the flow

Picture a river whose current is fixed in place: at every spot the water moves in a definite direction at a definite speed, and that pattern never changes with time. Drop a leaf anywhere and it traces out a path; drop a thousand leaves and the whole sheet of water slides downstream as one motion. The flow is the mathematical name for that single, total motion — the rule that takes any starting point and tells you where it has drifted to after any amount of time.

Precisely, for an autonomous system x' = f(x), the flow is a function usually written phi_t(x): give it a starting state x and an elapsed time t, and it returns the state the solution has reached, phi_t(x) = x(t) with x(0) = x. Two properties make it a flow and not just a list of solutions. First, phi_0(x) = x: zero time means you have not moved. Second, the semigroup law phi_(t+s) = phi_t composed with phi_s: flowing for time s and then for time t lands you exactly where flowing for time t+s would — time can be stitched together. So instead of solving the equation one initial condition at a time, the flow describes how the entire phase space is carried along all at once, like the river moving every leaf together.

This shift in viewpoint is the heart of dynamical systems. Once you have the flow you stop asking 'what is the formula for this one solution?' and start asking geometric questions: which sets are dragged into themselves, which points stay fixed, where do nearby starts end up far apart? The flow only makes clean sense for autonomous equations, where f does not depend on t — that time-independence is exactly what lets the same map phi_t work no matter when you start, and it is why the semigroup law holds.