Today I have more notes and insights from the Chaos Cookbook, Chapter 2. I'm trying to understand what characteristics are common to all chaotic systems and what it means for these systems to be chaotic. I'm also trying to find what links the ideas chaos, fractals, emergence, and complexity together. Chapter 2 deals with a specific type of chaotic entity: an iterated function.
"For a function to exhibit chaotic behavior, the function used has to be nonlinear." However, a nonlinear equation does not necessarily display chaotic behavior (example: y = x**2).
An iterated function is one whose value depends on the prior iteration. This is a form of mathematical feedback. Feedback is common in all chaotic systems.
There are two important parameters that describe an iterated function: the initial value and the number of iterations. Chaotic functions may be very sensitive to the initial value, meaning that very small changes to it will produce very drastic differences in the outcome of the iteration process.
Despite the sensitivity of the values of an iterated function to the initial value used, each graph may display common features.
The sequence of numbers obtained by iterating over a function is known as an orbit. Orbits may be stable, unstable, or chaotic.
A "lab notebook" about my life in science, programming, and philosophy.
