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Physics · L5 · Period-Doubling Route to Chaos

How the logistic map's long-term behavior evolves as the growth-rate parameter r increases from a stable fixed point through successive period-doublings into chaos.

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r < 1Population dies out (x → 0)1 < r < 3Single stable fixed point3 < r < 3.449Period-2 cycle3.449 < r < 3.544Period-4 cycle🔁 Cascade: period 8, 16, 32, ...(intervals shrink by Feigenbaum δ ≈ 4.669)🦋 r > r∞ ≈ 3.570Chaos: aperiodic, bounded, sensitive to initial conditions🔄 Narrow periodic windows(e.g. stable period-3 near r ≈ 3.83)

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