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Period Doubling Bifurcation

Journey into Chaos: From order to chaos through period doubling

Chaos Controls

System Parameters

3.0000
Stable (1.0) Period 2 (3.0) Chaos (4.0)
0.5000
Normal

System Selection

Visualization Options

Chaos Theory

Period Doubling

Route to Chaos

r ≈ 3.0

At r = 3.0, the fixed point loses stability and a period-2 orbit emerges. This is the first period doubling bifurcation.

Feigenbaum Constant

δ ≈ 4.669

Universal

The ratio between successive bifurcation intervals approaches a constant value, discovered by Mitchell Feigenbaum.

Logistic Map

Simple Chaos

xₙ₊₁ = r·xₙ(1-xₙ)

A simple quadratic recurrence that exhibits complex behavior including period doubling bifurcations and chaos.

Bifurcation Diagram

Order → Chaos

Fractal

The diagram shows how the system's stable states evolve as the parameter r increases, revealing the path to chaos.

Cobweb Diagram

Iteration: 0
Current Value
0.5000
Period Detected
Fixed Point

Bifurcation Diagram

Evolution of stable states as r increases from 2.5 to 4.0

Current r value
Period 2 Region
Feigenbaum δ: 4.669

Time Series

Period length: 1 (Fixed Point)

Chaos Indicator

Stable (λ = -0.50)

Mathematical Analysis

Logistic Map

xₙ₊₁ = r·xₙ(1-xₙ)

Where:

  • r ∈ [0, 4] is the growth parameter
  • xₙ ∈ [0, 1] is the population at step n
  • Fixed points satisfy x = r·x(1-x)

Stability Analysis

Fixed points:

x* = 0 and x* = 1 - 1/r

Derivative at x* = 1 - 1/r:

f'(x*) = 2 - r

Stability condition: |f'(x*)| < 1

Period doubling at r = 3 (f'(x*) = -1)