In Physics, “simple systems” usually have only a few moving parts and usually exhibit “linear” dynamics; that is to say that the behaviour of these simple systems usually follow very simple mathematical rules.

Most systems in the real world however are not simple, most system have many moving parts and are consequently quite complicated; so how do we come up with mathematical rules to deal with these types of systems?

Well generally speaking (when dealing with system of a large number of parts) what we tend to do is to compress the behaviour of all the individual things into a single “average” behaviour which usually exhibits linear dynamics. This is what I mean by “compressible systems”; these systems are mathematically compressible into neat differential equations, that describe simple linear dynamics.

This website is primarily about “Incompressible Systems” and their “Nonlinear Dynamics”, however before I move on to the real meat of this subject material; I should first discuss the core mathematics of linear dynamics. To that end this section covers the following topics…

- How
*“Linearization”*and*“Linear Dynamics”*have, to-date, been the cornerstone of all of mathematical science:*Linearization* - How the mathematics defines smooth
*“Continuous Change”*using an infinitesimally small*“Unit of Change”*:*Calculus* - How to define a mathematical
*“Step-Size of Growth and Decay”*using an infinitesimally small*“Unit of Linear Change”*:*What is “e” ?* - How to define a mathematical
*“Step-Size of**Oscillation”*using an infinitesimally small*“Unit of Angular Change”*:*Euler’s Identity*