Overview of Chapter Five

In Chapter 5, we study nonlinear systems. In many ways, this material is a continuation of the discussion that began in Chapter 2. However, we are now able to take advantage of the results of Chapter 3 to discuss linearization near an equilibrium point.

Section 5.1 introduces linearization. We feel that this section is one of the most important, as it relates the more traditional material on linear systems to nonlinear systems. In most semesters, this is the only section in Chapter 5 that we have time to cover.

Section 5.2 uses nullclines as a tool for qualitative analysis. This section is a nice application of qualitative techniques. Even though the techniques seem straightforward, it is surprising to see our students struggle with this material. Nevertheless, this material is appropriate for all students at this level.

The rest of the chapter covers special types of systems, and the level of sophistication is somewhat higher. Hamiltonian systems and systems with an integral are discussed in Section 5.3, and Section 5.4 considers systems with a Lyapunov function and gradient systems. Both sections are of particular interest to physics majors, and we teach these sections in our honors class. Section 5.5 discusses two examples of systems in three dimensions, a food chain model and the Lorenz system. In Section 5.6, we discuss nonlinear, periodically-forced systems and introduce return maps. This section is considerably more challenging than most other sections in the book. It is intended for advanced students.

DETools

In addition to HPGSystemSolver, Hu Hohn has written many special purpose demonstrations that go nicely with the examples in this chapter. Make sure that you take a look at:

5.1 Equilibrium Point Analysis

The technique of linearization of an equilibrium point of a system is introduced in this section. It can be viewed as a generalization of the linearization technique introduced at the end of Section 1.6. Using the classification of linear systems obtained in Chapter 3, equilibrium points of nonlinear systems are classified. The major difficulty in this section is to keep clear the distinction between nonlinear and linear systems. (Some students try to compute eigenvalues for equilibrium points of nonlinear systems.)

DETools

The HPGLinearizer tool is designed to help illustrate the material in this section. It is based on HPGSystemSolver, but it can also find equilibria numerically and determine their linearizations. In addition, the "Zoom In" feature of HPGSystemSolver is also useful here.

Comments on selected exercises

Exercises 1-4 practice linearization at the origin (dropping higher order terms).

In Exercise 5, solutions of a linear and nonlinear system at a saddle are compared in an example for which separatrices can be computed explicitly.

Exercises 7-16 ask for a classification of all equilibria. These systems also appear in Exercises 5-14 of Section 5.2.

Exercise 17 compares the linearization to the nonlinear system at a degenerate equilibrium point.

Exercises 18-26 use linearization to study bifurcation of equilibrium points in one-parameter families.

Exercises 27-30 are modeling problems where information regarding the linearization of the vector field at the origin is the only information given.

5.2 Qualitative Analysis

In this section we use the direction field, along with some numerics when necessary, to study the long-term behavior of solutions of nonlinear systems. The only new technique introduced is the location of nullclines in the phase plane. Unfortunately, many students are confused initially about the difference between nullclines and straight-line solutions.

Geometric analysis of this sort is particularly hard for students because it involves many steps and many different ideas and techniques. (They keep hoping you will just give them the magic bullet for understanding systems and are skeptical when you say there isn't one.) Extended projects are particularly helpful in making students realize that there is no template that leads to a complete description of a phase portrait.

DETools

All of the tools listed in the introduction can be used in this section, but ChemicalOscillator and CompetingSpecies are particularly well suited for analysis involving nullclines.

Comments on selected exercises

In Exercises 1-3, 5-14, and 16-20, we request a qualitative analysis of the given system. This analysis should go beyond what a student can print out from a good numerical solver. Note that Exercises 5-14 are the same systems as in Exercises 7-16 of Section 5.1 and Exercises 16-20 relate to the chemical reaction models created in Exercises 25-30 of Section 2.1.

Exercises 4 and 15 concern the general Volterra-Lotka models of a pair of species.

Exercises 21-23 study a nonlinear saddle.

5.3 Hamiltonian Systems

This section discusses systems with a conserved quantity (an integral of motion). Although it is an optional section, it may be of particular interest to students in physics and mechanics. Hamiltonian systems are introduced as a type of system for which the existence of a conserved quantity is automatic.

The nonlinear pendulum is covered in this section. This example can be covered in conjunction with Section 5.1, obtaining the period of oscillation from the linearization around the origin. For the ideal pendulum, the linearization at the origin is a case where linearization alone fails to describe the long-term behavior of the system.

