System

A collection of interrelated things...

Dynamical System

At the left is a picture of the physical view of a system. This system is composed of subsystems like the pendulum, weights, gears, face and hands. Each subsystem may itself be considered a system. For now let's focus on the pendulum system consisting of the pendulum weight, the rods attached to the weight and the pivot from which the rods are suspended.
As long as the pendulum is stopped, hanging straight down, the pendulum system is a static system and not very interesting. When we set the pendulum to swinging the system becomes a dynamical system. Why dynamical rather than dynamic? Nobody seems to know. Wikipedia intercepts any search for dynamic system and redirects you to dynamical system.

Dynamical systems include a fixed rule in mathematical terms that describes how the system changes as time passes. The rule for a pendulum may be stated in this form:

θ'' = -g/L*sin(x)-μ*θ'

This may be read, "The angular acceleration equals minus the acceleration of gravity over the pendulum length times the sine of the angle of the pendulum rod measured from the vertical minus the acceleration due to friction". The time dependence is burried in the θ'' and θ' terms. Don't worry about the what all this means or how to use this rule. We use these sorts of rules to identify systems and the computer uses the rules to create the images you will see.

Sometimes the statement of the rule is itself called the "dynamical system ", bypassing the physical view of the system altogether. Many dynamical systems, like electrons flowing in an electronic circuit for example, are not visible anyway.

What all dynamical systems have in common is that the state of the system changes as time passes. The fundamental things that change with time in a dynamical system are called state variables. For a pendulum the state is defined by the position and speed of the swinging weight. The position is measured by the angle the pendulum rod makes with the vertical. The speed is measured by the rate of change of the position with respect to time... how many degrees per second for example. Once the state variables are known at any time, they may be calculated by a process called integration for any other time through the mathematical rule mentioned above. Quite often we call a known set of state variables the initial conditions for a system.

Mathematical System

Now fasten your seat belt. We are coming to the bumpy part of this system discussion. Dynamical systems are included in a larger set of systems I will call mathematical systems. For our purposes mathematical systems are sets of rules, written in terms of
system variables, that define the
behavior
of the system. For thoses mathematical systems that are not dynamical systems, we follow the system development through a process called
iteration
One example of a mathematical system is x←x*(1-x). This is read, "multiply the system variable x by the quantity (1-x) and replace x with the result, repeating until a specified condition is met". Again don't worry about having to actually do this. The computer takes care of this in generating the images.
Mathematical systems are written in terms of system variables and other terms called parameters. In our pendulum system example, the symbols g and L are explicit parameters. Time, usually symbolized by t, is an implicit parameter, hidden as I said, in the θ'' and θ' terms. If mathematical systems include time explicitly or implicitly as a parameter they fall in the realm of dynamical systems. Mathematical systems may reflect a physical reality or may exist only as mathematical entities. It may be a stretch to include systems other than dynamical in "Nature's Hidden Art". I hold the view that mathematics is just another aspect of nature, subject to exploration as much as are the hidden regions of the physical world.

Many of the most interesting mathematical systems involve mathematical entities called complex numbers. Most of us are familiar with the concept of number as a way to enumerate objects. Beginning with the counting numbers, 1,2,3...up to very many. Mathematicians have tacked on fractions, negative numbers, zero, rational numbers, irrational numbers, real numbers, imaginary numbers, complex numbers and the list goes on. What mathematicians do for a living is to continue defining larger and more complicated objects together with rules for manipulating them, similar to the rules for adding and multiplying counting numbers.

Complex numbers have their own rules for addition and mutiplication allowing us to write expression like z←z*z+c defining a mathematical system, where z and c are complex numbers. This may be read, "multiply the complex number z by itself, add another complex number c and replace z with the result, repeating until a specified condition is reached". When this mathematical system is illuminated by the escape time algorithm one of the most complex objects in nature, the Mandelbrot set boundary, becomes visible.