Questions: Iterated Maps and the Logistic Map

Question 1

The logistic map x_{n+1} = rx_n(1 - x_n) has a fixed point at x* = 1 - 1/r for r > 1. This fixed point is stable when |f'(x*)| < 1. For what range of r is it stable?

• A. 1 < r < 2
• B. 1 < r < 3
• C. All r > 1
• D. r > 3

Answer: (B)

f'(x) = r(1 - 2x). At x* = 1 - 1/r, f'(x*) = r(1 - 2(1 - 1/r)) = r(2/r - 1) = 2 - r. Stability requires |2 - r| < 1, which gives 1 < r < 3. At r = 3, the fixed point loses stability (|f'(x*)| = 1) and a period-2 cycle is born via a period-doubling bifurcation. Below r = 1, the fixed point at 0 is the only attractor.

Question 2

Why can a one-dimensional map like the logistic map exhibit chaos, even though a one-dimensional continuous ODE ẋ = f(x) cannot?

• A. One-dimensional maps can't actually be chaotic — only higher-dimensional maps can
• B. The discrete time step allows the map to 'jump' over barriers that would block a continuous trajectory. In a continuous 1D flow, a trajectory moving right can't reverse without passing through a fixed point. In a discrete map, x_{n+1} can be anywhere the map sends it — there's no continuity constraint between successive states that prevents folding.
• C. One-dimensional ODEs can also be chaotic; the claim that they can't is incorrect
• D. The logistic map is actually a hidden two-dimensional system

Answer: (B)

In a continuous 1D flow, the uniqueness theorem and the intermediate value theorem trap trajectories — they can't cross fixed points, so they must monotonically approach a fixed point or diverge. A discrete map has no such constraint: f can fold the interval back on itself (like x → 4x(1-x), which maps [0,1] → [0,1] non-monotonically). This folding is the mechanism that creates chaos. The map stretches (the slope |f'| > 1 in some regions) and folds (the interval gets mapped non-monotonically), producing sensitive dependence and mixing in just one dimension.

Question 3

At r = 4, the logistic map x_{n+1} = 4x(1-x) maps [0,1] onto [0,1] and is fully chaotic. The trajectory of almost every initial condition is dense in [0,1].

• T. True
• F. False

Answer: True

At r = 4, the logistic map is conjugate to the tent map and to the doubling map, both known to be fully chaotic on [0,1]. The Lyapunov exponent is λ = ln 2 ≈ 0.693 > 0. Almost every orbit (in the measure-theoretic sense) is dense in [0,1] — it visits every subinterval, no matter how small, given enough iterations. The invariant measure is the arcsine distribution ρ(x) = 1/(π√(x(1-x))), which concentrates near 0 and 1 where the map spends more time.

Question 4

What is a cobweb diagram and how does it help visualize the dynamics of an iterated map?

Model answer: A cobweb diagram plots y = f(x) and y = x on the same axes. Starting from x₀ on the x-axis, draw vertically to the curve y = f(x₀), then horizontally to the line y = x (which converts the output back to an input). Repeat. The resulting staircase or spiral pattern shows whether the iteration converges to a fixed point (spiral inward), diverges (spiral outward), or exhibits more complex behavior. Fixed points occur where y = f(x) crosses y = x. The cobweb makes stability visually apparent: if |f'(x*)| < 1, the cobweb spirals in; if |f'(x*)| > 1, it spirals out.

For the logistic map with r = 2.5, the cobweb spirals into the stable fixed point x* = 0.6. For r = 3.2, it oscillates between two values (period-2 orbit). For r = 4, the cobweb bounces chaotically throughout [0,1]. The cobweb diagram is the discrete-time analog of a phase portrait — it shows the complete qualitative dynamics at a glance.