We began by saying that most differential equations cannot be solved, and that the art is to understand them anyway. We asked three questions: does the present determine the future, what can we know without solving, and what happens in the long run? Here is what the chapter’s eight questions showed.
1. Does the present determine the future?
Usually, but not always. In Section 8.1, the equation \((y')^2 = 4y\) had infinitely many solutions through every point of the \(x\)-axis: a solution could rest at \(y = 0\) for as long as it liked and then leave. The culprit was the square root, whose infinite slope at zero breaks the Lipschitz condition of the Picard–Lindelöf theorem. The same mechanism explains why a leaking bucket cannot tell you when it emptied, and it lies behind Norton’s dome, a puzzle about determinism in classical mechanics.
Even when the future is determined, it may not be predictable. The logistic map of Section 8.7 has an exact formula, \(u_n = \sin^2(2^n\theta)\), and yet every step doubles any error in the starting value. Determinism in principle is not predictability in practice.
2. What can we know without solving?
A great deal.
Energy. In Section 8.2, multiplying the equation by \(y'\) produced a quantity that could not increase, and that bounded the solution of an equation nobody can solve. Such quantities are called Lyapunov functions, and they are the main tool for proving stability.
Equilibria and stability. In Section 8.3, the sign of the growth rate predicted the fate of a fish population, and showed that harvesting beyond a threshold causes sudden collapse.
The right coordinates. In Section 8.4, polar coordinates turned a coupled system into a single equation for \(r\), showing every solution spiralling onto a circle. In Section 8.5, symmetry reduced four chasing dogs to one, and dividing to eliminate time gave the shape of their paths.
Structure from an identity. In Section 8.6, the Wronskian showed that the solutions of an equation are orthogonal, without integrating a single one of them.
Counting and covering. In Section 8.8, counting the pieces of a graph found every cycle, and the intermediate value theorem proved the existence of cycles of every length.
3. What happens in the long run?
The chapter met every kind of long-term behaviour. Solutions settled to stable equilibria (Section 8.3), approached a repeating cycle (Section 8.4), spiralled into a point in finite time (Section 8.5), or behaved chaotically (Section 8.7). For the logistic map, a single parameter moves the system through all of these: a fixed point, a cascade of period doublings governed by Feigenbaum’s universal constant, and finally chaos, with windows of order such as the \(3\)-cycle. And Section 8.8 showed that once a \(3\)-cycle appears, cycles of every length must exist, the first step in Sharkovskii’s ordering of the integers.
4. A toolkit for differential equations and dynamics
| Method | Where you used it |
|---|---|
| Looking for singular solutions | Non-uniqueness (8.1) |
| A quantity that cannot increase | Bounding (8.2) |
| Sign of the growth rate | Harvesting (8.3) |
| Coordinates that match the symmetry | Limit cycle (8.4); pursuit (8.5) |
| Substitution to linearise | Bernoulli equation (8.4) |
| The Wronskian | Orthogonality (8.6) |
| Conjugacy | Logistic map (8.7) |
| Induction | \(u_n = \sin^2(2^n\theta)\) (8.7) |
| Counting pieces of a graph | Cycles of \(\mathrm{f}^n\) (8.8) |
| Intermediate value theorem | Period three (8.8) |
Where next
Energy and conservation laws return in mechanics, starting with Section 10.1.
Linear differential equations with several unknowns, solved by eigenvectors, govern vibrating systems in Section 11.2; the nonlinear pendulum, which cannot be solved in elementary functions, is in Section 11.3.
Iterating a map to find its fixed points decides whether a family name dies out in Section 12.15, and long-run behaviour of random sequences is the subject of Section 12.16.
Steven Strogatz, Nonlinear Dynamics and Chaos: the best first book on dynamical systems, full of applications and pictures. Strogatz’s recorded lectures based on it are free online.
James Gleick, Chaos: Making a New Science: the story of how chaos was discovered.
George Simmons, Differential Equations with Applications and Historical Notes: a classic first course, rich in history.
Robert Devaney, A First Course in Chaotic Dynamical Systems: iterated maps, with proofs, at first-year level.
Laplace imagined that, knowing the present exactly, one could compute the whole future. This chapter has shown where that dream stops. Uniqueness can fail; and where it holds, chaos can amplify the smallest uncertainty until prediction is impossible. But it has also shown that a great deal can be understood without a formula: bounds, equilibria, cycles and their stability, and the structure of chaos itself. Moving from “solve the equation” to “understand the solutions” is one of the most important shifts in thinking that a university course in mathematics asks of you.