Consider the logistic equation (15.35) for the special case a = 1, which is large enough to ensure that chaos has set in.

(a) Make the substitution x_n = sin²πθ_n, and show that the logistic equation can be expressed in the form θ_n+1= 2θ_n (mod 1); that is, θ_n+1 equals the fractional part of 2θ_n.

(b) Write θ_n as a “binimal” (binary decimal). For example, 11/16 = 1/2 + 0/4 + 1/8 + 1/16 has the binary decimal form 0.1011. Explain what happens to this number in each successive iteration.

(c) Now suppose that an error is made in the ith digit of the starting binimal. When will it cause a major error in the predicted value of x_n?

(d) If the error after n iterations is written ∈_n, show that the Lyapunov exponent p defined by

is ln 2 (so ∈_n ≈ 2ⁿ∈₀ for large enough n). Lyapunov exponents play an important role in the theory of dynamical systems.

Equation (15.35)

P = lim In n n - En 0 (15.38)