Chapter 4 is a continuation of the study of nonlinear systems started in Chapter 2. In particular, the idea of linearization near an equilibrium point is introduced and added to the other techniques already studied for the analysis of systems. The level of sophistication required for this chapter, particularly the last three sections, is somewhat higher than what has come before. We feel that Section 4.1 is one of the most important, as it relates the more traditional material on linear systems to nonlinear systems. The remaining sections can be safely skipped.
Section 4.1 introduces the idea of linearization of an equilibrium point of a system. The rest of the chapter covers special types of systems. Hamiltonian systems and systems with an integral are discussed in Section 4.2, which may be of particular interest to physics majors. Section 4.3 considers systems with a Liapounov function and gradient systems. Section 4.4 discusses two examples of systems in three dimensions, a food chain model and the Lorenz system.
4.1 Equilibrium Point Analysis
The technique of linearization of an equilibrium point of a system is introduced in this section. This can be compared to linearization for phase lines introduced at the end of Section 1.6. By 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--5 practice linearization at the origin (dropping higher order terms).
In Exercise 6, solutions of a linear and nonlinear system at a saddle are compared in an example for which separatrices can be computed explicitly.
Exercises 7--20 ask for phase plane analysis using all the tools available (including linearization). The complete answer is quite involved.
Exercise 21 compares the linearization to the nonlinear system for a degenerate equilibrium point.
Exercises 22--30 use linearization to study bifurcation of equilibrium points in one parameter families.
Exercises 31--34 are modeling problems where only knowledge of the linearization of the vector field at the origin is given.
4.2 Hamiltonian Systems
This section discusses systems with a conserved quantity or integral of motion. It may be safely skipped, but 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 4.1, 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, information about the approximate period of oscillations.
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 4.3).
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 a position dependent change of time scale).
Exercises 18--20 study examples of bifurcations of Hamiltonian systems.
4.3 Dissipative Systems
Adding friction or damping to the pendulum or the harmonic oscillator turns the energy function into a Liapounov function. These two examples are discussed in this section and a rather technical definition of Liapounov 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 Liapounov 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 Liapounov and using this function to sketch the phase plane.
Exercises 4--11 study the damped pendulum using the energy as a Liapounov function.
Exercises 12--22 study gradient systems with Exercises 21 and 22 comparing gradient and Hamiltonian systems.
4.4 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 2.7 (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 which agrees with numerically generated pictures. The Lorenz system is studied more carefully at the end of Chapter 6.
Comments on selected exercises
All the exercises involve the equilibrium points of the food chain model.
Comments on the Labs
Lab 4.1: 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 4.2 and 4.3 are covered, then interpreting the system using techniques from these systems can be included.
Lab 4.2: Higher Order Approximations of the Pendulum
This is the same system as the soft spring in Lab 4.1 and the swaying building of Section 2.4, but couched in different language.
Lab 4.3: Off Line Springs
This lab is quite difficult. Comparison of results with physical intuition (common sense) should be encouraged.
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Version 1.1. May, 1996.