Chapter 10.5, Problem 7E

Interpretation Introduction

Interpretation:

To show that analytically the graph in Figure $\text{10}\text{.5}\text{.2}$ suggest that $\text{λ = 0}$ at each period-doubling bifurcation value is correct.

Concept Introduction:

• ➢ A logistic map can exhibit aperiodic orbits for certain parameter values.

• ➢ Given some initial condition ${\text{x}}_{\text{0}}$, consider the nearby point ${\text{x}}_{\text{0}}{\text{+δ}}_{\text{0}}$, where the initial separation ${\text{δ}}_{\text{0}}$ is extremely small. Let ${\text{δ}}_{\text{n}}$ be the separation after $\text{n}$ iterates. If ${\text{|δ}}_{\text{n}}\left|\approx \right|{\text{δ}}_{\text{0}}{\text{| e}}^{\text{n λ}}$, then $\text{λ}$ is called the Liapunov exponent.

• ➢ Apositive Liapunov exponent is a signature of chaos.