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xn+1=rxn(1−xn)x_{n+1} = r x_n (1 - x_n)xn+1​=rxn​(1−xn​)

xn+1x_{n+1}xn+1​

,

rrr

,

xnx_nxn​

,

1−xn1 - x_n1−xn​

Logistic Map

Click on formula components below to explore their properties

Full Formula Properties

Category: Chaos Theory

👶

Baby Fast Definition

Imagine rabbits in a field: if they breed too fast the next year’s count becomes totally unpredictable even though the rule is simple.

This rule captures how a population breeds, competes for food, then rests. Despite its simplicity, nudging the growth rate rrr just a little can flip predictable cycles into endless, never-repeating chaos.

Role:

A simple rule that shows how order can turn into chaos

Domain:

Input: current population fraction xn∈[0,1]x_n \in [0,1]xn​∈[0,1]; Output: next population fraction xn+1∈[0,1]x_{n+1} \in [0,1]xn+1​∈[0,1]

Binding:

The growth rate rrr ties this generation’s population to the next

Variance:

Tiny changes in xnx_nxn​ or rrr explode into wildly different futures

Geometric:

Points bounce along a parabolic tent–shaped curve

Invariant:

The interval [0,1][0,1][0,1] stays bounded forever

Limits:

For r<3r < 3r<3 the orbit settles; near r≈3.57r \approx 3.57r≈3.57 it becomes chaotic

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