🦋 Logistic Map - Chaos Theory

Demonstrating Sensitive Dependence on Initial Conditions

xn+1 = r × xn × (1 - xn)
Time Series
Bifurcation Diagram
Cobweb Plot

Time Series - Sensitive Dependence

xâ‚€ = 0.5
x₀ + δ = 0.501
x₀ - δ = 0.499

Bifurcation Diagram

Shows how the system behavior changes with growth rate r

Cobweb Plot

Visualizes the iteration process

What is Sensitive Dependence on Initial Conditions?

The Butterfly Effect: Tiny differences in starting conditions lead to dramatically different outcomes. In the logistic map, changing xâ‚€ by just 0.001 can result in completely different trajectories after a few iterations.

The Logistic Map: Originally used to model population growth, this simple equation exhibits complex behavior:

Try this: Set r = 3.9 and watch how three nearly identical starting points diverge wildly!