# Exercises for Chaos Under Control

## Chapter 5: The Logistic Map

1.[N] Consider the logistic map with s = 4, xn+1 = 4*xn*(1 - xn). This system is deterministic -- if we know xn exactly, then we can find xn+1 exactly. Starting with some x0 between 0 and 1 (and not including 1/2), use a calculator to generate x1, ..., x10, and record your answers to the fourth decimal place. Now change x0 in the third decimal place, generate a new sequence x1, ..., x10, and record your answers to the fourth decimal place. Comparing the last term of the new sequence with the last term of the old sequence, comment on the accuracy with which the initial value must be known to make an accurate prediction of x10.

2.[N] For the logistic map with s = 4, start with x0 = 0.0001 and use a calculator to generate x1, ..., x10. For the tent map with s = 2, start with x0 = 0.0001 and use your calculator to generate x1, ..., x10. Compare these answers. What can you conclude by comparing these sequences?

3.[C] For s = 0.999, both the tent map and the logistic map have attracting fixed points at x = 0, while for s > 1, neither do. For a starting value of x0 = 0.5, compare the number of iterations needed to converge for these two maps using TENT/HISTOGRAM and LOGISTIC/HISTOGRAM. Why do think there is such a difference?