MM504 Lecture 33b: Chaos and order, Self-symmetry, Mandelbrot fractals

There is a mistake in the first graph. It should have been between r vs saturated value of population Xn.
Chaos theory is the study of complex and unpredictable systems that are highly sensitive to initial conditions. Chaos theory has applications in various fields, including physics, mathematics, economics, and biology. Mandelbrot fractals are a type of complex, self-similar geometric pattern that is generated through iterations of a simple mathematical equation. Mandelbrot fractals exhibit intricate and often beautiful patterns, which have made them a popular subject in art, computer graphics, and popular culture. In population dynamics, growth rate versus population random behavior refers to the tendency of populations to experience unpredictable fluctuations in growth rate due to factors such as environmental changes, competition, and random events. This can lead to population booms and crashes, which have significant ecological and economic impacts. The study of chaotic and random behavior has important practical implications, including the prediction and management of complex systems and the development of new technologies and models.