This bifurcation diagram reveals the route to chaos in the logistic map. Each vertical slice represents a specific growth rate r, with dots showing the long-term population values.

• Stable region (r < 3): Single dots show the population converges to one fixed value
• Period-doubling cascade (3 < r < 3.57): The number of stable points doubles repeatedly: 2, 4, 8, 16...
• Chaotic region (r > 3.57): Dense vertical bands indicate the population never settles, bouncing unpredictably
• Notice the narrow white "windows" in the chaotic region—brief moments of order amid chaos