Section 5.1 introduces the idea of linearization of an equilibrium point of a system, and Section 5.2 uses nullclines as a tool for qualitative analysis. The rest of the chapter covers special types of systems. Hamiltonian systems and systems with an integral are discussed in Section 5.3, which may be of particular interest to physics majors. Section 5.4 considers systems with a Lyapunov function and gradient systems. Section 5.5 discusses two examples of systems in three dimensions, a food chain model and the Lorenz system. In Section 5.5 we discuss other periodically forced systems and introduce return maps. This section is long and considerably more challenging than the previous sections. It is intended for advanced students.
5.1 Equilibrium Point Analysis
The technique of linearization of an equilibrium point of a system is introduced in this section. This technique is 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. (Many students try to compute eigenvalues for equilibrium points of nonlinear systems.)
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 only knowledge of the linearization of the vector field at the origin is 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 plane.
Comments on selected exercises
In Exercises 1--3, 5--14, and 16--20, a qualitative analysis of the given system is requested. This analysis should go beyond what a student can print out from a good numerical solver. Exercises 5--14 are the same systems as in Exercises 7--16 of Section 5.1. 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 can be safely skipped, 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.
Comments on selected exercises
Exercises 1--3 involve checking that a given system is Hamiltonian (given H) and using H to sketch the phase plane.
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 study rescaling the vector field so that a system is Hamiltonian. (This can also be considered to be a position-dependent change of time scale.)
Exercises 18--20 study examples of bifurcations of Hamiltonian systems.
5.4 Dissipative Systems
Adding friction or damping to the pendulum or the harmonic oscillator turns the energy function into a Lyapunov function. These two examples are discussed in this section and a rather technical definition of Lyapunov function (which applies to the damped pendulum and oscillator) is given. Gradient systems are introduced as a type of system for which the existence of a Lyapunov function is automatic. These lead naturally to applications where the motion is in the direction of steepest ascent.
Comments on selected exercises
Exercises 1--3 involve checking that a given function is Lyapunov and using this function to sketch the phase plane.
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.
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 equations. This example builds on Section 2.5 (which can be covered quickly now if it was skipped earlier). An analysis of the equilibrium points is used to motivate a qualitative description of solutions that agrees with numerically-generated pictures. The Lorenz system is studied more carefully at the end of Chapter 8.
Comments on selected exercises
All the exercises involve the equilibrium points of the food chain model.
5.6 Periodic Forcing of Nonlinear Systems and Chaos
This section looks at periodically forced nonlinear systems. An attempt is made to introduce Poincare return maps. This is a long and difficult section, requiring considerable mathematical maturity. A more accessible introduction to recent results in the study of dynamical systems, e.g., chaos, is found in Chapter 8.
Comments on selected exercises
One problem with trying to present this material at this level is coming up with reasonable exercises that can be done without very fancy technology. Exercises 1--9 try to reinforce the relationship between the graphs of components of solutions and the return map pictures.
Comments on the Labs
Lab 5.1. Hard and Soft Springs
This lab studies the harmonic oscillator with modified restoring force term (hard and soft springs). The soft spring case is the same as the swaying building model considered in Section 2.4. If Sections 5.3 and 5.4 are covered, then interpreting the system using techniques from these sections can be included.
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.