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Logistic Map — Definition, Formula & Examples

The logistic map is a recurrence relation that takes a number between 0 and 1, applies a simple quadratic rule, and feeds the output back in as the next input. Despite its simplicity, it can produce stable fixed points, periodic cycles, or fully chaotic sequences depending on a single parameter.

The logistic map is the discrete dynamical system defined by xn+1=r xn(1−xn)x_{n+1} = r\,x_n(1 - x_n), where xn∈[0,1]x_n \in [0,1] represents the state at iteration nn and r∈[0,4]r \in [0,4] is a bifurcation parameter. For r<1r < 1 the orbit converges to 0; for 1<r<31 < r < 3 it converges to a nonzero fixed point; for r>3r > 3 a cascade of period-doubling bifurcations occurs, leading to deterministic chaos for most values near r≈3.57r \approx 3.57 and beyond.

Key Formula

xn+1=r xn (1−xn)x_{n+1} = r\,x_n\,(1 - x_n)
Where:
  • xnx_n = State at iteration n, a value in [0, 1]
  • rr = Bifurcation parameter, typically in [0, 4]
  • xn+1x_{n+1} = State at the next iteration

How It Works

Pick a parameter rr and an initial value x0x_0 between 0 and 1. Repeatedly apply the rule xn+1=r xn(1−xn)x_{n+1} = r\,x_n(1-x_n) to generate a sequence. For small rr, the sequence settles to a single value (a fixed point). As you increase rr past 3, the long-term behavior alternates between 2 values, then 4, then 8, and so on — this is period doubling. Beyond approximately r=3.56995r = 3.56995, the orbit becomes chaotic for most values of rr, meaning tiny changes in x0x_0 lead to wildly different sequences. This sensitivity to initial conditions is a hallmark of chaos.

Worked Example

Problem: Let r = 2.5 and x₀ = 0.4. Compute the first three iterates of the logistic map.
Iteration 1: Apply the formula with x₀ = 0.4.
x1=2.5×0.4×(1−0.4)=2.5×0.4×0.6=0.6x_1 = 2.5 \times 0.4 \times (1 - 0.4) = 2.5 \times 0.4 \times 0.6 = 0.6
Iteration 2: Now use x₁ = 0.6.
x2=2.5×0.6×(1−0.6)=2.5×0.6×0.4=0.6x_2 = 2.5 \times 0.6 \times (1 - 0.6) = 2.5 \times 0.6 \times 0.4 = 0.6
Iteration 3: Since x₂ = 0.6, the same calculation repeats.
x3=2.5×0.6×0.4=0.6x_3 = 2.5 \times 0.6 \times 0.4 = 0.6
Answer: The sequence reaches the fixed point x* = 0.6 after just one iteration. This is expected because for r = 2.5 (which lies between 1 and 3), the logistic map converges to the fixed point x* = 1 − 1/r = 1 − 0.4 = 0.6.

Why It Matters

The logistic map is a canonical example in nonlinear dynamics courses, demonstrating how deterministic rules produce unpredictable behavior. It appears in population biology (modeling boom-bust cycles), cryptography (pseudo-random number generation), and serves as the simplest gateway to understanding the Feigenbaum constants and universality in chaotic systems.

Common Mistakes

Mistake: Using an initial value outside [0, 1] or a parameter r > 4 and expecting bounded behavior.
Correction: For r > 4, most orbits escape to negative infinity. The logistic map's rich dynamics occur specifically for r ∈ [0, 4] with x₀ ∈ (0, 1).