An Easy Learning Guide - Chaos Theory
This is a basic learning and revision guide to chaos theory. I’ve tried to make this article easy to follow by using my own learning techniques.
The Meaning of Chaos Theory
- The meaning of the word “chaos" as it is generally used today is: a state of confusion lacking any order.
- The term “chaos theory” used in physics refers to: an apparent lack of order in a system that nevertheless obeys particular laws and rules.
- It is also described as an apparent randomness that results from complex systems and their interactions with other systems.
- This condition (an inherent lack of predictability in some physical systems) was discovered by physicist Henri Poincare in the early twentieth century.
Relevant Words and Their Definitions
- Uncertainty Principle: A statement relating to quantum mechanics that asserts it is impossible to measure two properties of a quantum object (e.g. position/momentum or energy/time) at the same time with infinite precision.
- Self Similarity: Allows molecules, crystals and more to mimic their own shape in the thing they make (e.g. a snowflake).
- Complex Systems: These often look to settle in one specific situation, static (attractor) or dynamic (strange attractor).
- Attractor: Represents a state in a chaotic system that seems to be responsible for helping that system settle down.
- Strange Attractor: Represents a system which runs from event to event without ever settling down.
- Generator: Elements in a system that seem to be responsible for chaotic behaviour in that system.
- The unpredictability of all areas of nature is what chaos theory examines.
- Chaos theory is a branch of mathematics that looks at complex systems whose behaviour is extremely sensitive to minor changes in conditions. Small alterations can give rise to strikingly great consequences.
- Complex systems appear to move through a form of cycle, but these cycles are rarely necessarily duplicated or repeated.
- Although these systems can seem straightforward they are very sensitive to the starting conditions which can lead to seemingly random effects.
- These complex systems have so many elements that move (motions) that computers are required to calculate all the varying possibilities. This is the reason chaos theory did not appear before the second half of the twentieth century.
- An example of a complex system that chaos theory helped comprehend is earth’s weather systems. Although even with the largest computers now available the weather can only be forecast a few days ahead.
- Even if the weather was perfectly measured a small change can make the prediction completely wrong. A butterfly can make enough wind with its wings to change a chaotic system. This chaotic system is sometimes known as the butterfly effect.
- Systems, no matter how complicated they are, rely on an underlying order.
- Very simple or very small systems or events can cause very complex behavioural patterns or occurrences.
- Newton’s law of physics assumes that (at least theoretically) that the more accurate and precise the measurements of any condition then the more accurate and precise the predictions will be of any future or past condition.
- This assumption, in theory, stated that it was possible to make almost perfect predictions about the behaviour of any physical system.
- Physicist Henri Poincare proved mathematically that even if the initial measurements could be a million times more accurate the uncertainty of prediction does not lessen but remained massive.
- When Henri Poincare was working on a problem (@ 1890’s) of interactions between three planets and how they affect each other he considered that since gravitational laws were well known the solution should be straightforward.
- However the results were so unexpected that he gave up his work stating “the results are so bizarre that I cannot bear to contemplate them”.
- The impossibility of being able to absolutely define initial measurements meant that predictability of chaotic complex systems resulted in predictions almost no better than if these predictions had been randomly selected.
The Butterfly Effect
- "Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?" (Edward Norton Lorenz, Theoretical Meteorologist)
- Lorenz quoted in a paper in 1963 an unnamed meteorologist’s assertion that if chaos theory were true then a single flap of a seagull’s wings would be enough to alter the course of all future weather systems on earth.
- Lorenz had studied that idea for his talk in 1972 in which he stated that the flap of a butterfly’s wings affecting weather systems illustrated the impossibility of making precise predictions for any complex system where you cannot measure precisely the effect of all other conditions affecting the system.
- Certain patterns exist within chaos that can be found and therefore analyzed.
- Certain features (generators) of a system seem to be able to create chaotic behaviour.
- Very small differences in a generator can result in very large differences in a system further on in time (the butterfly effect).
- Elements (attractors) in chaotic behaviour sometimes settle down to form predictable behaviour in a more understandable pattern.
Running a marble down a tilted board filled with obstacles. The marble will take a certain path down the board hitting obstacles on its journey. Dropping the marble a second time will result in it most likely taking a different route. Any minute differences at the start will because it to behave differently on its path down the board.
In a game of billiards or pool no matter how consistent you think you are with the break shot the smallest of differences in the speed and angle when you strike the ball will result in the pack of balls moving in random directions each time. The smallest of differences are producing massive effects to the system. i.e. The butterfly effect.
A Final Thought
Trying to put even the basics of chaos theory and its Laws into easy to understand (by me) bite-sizes tested my rudimentary writing skills to the limit.
If you are studying and learning all about chaos theory then good on you and I wish you well.
If there are any mistakes please let me know.
© 2018 Brian OldWolf