# 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:

- The starting value \(Y_0\) at \(i=0\), also called the
*starting value*, or the*initial conditions*. - 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).