues and eigenvectors, so we find that most students learn something about linear algebra from our presentation.
There are two examples that reappear frequently throughout the chapter. They are the harmonic oscillator and Paul and Bob's CD stores. The purpose of the CD store 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 enjoy 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 the degenerate cases. Section 3.6 is really a summary section, and it uses the classification of equilibria to study the bifurcations found in various one-parameter families of systems. In preparation for a more detailed discussion of the Lorenz equations, we briefly discuss three-dimensional linear systems in Section 3.7.
3.1 Examples of Linear Systems and The Linearity Principle
In this section we introduce matrix notation for linear systems. Most of our students have previously seen 2 by 2 matrices (frequently in high school). The determinant of a 2 by 2 matrix arises naturally when checking for equilibrium points of a linear system.
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. Ideally, 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 has an important typo. The condition stated should be Det A =0. See the errata
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--29, 30, 33 and 34 concern the Linearity Principle. Exercises 26 and 27 involve the same initial-value problem, so the answers to parts (c)--(e) are indentical. However, the work in parts (a) and (b) are different. Making this observation in class after the students have done both exercises can lead to a valuable discussion. In addition, parts (d) and (e) of Exercises 26--29 require graphing solutions. If technology is used to produce the graphs, then the students will have no problems. However, if technology is not used, these two parts may be best left until more geometry is discussed in subsequent sections. (On the other hand, it is important for students to learn that they can produce a rough graph of x(t) and y(t) from a solution curve, and we practice this skill repeatedly throughout the chapter.)
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 for Linear Systems
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 plane if the eigenvalues of the linear system are real and distinct.
In Section 3.2, we begin with the geometric observation that some 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.
The "lucky-guess" method (guessing an exponential) for solving the harmonic oscillator is also discussed in this section, and the characteristic polynomial for the second-order equation is compared to the characteristic polynomial for the corresponding system. This connection is so important that it should become second nature to the students.
Comments on selected exercises
Exercises 1--8 ask for a qualitative analysis of linear systems using the direction field. We expect that the students will generate the direction fields using some type of technology, but then they should locate the straight-line solutions by inspection.
Exercises 9--16 ask for an algebraic analysis of the systems that were examined geometrically in Exercises 1--8. It is useful to make the connection between the algebraic and geometric approaches.
Exercises 17--22 ask for the general solution using the method of eigenvalues and eigenvectors.
Exercises 23--28 ask for particular solutions for the systems presented in Exercises 17--22. (Perhaps the authors should learn more about the natural ordering associated to the alphabet.) Also, the third bullet in the directions is not very clear. We want the students to generate a rough sketch of the x(t)- and y(t)-graphs from the solution curves in the phase plane. Then they should compare their graphs to the ones obtained from the solution to the initial-value problem. Although sketching the graph from the solution curve in the phase plane is not exact, it is much quicker than deriving explicit solutions and then graphing them. The algebra involved in Exercises 27 and 28 is "involved." In fact, there is a typo in one term of the answer to part (c) of Exercise 27 in the back of the book. The 2 + \sqrt{33} in one of the denominators should be 2\sqrt{33}. See errata.
Exercises 29--31 are more theoretical. The students are asked to make general observations regarding eigenvalues of matrices of certain special forms.
Exercises 32--38 involve second-order equations. After the students do Exercise 32, it is useful to note how quickly one can derive the characteristic polynomial from the second-order equation. Also, the result about negative eigenvalues has important implications for physical systems such as RLC circuits and mass-spring systems. Exercises 33--36 involve solving initial-value problems. After the students do these exercises, it is useful to note why one immediately knows the eigenvectors from the eigenvalues. In Exercise 38, students use the corresponding first-order system to compute approximate solutions to the harmonic oscillator.
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. Both the harmonic oscillator (overdamped case) and the CD stores reappear as examples.
Unfortunately, we interchanged the roles of V_1 and V_2 in the discussion of the phase portrait of the harmonic oscillator on pages 270 and 271. This will be confusing to students. See the errata.
Comments on selected exercises
Exercises 1--4 involve phase planes corresponding to some of the systems that arose in the exercise set for Section 3.2. Exercise 5--8 pair up with Exercises 1--4. We expect the students to be proficient at graphing x(t) and y(t) from the phase portrait.
Exercises 9--12 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 4.1.
The CD store returns in Exercises 13 and 14. Now the students have more tools at their disposal.
In Exercises 15--18, the harmonic oscillator returns. Again, students should be able to provide rough sketches of y(t) directly from the phase plane. Unfortunately, the graph of y(t) in part (d) of Exercise 15 is wrong. See the errata.
Ignore the parenthetical comment in the statement of Exercise 19.
