Chapter 3 Introduction to the Mathematics of Change

The simplest non-trivial iterative change process can be described by the following difference equation:

\[ Y_{i+1} = a*Y_i \]

The equation describes the way in which the value of \(Y\) changes between two adjacent, discrete moments in time (hence the term difference equation, or recurrence relation). There are two parameters resembling an intercept and a slope:

  1. The starting value \(Y_0\) at \(i=0\), also called the starting value, or the initial conditions.
  2. A rule for incrementing time, here the change in \(Y\) takes place over a discrete time step of 1: \(i+1\).

The values taken on by variable \(Y\) are considered to represent the states quantifiable observable alternative ways to describe the change of states :

  • A dynamical rule describing the propagation of the states of a system observable measured by the values of variable Y through discrete time.
  • A dynamic law describing the time-evolution of the states of a system observable measured by the variable Y.

These descriptions all refer to the change processes that govern system observables (properties of dynamical systems that can be observed through measurement).