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An iterated map x_{n+1} = f(x_n) defines a discrete dynamical system where the state updates in steps rather than flowing continuously. The logistic map x_{n+1} = rx_n(1 - x_n) is the simplest example that exhibits the full range of nonlinear phenomena: fixed points, period doubling, chaos, and periodic windows — all in one dimension. Unlike continuous flows, where chaos requires at least three dimensions (by Poincare-Bendixson), discrete maps can be chaotic in just one dimension because the non-crossing constraint doesn't apply.
Continuous flows and discrete maps are the two fundamental types of dynamical systems. You've been studying flows — systems where time is continuous and the state evolves according to differential equations. Iterated maps are the discrete counterpart: the state updates in discrete steps according to x_{n+1} = f(x_n). Maps arise naturally as Poincare sections of continuous flows (sample the flow once per cycle), as models in ecology (non-overlapping generations), and as mathematical laboratories for chaos (because they can exhibit chaos in just one dimension, making visualization and analysis far simpler than 3D flows).
The logistic map x_{n+1} = rx_n(1 - x_n) is the central example. It models population growth with a carrying capacity: x represents the population fraction (between 0 and 1), r is the growth rate, and the (1 - x) factor models crowding. For small r, the population settles to a stable equilibrium — the map has an attracting fixed point. As r increases past 3, this fixed point loses stability and a period-2 cycle appears: the population oscillates between two values every other generation. Further increase yields period-4, period-8, and so on — a cascade of period-doubling bifurcations that accelerates and culminates in chaos at r ≈ 3.57.
The transition from order to chaos in the logistic map is remarkable for its universality and its richness. Beyond the onset of chaos, the bifurcation diagram shows periodic windows — islands of order within the chaos where the system temporarily locks into periodic behavior. The most prominent is the period-3 window near r ≈ 3.83. The theorem "period 3 implies chaos" (Sharkovskii's theorem, generalized) says that a one-dimensional continuous map with a period-3 orbit must have periodic orbits of every integer period — and must also have uncountably many chaotic orbits. The logistic map packs an astonishing amount of mathematical structure into a single one-dimensional equation.
What makes maps different from flows is the absence of the continuity constraint that prevents trajectory crossing. In a one-dimensional flow, a trajectory moving to the right cannot reverse direction without hitting a fixed point (where ẋ = 0). In a map, x_{n+1} can be anywhere — the map can fold the interval back on itself, sending nearby points far apart and distant points close together. This folding is the discrete-time analog of stretching and folding in continuous flows, and it is what makes one-dimensional chaos possible. The logistic map at r = 4 is fully chaotic: its Lyapunov exponent is ln 2, every orbit is either periodic or dense in [0,1], and nearby initial conditions diverge at an exponential rate.