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.
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 some students struggle with this material.
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 appropriate for 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:
CompetingSpecies, Duffing, HMSGlider, OscillatingChemicalReactions, Pendulums, PredatorPrey , VanderPol.
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. (Do not try to compute eigenvalues for equilibrium points of nonlinear systems.)
DETools:
The "Zoom In" feature of HPGSystemSolver is particularly handy when discussing linearization.
Comments on selected exercises:
Exercises 1,4 practice linearization at the origin (dropping higher order terms).
Exercise 5; solutions of a linear and nonlinear system at a saddle are compared in an example for which separatrices can be computed explicitly.
Exercises7,9,11 ask for a classification of all equilibria. These systems also appear in Ex. 5-14 of Section 5.2.
Exercises 21,23 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 Competing Species and Oscillating Chemical Reactions are particularly well suited for analysis involving nullclines.
Comments on selected exercises:
Exercises 1,3, 5,7,9 and 17,19 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 5,7,9 are the same systems as in Exercises 7,9,11 of Section 5.1 and Exercises 16-20 relate to the chemical reaction models created in Exercises 25-30 of Section 2.1.
Exercises 21-23 study a nonlinear saddle.
Comments on the Labs
Lab 5.1. Hard and Soft Springs:
This lab studies the harmonic oscillator with a modified restoring force (both hard and soft springs). The soft spring is identical to the model of the swaying building considered in Section 2.4. This lab uses techniques from Section 5.1 (but techniques from Sections 5.3 and 5.4 could be used 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, it is possible to complete the 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.
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 it is recommend doing the lab without this tool, then comparing to the tool. It is interesting to note that this model can be traced back to 1908.