Plot Logistic Map and Analyze Chaotic Behavior

In this experiment, we will investigate and demonstrate the plotting and chaotic behavior of the logistic map. The logistic map is a mathematical model used to describe the behavior of nonlinear dynamical systems. We will discuss the concept and underlying mathematical principles of the logistic map in detail, then demonstrate how to implement the iterative calculation algorithm using code. The implementation typically involves initializing a population value and iterating the equation x_n+1 = r * x_n (1 - x_n) for various growth rate parameters (r). We will generate bifurcation diagrams by sweeping r values from 2.5 to 4.0 with step increments, plotting the stable population values after transients die out. The code will highlight chaotic regimes where tiny changes in initial conditions lead to dramatically different outcomes, showcasing sensitivity to initial conditions - a hallmark of chaotic systems. Through this experiment, we will help you better understand the significance of the logistic map and its applications across various scientific domains including population biology, physics, and cryptography.