Rules for Change

At school, a differential equation is something to be solved. You separate the variables, or find an integrating factor, or guess a solution of the form \(e^{\lambda x}\), and at the end you have a formula. The equations in textbooks are chosen because this works. Unfortunately, almost every differential equation that comes from the real world (the motion of three planets, the weather, a population of predators and prey, a swinging pendulum with a large amplitude) has no solution that can be written down with the functions we know.

This chapter is about what to do instead. Rather than asking “what is the formula?”, it asks questions that can be answered without one. Is there a solution at all, and is there only one? Does it stay bounded? Does it settle down to a steady state, or to a repeating cycle, or does it wander for ever? What does it do in the long run? This change of viewpoint, from solving to understanding, was begun by Henri Poincaré in the 1880s, and it created the subject now called dynamical systems.

Why differential equations matter

Differential equations are the language of science. Newton’s second law is a differential equation; so are Maxwell’s equations of electromagnetism, the Schrödinger equation of quantum mechanics, and the equations of fluid flow used to forecast the weather and design aircraft. Epidemiologists model the spread of a disease with them, ecologists the populations of fish stocks, economists the growth of an economy, and engineers the response of every control system from a thermostat to an autopilot. Whenever a quantity changes at a rate that depends on its current state, there is a differential equation, and usually one that nobody can solve exactly.

Alongside these continuous models are discrete ones, where time moves in steps: the population this year determines the population next year. These iterated maps are simpler to write down, but the last two sections show that they can behave just as wildly. One of the great discoveries of the 1970s was that very simple rules can produce chaos: behaviour that is completely determined, yet in practice unpredictable.

What you will meet at university

A first-year course on differential equations covers the standard methods of solution, but it also introduces the questions of this chapter. You will meet existence and uniqueness theorems, which tell you when a solution exists and when it is determined by its starting value. You will learn to sketch phase portraits, pictures of all solutions at once, and to classify equilibria as stable or unstable by linearising. Later courses go much further: nonlinear dynamics studies limit cycles, bifurcations and chaos; mathematical biology applies them to populations and epidemics; and the theory of partial differential equations, Fourier series and orthogonal functions (glimpsed in Section 8.6) underpins quantum mechanics and much of applied mathematics.

The questions that run through this chapter

The chapter begins with uniqueness and what happens when it fails (Section 8.1), then develops tools for understanding equations without solving them: energy-like quantities that cannot increase, equilibria and their stability, and coordinates matched to the symmetry of the problem. Two sections apply these ideas to physical problems: dogs chasing each other in a square, and the quantum harmonic oscillator. The chapter ends with discrete maps, where a formula reveals chaos, and where a single cycle of length three forces cycles of every length.

Ideas in this chapter

How to read this chapter

Before solving anything, sketch. For a differential equation \(y' = F(y)\), mark where \(F\) is positive, negative and zero, and draw arrows on the \(y\)-axis; for a map \(x \mapsto \mathrm{f}(x)\), draw its graph with the line \(y = x\) and trace a few sequences by the cobweb method. These pictures take a minute, and they will usually tell you what the answer should look like before the algebra begins.