Recasting nonlinear differential equations as S-systems: a canonical nonlinear form

An enormous variety of nonlinear differential equations and functions have been recast exactly in the canonical form called an S-system. This is a system of nonlinear ordinary differential equations, each with the same structure: the change in a variable is equal to a difference of products of power-law functions. We review the development of S-systems, prove that the minimum for the range of equations that can be recast as S-systems consists of all equations composed of elementary functions and nested elementary functions of elementary functions, give a detailed example of the recasting process, and discuss the theoretical and practical implications. Among the latter is the ability to solve numerically nonlinear ordinary differential equations in their S-system form significantly faster than in their original form through utilization of a specially designed algorithm.