DETools

The following five tools are especially useful in this section:

Comments on selected exercises

Exercises 1-3 involve checking that a given system is Hamiltonian given H. Then H is used to sketch the phase portrait.

Exercises 4-8 study the ideal pendulum. (For the damped pendulum, see Exercises 4-11 of Section 5.4.)

Exercises 9-14 involve determining if a given system is Hamiltonian and determination of the Hamiltonian function.

Exercises 15 and 16 introduce the technique of rescaling a vector field so that the system obtained is Hamiltonian. This rescaling can be thought of as a position-dependent change of time scale.

Exercises 18-20 study examples of bifurcations of Hamiltonian systems.

5.4 Dissipative Systems

Adding damping to the pendulum or the harmonic oscillator turns the energy function into a Lyapunov function. In this section, we discuss Lyapunov functions. We also discuss gradient systems as a type of system that automatically has a Lyapunov function. Gradient systems lead naturally to applications where the motion is in the direction of steepest ascent.

DETools

Four of the tools cited in Section 5.3 are also helpful here with different parameter values:

Comments on selected exercises

Exercises 1-3 involve checking that a given function is Lyapunov and using this function to sketch the phase portrait.

Exercises 4-11 study the damped pendulum using the energy as a Lyapunov function.

Exercises 12-22 study gradient systems with Exercises 21 and 22 comparing gradient and Hamiltonian systems.

Exercises 23 and 24 involve the tuned-mass damper discussed in the section.

5.5 Nonlinear Systems in Three Dimensions

Two examples of three-dimensional systems are discussed in this section. The first is a food chain model with three species. The system has one equilibrium point where the three species coexist, and this point is a sink. The dependence of this point on parameters is considered.

The second example is the Lorenz system, first introduced in Section 2.5. (Section 2.5 can be covered quickly now if it was skipped earlier.) An analysis of the equilibrium points yields a qualitative description of solutions which agrees with numerically-generated pictures. The Lorenz system is studied again at the end of Chapter 8.

DETools

The LorenzEquations tool helps students visualize the Lorenz attractor in three dimensions.

Comments on selected exercises

All of the exercises involve the equilibrium points of the food chain model.

5.6 Periodic Forcing of Nonlinear Systems and Chaos

This section considers periodically forced nonlinear systems and Poincaré return maps. This is a difficult section that requires considerable mathematical maturity. In Chapter 8, we provide a more accessible introduction to recent results in the study of dynamical systems, e.g., chaos.

DETools

The Duffing and Pendulums tools illustrate some of the mathematical issues involved.

Comments on selected exercises

One problem with presenting this material at this level is the lack of reasonable exercises that can be done without fancy technology. Exercises 1-9 reinforce the relationship between the graphs of solutions and the return map pictures.

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 11-14 are true/false problems. We always expect our students to justify their answers.

Exercises 15-18 involve linearization, qualitative analysis via nullclines, and phase portraits for a collection of nonlinear systems.

Exercises 19-24 result in a complete analysis of a two-parameter family of systems.

In Exercise 25, most of the techniques of this chapter are applied to a system.

Exercise 26 involves linearization of equilibrium points that come from a nonlinear second-order equation.

Exercises 27 and 28 involve the manner in which some of the notions introduced in this chapter relate to 2x2 linear systems.

Exercise 29 involves linearization for the simple model of the glider that was introduced in Chapter 2.

Comments on the Labs

Lab 5.1. Hard and Soft Springs

This lab studies the harmonic oscillator with a modified restoring force (either hard and soft springs). The soft spring is identical to the model of the swaying building considered in Section 2.4. This lab can be assigned immediately after Section 5.1, but if Sections 5.3 and 5.4 are covered, then the students should be expected to use techniques from those sections as well.

Lab 5.2. Higher Order Approximations of the Pendulum

This is the same system as the soft spring in Lab 5.1 and the swaying building of Section 2.4 but couched in different language.

Lab 5.3. A Family of Predator-Prey Equations

Students are asked to analyze a one-parameter family of predator-prey equations. In fact, they can complete their analysis without the use of a computer, and this lab does a good job of distinguishing those students who can use both a computer and a pencil. This lab has been one of our most successful.

Lab 5.4. The Glider

This lab is based on a simple model of a glider. Students are asked to analyze the motion of the glider with or without drag. Hu Hohn based his HMSGlider tool on this model, but we recommend that you assign the lab without mentioning that tool to the students. Show them the tool after they have submitted their reports. It is interesting to note that this model has been traced back to 1908.