One can view dynamics as the study of iteration processes. Iteration involves
taking the output of a function and feeding it back as input. For example,
suppose that you input a number and repeatedly press the cosine button on a
pocket calculator. The number starting with the digits 0.739 eventually appears
and remains on the display. This is explained by the technique of graphical
analysis discussed in Chapter 1. Iteration processes are at the heart of many
algorithms. They are used to numerically approximate solutions to ordinary
and partial differential equations, and to numerically solve linear and
nonlinear systems of equations.
Each iteration procedure generates a discrete dynamical system. The word
"discrete" refers to fixing a time step and describing the state of a physical
system at discrete instants of time that are integer multiples of the time step.
However, the phase space or state space of the system is usually a connected
subset of a Euclidean space. Mathematical models of continuously evolving
physical systems can also be viewed as discrete dynamical systems by fixing a
unit of time.