Explore recursive dynamics and chaos theory using the Logistic Map. Witness how simple rules lead to complex, unpredictable behavior.
Dynamics Config
Iteration Sequence
0.500884Final State
Convergence Point
Calculated Values (xₙ)
n=0
0.5000
n=1
0.8750
n=2
0.3828
n=3
0.8269
n=4
0.5009
n=5
0.8750
n=6
0.3828
n=7
0.8269
n=8
0.5009
n=9
0.8750
n=10
0.3828
n=11
0.8269
n=12
0.5009
n=13
0.8750
n=14
0.3828
n=15
0.8269
n=16
0.5009
n=17
0.8750
n=18
0.3828
n=19
0.8269
n=20
0.5009
BehaviorStable
Initial0.5
About this calculator
Iteration and the Logistic Map
Function iteration involves repeatedly applying the same function to a value. The output of one step becomes the input for the next. The Logistic Map is a classic example of how chaos can arise from simplicity.
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Pro Tips
Stability: For r < 3, the system stabilizes to a fixed point.
Chaos: As r increases beyond 3.57, the system becomes chaotic and unpredictable.
Feedback: Iteration is a form of feedback where nature 'recycles' its state.
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Fun Facts
"The Logistic Map was popularised by Robert May in 1976 as a model for biological populations."
"Chaos theory is the study of systems that are highly sensitive to initial conditions (The 'Butterfly Effect')."
"Function iteration is the core engine behind fractal generators like the Mandelbrot Set."
