The logistic map is a classic example of how simple nonlinear dynamical equations can produce complex, chaotic behavior. It is defined by the recurrence relation: \[ x_{n+1} = r x_n (1 - x_n) \] where \(x_n\) is a number between 0 and 1, and \(r\) is a positive constant representing the growth rate. Behavior Across Parameter Values Depending on the value of \(r\), the logistic map exhibits very different behaviors:...