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The Logistic Map

Time Evolution

Bifurcation Diagram

Controls & Parameters

3.500
Current value: x = 0.5000

Key Values

  • r < 3.0: Stable point
  • r ≈ 3.0: First bifurcation
  • r ≈ 3.45: Second bifurcation
  • r ≈ 3.54: Chaos begins
  • r = 4.0: Full chaos

Understanding the Logistic Map

The Equation

xn+1 = rxn(1 - xn)

Where r is the growth rate and x represents the population (between 0 and 1)

Key Values of r

  • r < 3.0: Population stabilizes
  • r ≈ 3.0: First period-doubling
  • r ≈ 3.45: Second period-doubling
  • r ≈ 3.54: Period-doubling cascade
  • r > 3.57: Chaos begins
  • r = 4.0: Full chaos

Observations

  • System becomes chaotic through period doubling
  • Windows of stability exist within chaos
  • Behavior is deterministic but unpredictable
  • Small changes in r lead to dramatic changes in behavior