Nonlinear Ordinary Differential Equations

R. Grimshaw

• Publisher: Routledge
• Year: 2017
• ISBN:
  9781351428088
• Edition: 1
• Topic: Mathematics
• Subtopic: Mathematics General
• Index: Mathematics

Ordinary differential equations have long been an important area of study because of their wide application in physics, engineering, biology, chemistry, ecology, and economics. Based on a series of lectures given at the Universities of Melbourne and New South Wales in Australia, Nonlinear Ordinary Differential Equations takes the reader from basic elementary notions to the point where the exciting and fascinating developments in the theory of nonlinear differential equations can be understood and appreciated.
Each chapter is self-contained, and includes a selection of problems together with some detailed workings within the main text. Nonlinear Ordinary Differential Equations helps develop an understanding of the subtle and sometimes unexpected properties of nonlinear systems and simultaneously introduces practical analytical techniques to analyze nonlinear phenomena. This excellent book gives a structured, systematic, and rigorous development of the basic theory from elementary concepts to a point where readers can utilize ideas in nonlinear differential equations.

CHAPTER ONE

INTRODUCTION

1.1 Preliminary notions

Ordinary differential equations involve an independent variable, t, and a dependent variable, x, which is to be a function of t so that x = x(t). We shall denote the derivative of x with respect to t by x’ SO that

$x^{\prime }=\frac{\text{d}x}{\text{d}t}.$

In this section x will be a scalar variable, although later, in section 1.2 and elsewhere in this text, we shall allow x to be a vector. The simplest general form for a differential equation that we can pose is

$x^{\prime }=f\left(x,t\right),$

where f(x, t) is a specified function of x and t. This is said to be a first-order differential equation where the terminology order refers to the highest derivative of x which appears in the equation. The physical interpretation of the variables x and t depends of course on the physical context from which the differential equation arises. However, it is often the case that t corresponds to the time and then the differential equation describes the evolution of some dynamical process as t increases.

A simple example of a first-order differential equation is

$x^{\prime }=\mu x,$

which can be used to describe the growth of a population, when the growth rate is assumed to be proportional to the population itself, the factor of proportionality being the constant μ. This equation has the solution

$x=C{\text{e}}^{\mu t},$

which contains an arbitrary constant, C, and describes the exponential growth of x(t). In general, the solution of any first-order differential equation will contain an arbitrary constant. Hence an extra condition is needed to characterize a solution completely, and often this will be the initial condition

$x\left(t_0\right)=x_0,$

where x₀ and t₀ are specified. For the exponential equation discussed above, we see that x₀ = C e^μt₀ and so

$x=x_0\exp \mu \left(t-t_0\right).$

For specified values of x₀ and t₀ the solution is now unique. Further we note that the solut...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• PREFACE
• 1 INTRODUCTION
• 2 LINEAR EQUATIONS
• 3 LINEAR EQUATIONS WITH PERIODIC COEFFICIENTS
• 4 STABILITY
• 5 PLANE AUTONOMOUS SYTEMS
• 6 PERIODIC SOLUTIONS OF PLANE AUTONOMOUS SYSTEMS
• 7 PERTURBATION METHODS FOR PERIODIC SOLUTIONS
• 8 PERTURBATION METHODS FOR FORCED OSCILLATIONS
• 9 AVERAGING METHODS
• 10 ELEMENTARY BIFURCATION THEORY
• 11 HAMILTONIAN SYSTEMS
• ANSWERS TO SELECTED PROBLEMS
• REFERENCES
• INDEX