Contents of "From Calculus to Chaos"





From Calculus to Chaos

An introduction to dynamics




Contents





1. Introduction

1.1 The beginning of dynamics
1.2 From calculus to chaos

2. A brief review of calculus

2.1 Introduction
2.2 Some elementary results
2.3 Taylor series
2.4 The functions exp x and log x
Exercises

3. Ordinary differential equations

3.1 Introduction
3.2 First-order linear equations
3.3 First-order nonlinear equations
3.4 Second-order linear equations
3.5 Second-order nonlinear equations
3.6 Phase space
Exercises

4. Computer solution methods

4.1 Introduction
4.2 Euler's method
4.3 Computer implementation of Euler's method
4.4 Systems of differential equations
4.5 More accurate step-by-step methods
Exercises

5. Elementary oscillations

5.1 Introduction
5.2 The linear oscillator
5.3 Multiple modes of oscillation
5.4 Coupled oscillators
5.5 Nonlinear oscillations
Exercises

6. Planetary motion

6.1 Introduction
6.2 Equations of motion under a central force
6.3 The equal-area rule
6.4 Differential equation for the orbit
6.5 Orbits under an inverse-square law
6.6 A numerical approach
6.7 The two-body problem
6.8 The three-body problem
Exercises

7. Waves and diffusion

7.1 Introduction
7.2 Wave motion
7.3 The diffusion equation
Exercises

8. The best of all possible worlds?

8.1 Introduction
8.2 The concept of 'action'
8.3 The calculus of variations
8.4 Lagrange's equations of motion
Exercises

9. Fluid flow

9.1 Introduction
9.2 The geometry of fluid motion
9.3 The equations of viscous flow
9.4 Very viscous flow
9.5 The case of small viscosity
Exercises

10. Instability and catastrophe

10.1 Introduction
10.2 Linear stability theory
10.3 Multiple solutions; bifurcation
10.4 Sudden changes of state
10.5 Imperfection and catastrophe
10.6 Instability of motion
Exercises

11. Nonlinear oscillations and chaos

11.1 Introduction
11.2 Limit cycles; the van der Pol equation
11.3 Conditions for chaos
11.4 The Lorenz equations
11.5 Chaotic mixing: stretch and fold
11.6 One route to chaos: period-doubling
11.7 Multiple solutions and 'jumps'
Exercises

12. The not-so-simple pendulum

12.1 Introduction
12.2 Pendulums from the past
12.3 A vibrated pendulum
12.4 Chaotic pendulums
12.5 Not quite the Indian Rope Trick
Exercises

Further reading

Appendix A: Elementary programming in QBasic

A.1 Introduction
A.2 Getting started
A.3 Mathematical variables, operations and functions
A.4 Program loops
A.5 Graphics

Appendix B: Ten programs for exploring dynamics

Solutions to the exercises

Index



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