Exercises 21 and 22 have been very successful examination questions to test to see if students understand the relationship between the phase plane and the graphs of x(t) and y(t).
The estimate in Exercise 23 shows the kind of quantitative information that the eigenvalues represent.
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.
Again, it is stressed that a great deal of qualitative (and quantitative) information can be obtained from the eigenvalues alone. When sketching phase planes, the easiest way to tell which way solutions spiral (clockwise or counter-clockwise) is by looking at (a vector of) the vector field.
The undamped and underdamped harmonic oscillator and the CD stores reappear.
Comments on selected exercises
Exercises 1 and 2 consider Euler's formula for complex exponentials and real and imaginary parts of complex vectors.
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. Exercises 23--26 also require the derivation of closed form solutions for second-order equations. 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 32 gives the students practice working with the graphs that arise if the eigenvalues are complex. NOTE: The answer in the back of the book to Exercise 15 will provoke discussion, as one of the given graphs is very, very close to having constant natural period. See the errata.
Exercises 16--20 are theoretically-oriented problems that consider certain points that arise in the case of complex eigenvalues.
Exercises 21 and 22 connect our approach with that which is more common in engineering courses (amplitude, phase, \dots). Unfortunately, in the hint to Exercise 22, we interchanged k_1 and k_2, so the hint is incorrect as stated.
Exercises 27--29 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.
Exercises 30 and 31 concern second-order linear equations and the "Method of the Lucky Guess" in the case of complex eigenvalues. It is important to make this connection if your students are studying second-order equations in other courses.
Exercise 33 is an essay on why there are no spiral saddles in two dimensions.
3.5 The Special Cases (Repeated and Zero Eigenvalues)
In this section the atypical cases are covered. The amount of time spent on this section varies widely depending on how much detail you want the students to master. With the possible exception of the critically-damped harmonic oscillator, 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.
Comments on selected exercises
Exercises 1--4 and 5--8 form another pair of parallel groups of exercises. Again qualitative information is determined initially, and closed form solutions are obtained later.
Exercises 20--22, 24--25, and 27 are also computational, involving double roots, critically-damped oscillators and zero eigenvalues. Exercises 24 and 25 are a pair similar to Exercises 1--8.
Exercises 9--14, 23, and 26 are theoretical. Exercises 15--17 consider harmonic oscillators again, and Exercises 18--19 connect our approach with the more traditional treatment of second-order equations with repeated eigenvalues.
3.6 One Parameter Families of Linear Systems
The study of one-parameter families of linear systems is used as a way to review the classification of linear systems by eigenvalues. This approach avoids the student's desire to memorize a list of cases (see Exercises 1 and 2, however). It also reinforces the role of the atypical linear systems as dividing cases between the major types of systems. The harmonic oscillator, with its damping coefficient as a parameter, is considered in the same light. We have not included the usual tr A vs. det A diagram. Rather, in Lab 3.1, students can construct their own version of this picture.
The income/interest rates model at the end of the section is quite technical but considerably more realistic than the CD store story (which is probably starting to wear thin by this point in the chapter).
Comments on selected exercises
Exercises 1 and 2 ask for a table of the possible cases of linear systems and harmonic oscillators respectively. When we were testing these materials, we had these tables in the text of the section, but we decided to change the orientation of the section based on feedback we got from our test sites. (Remember: keep emailing those cards and letters to \texttt{odes@math.bu.edu}. We read and think about every message.)
Exercises 3--7 and 8--12 are a parallel grouping of problems involving linear systems. Exercises 13--20 and 21-28 are a parallel grouping of problems involving harmonic oscillators. We hope that, by the end of these pairings, 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.
Exercises 29--32 consider one-parameter families of certain systems. Each exercise takes a fair amount of time to analyze in complete detail.
Exercises 35--42 study models that are linear systems. We have always enjoyed discussing Exercise 36 with our students.
3.7 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 some background in linear algebra may find this section particularly difficult. 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 4.4.
There is an error on the bottom of page 331: the printed differential equation is wrong. See the errata.
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 not easy to visualize the phase space or 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).
Comments on the Labs
Lab 3.1: Bifurcation in Linear Systems
Students find this lab difficult but very useful in understanding linear systems. Some students attempt to discover the diagram necessary for Question 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.
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 classes. The dependent variables are voltage over the capacitor and the current. In most circuit theory texts, it 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. It can be assigned after Section 2.4
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: Find Your Own Harmonic Oscillator
Question 1 of this lab can also be assigned as a "mini-lab" or homework essay. Some students have some difficulty separating external from restoring forces. Also, some students identify anything that oscillates as a harmonic oscillator. On the other hand, some very clever models will be produced.
Lab 3.5: 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.)
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Version 1.1. May, 1996.