It is very unfortunate that the term “chaos” was chosen for the name of the topic we are going to discuss.  To most people, the word “chaos” means confusion, disorder, or randomness.  However, the term “chaos” (more precisely “mathematical chaos”) has an entirely different meaning to scientists, engineers and mathematicians. 

We will discuss the essential features of mathematical chaos in greater detail later but, in the meantime, you need to understand that mathematical chaos involves neither disorder nor randomness.  It is completely deterministic.  This means that the behavior of any chaotic system is governed by mathematical equations and not by randomness or chance. 

Let me explain what I mean through an experiment which we all can readily visualize.  If I throw a ball, its motion is deterministic in that Newton’s Laws of Motion will determine where that ball will land once I release it. 

If I throw the ball repeatedly and try my best to throw it in exactly the same way each time, the ball will land in approximately the same location.  The slight variations in where the ball lands will be due to the fact that there were slight differences in the angle of my arm and the velocity of the ball when I released it.  The velocity and the angle at which I release the ball are called the “initial conditions” for this event.

However, it turns out that there are numerous things in nature that, no matter how carefully I try to repeat the experiment exactly, the final results each time will be vastly different from what they were previously.  This is due to the fact that I cannot repeat the initial conditions with infinite precision.  

This phenomenon is called “extreme sensitivity to initial conditions” and is characteristic of all examples of mathematical chaos.  We will talk more about this “extreme sensitivity” in our subsequent discussions.

From now on, whenever I use the word “chaos,” please know that I mean “mathematical chaos.”



Investigations have shown that chaos can be observed in virtually all aspects of life including biology, geology, chemistry, finance, psychology, medicine, all branches of engineering and physics, economics, and countless other areas. 

The list is endless and the examples are fascinating.  The frequent occurrence of chaos is due to the fact that our world is fundamentally nonlinear. Many nonlinear systems give rise to chaos.

For just one example of the current widespread interest in chaos, do an internet search on the topic “chaos in biology” and you will find many thousands of websites with information on that subject.  The study of chaos is helping us to understand many things that previously were unknown or were inexplicable.