Most of maths asks what is the answer? A dynamical system asks a stranger, more alive question: what happens next — and next, and next, forever? Give it a rule for how a state changes moment to moment, and it tells you the whole future: the swing of a pendulum, the boom and bust of a population, the weather, the rhythm of a heartbeat, the orbit of a moon. It is the mathematics of change that feeds on itself.
And hiding inside those innocent-looking rules is one of the great surprises of the twentieth century: chaos. A system can be perfectly deterministic — no randomness anywhere — and yet be utterly unpredictable, because the tiniest nudge today explodes into a wholly different tomorrow. This course is the story of how simple rules give rise to endless, beautiful complexity.
One picture runs through everything here. Forget solving for a formula; instead, imagine the state of a system as a point, and the rule as a flow that carries that point along. Ask not "what is the value at time t?" but "where does the flow go?" — does it settle to rest, circle forever, or wander unpredictably? This shift from formulas to geometry in phase space is what lets us understand systems we can never solve on paper, and it is the thread from a lazy fixed point all the way to the Lorenz butterfly.
This course climbs in three stages, from calm to chaos.
We begin at the calmest place of all — a system sitting still. But even stillness has a secret: some resting points are stable, and some are traps waiting to fling you away. That difference is where the whole subject is born.