Nonlinear Dynamics & ChaosMaps & BifurcationsLesson 2.1
Beginner lesson

The Logistic Map

Iterate xₙ₊₁ = rxₙ(1−xₙ), build a cobweb plot, and watch fixed points become cycles and chaos.

10 min Live experiment Knowledge check
Lesson statusNot completed
Learning objectives

By the end of this lesson, you should be able to:

  • Iterate the logistic map xₙ₊₁=rxₙ(1−xₙ).
  • Use a cobweb diagram to visualize iteration.
  • Relate parameter r to long-term behavior.
Prerequisites: Fixed Points and Stability
EXPERIMENT Interactive dynamics laboratory
Perturb → Evolve → Compare
Change one parameter at a time. Predict the qualitative behavior before running or moving the control.

A simple nonlinear map

The logistic map is \(x_{n+1}=r x_n(1-x_n)\), usually with \(0\le x_n\le1\). Despite its simplicity, changing \(r\) produces a remarkable range of long-term behaviors.

Reading a cobweb plot

The curve \(y=f(x)\) gives the next state, while the diagonal \(y=x\) carries that output forward as the next input. Repeating vertical and horizontal steps draws the iteration geometrically.

Parameter-dependent behavior

As \(r\) increases, the system moves through stable fixed points, periodic cycles, period doubling, and chaotic regimes.

Key takeaways

What should stay with you

  • A simple nonlinear map can generate fixed points, cycles, and chaos.
  • Cobweb plots make iteration visible.
  • The control parameter r changes the qualitative regime of the system.
Knowledge check

In a cobweb plot, why do we alternate between the curve y=f(x) and the diagonal y=x?