Differential Equations, Bifurcations and Chaos

Inhaltsverzeichnis

1. Frontmatter

2. Chapter 1. Introduction

   Paul C. Matthews

   Abstract

   This short introductory and motivational chapter explains how differential equations are the natural mathematical language for the description of a wide range of natural phenomena, from the motion of the planets to that of fundamental particles, and how they can be used to model and understand developing processes such as population growth or the spread of an epidemic. In view of the unfortunate fact that most interesting differential equations cannot be solved analytically, the idea of a qualitative or descriptive approach is introduced. This method is the main technique used in this book.

3. Chapter 2. Analytical Methods for Differential Equations

   Paul C. Matthews

   Abstract

   This chapter gives some important basic definitions regarding differential equations, and describes how they are classified into different types, according to their order and whether they are linear or nonlinear. Standard analytical methods for solving certain classes of differential equations are introduced, and several worked examples are included. These types of equations include first-order separable equations, first-order linear equations, and second-order linear equations with constant coefficients. This is not the main theme of this book, but it is essential background material that will be needed for the later chapters.

4. Chapter 3. Qualitative Methods for First-Order Differential Equations

   Paul C. Matthews

   Abstract

   This chapter introduces the idea of the qualitative approach to differential equations. This means that rather than find an explicit solution, which is often not possible, the aim is to describe the behaviour of solutions to the equation, and sketch the possible solutions for different initial conditions. This approach will be the main theme of this book. For first-order differential equations, the qualitative method involves the construction of a phase line, which gives a visual representation of the behaviour of solutions of the equation.

5. Chapter 4. Second-Order Linear Systems

   Paul C. Matthews

   Abstract

   This chapter extends the qualitative approach to differential equations to second-order linear differential equations. These can be classified into several different types. Each type can be illustrated by a phase plane diagram, which is a two-dimensional analog of the phase line studied in the previous chapter, and provides a geometrical picture of the behaviour of the system.

6. Chapter 5. Second-Order Nonlinear Systems

   Paul C. Matthews

   Abstract

   This chapter discusses qualitative methods for systems of two coupled first-order nonlinear differential equations. In most cases, these systems cannot be solved analytically. As in Chap. 3, the aim is to describe the behaviour of the system and sketch solutions, but in this case the solutions are visualised using a two-dimensional phase plane. There are several useful techniques that help with the construction of the phase plane. Applications, described through worked examples, include models for competition between different species, the motion of a pendulum, and the growth and eventual decline of an epidemic.

7. Chapter 6. Bifurcations

   Paul C. Matthews

   Abstract

   Bifurcation theory is concerned with systems that have a parameter that can be varied. As the parameter changes, the qualitative behaviour of the system, for example the number of fixed points or their stability, will normally stay the same if the variation is small. But at certain parameter values, the behaviour of the system may change significantly. This is called a bifurcation. It turns out that there are only a few standard types of bifurcation that commonly occur across different systems. This chapter aims to explain why this is the case, and describes these standard types of bifurcation.

8. Chapter 7. Difference Equations

   Paul C. Matthews

   Abstract

   Difference equations can be thought of as the discrete analog of differential equations. They can be used to model systems that evolve in discrete steps, but they also arise directly from differential equations, both in the context of numerical solutions and in the investigation of the stability of periodic solutions. Some of the instabilities and bifurcations of fixed points of difference equations are analogous to those for differential equations, but one type is not—the flip or period-doubling bifurcation, which leads to a solution that alternates between two discrete values.

9. Chapter 8. Chaos

   Paul C. Matthews

   Abstract

   One of the most remarkable mathematical developments of the twentieth century was the understanding of what has become known as chaos. Nonlinear differential equations of order three or higher can show a behaviour that appears almost random, even though the equations are deterministic. Trajectories can diverge from each other exponentially, but thanks to a kind of repeated folding action, the solutions remain in a finite region of phase space. These solutions are described as chaotic. Chaos is difficult to analyse in differential equations, but it is much simpler to study and understand in discrete systems, or difference equations, which are closely related to differential equations.

10. Chapter 9. Solutions to Odd-Numbered Exercises

    Paul C. Matthews

    Abstract

    This chapter contains solutions to the odd-numbered exercises. Solutions to the even-numbered exercises are available as Electronic Supplementary Material.

11. Backmatter