Flows and Maps
The two ways time can pass — continuously, as a flow of differential equations, or in discrete steps, as an iterated map.
Flows and Maps
Dynamical systems come in two flavors, set by how time advances. In a flow, time is continuous and the rule is a differential equation, \dot x = f(x); the state slides smoothly along a trajectory. In a map, time ticks in discrete steps and the rule is an iteration, x_{n+1} = F(x_n); the state hops from one value to the next. Both describe a Dynamical System evolving in State Space — they are two dialects of the same language.
Flows: continuous time
A flow integrates a velocity field. Drop a point anywhere and it follows the arrows of the Phase Portrait, tracing a smooth curve that never crosses itself. Planetary orbits, the swing of a Pendulum, the convection of the Lorenz System — all are flows.
Maps: discrete time
A map applies its rule over and over. Models that are naturally step-by-step — a population censused once a year, the Logistic Map, the return of an orbit — live here. Maps are easy to iterate and can be wildly complex in one dimension, where a flow would need at least three.
The bridge: Poincaré sections
The two pictures are deeply connected. Take a flow and record the state only each time it pierces a chosen surface, and you have reduced it to a map — a Poincaré Section. This trades a continuous trajectory for a sequence of points, dropping the dimension by one while preserving the essential dynamics. It is the standard way to find the Limit Cycles and Strange Attractors hidden inside a flow.