Chaos theory is very beautiful in its conceptualization, almost mystical, that's why Michael Crichton used it in his novel about the dinosaurs. It is definitely mind-blowing, but not mystical, if you understand the basic components of the Chaos Theory. Initially, Chaos theory grew out of hard sciences, Physics, Hydrodynamics, Meteorology, but eventually found its way to softer sciences, such as psychology and sociology. It hasn't been as effective in social sciences, but that is because we don't have the Mathematics or computational power to describe human or social behavior. I have a historic example of Chaos Theory in action, it has nothing to do with the D&D, and I might put it here, based on the feedback that I get from the readers. Chaos Theory has also come in handy when it comes to analysis of disasters and major accidents - airplane crashes, nuclear meltdowns, etc. We are going to see how the Chaos Theory can spice the D&D adventure writing. First, some definitions.
ENTROPY vs. ENTHALPY (Order)
Order is the sand castle you build, entropy is the wave, that smashes it. This is similar to LAW vs CHAOS in D&D, which are defined in the context of the social order, and have moral connotations, Chaos being equivalent to Evil in some D&D games.
A complex system is assumed, a population of a single specie in ecology, a storm system, a commercial jet plane enroute to a destination, a weather system. Also a mind of a single individual, a classroom, a platoon can also be assumed. In D&D, we can assume a Dungeon (which has ecological balance, which the players can disrupt, a Kingdom, a Thieves' Guild, King's Court, warring sides in conflict, and political factions can all be considered systems.
ATTRACTORS AND REPELLERS
An Attractor is a value, a number or a set of numbers, which the system is striving to reach. In human terms, a river flows to the sea, a hurricane is traveling in a certain direction out to sea or away from it. A human being can be on a self-destructive oath with drugs, a thief - bound for gallows, a knight - bound for glory. These are all attractors and in a sandbox campaign, this is the PULL, that to which the players can strive - treasure, power, victory, experience levels, etc.
A Repeller is the opposite of the Attractor, it is something that people a running from. A population of animals might be avoiding famine, a person can be running from poverty, a group of survivors can be running from the old life, that they lost. A Wizard might be running from the experiences that he or she had as an apprentice.
This is the key and the best known part of the Chaos Theory. These are the non linear outcomes, to which the system is striving. Mathematically, the system goals can be at two places at the same time, that is because the mathematical function describing Strange Attractors are Fractals, which are the equations of the Chaos Theory, which describe patters in the random and the destructive events. What are strange attractors in human terms? It is the result of an explosive event or an incident, which utterly destroys the system, and brings about total chaos. An Outcome of a bombing or a disaster. In human terms, Strange attractor is the action of an individual assigning meaning and acting upon it. You see a picture or a situation, and you might make nothing of it. Another person might see the same situation, and because of their totality of training, ability, and experience might see something profound in it, or they might be insane and the same meaningless situation can trigger a breakdown in them. In terms of planes, trains, and historic events, a Strange attractor is the result of a catastrophic failure. In terms of D&D, either the player characters or their nemeses are Strange Attractors, depending whether players are trying to destroy Ancient Evil or protect a community from Ancient Evil being unleashed.
The classic definition is that the flutter of the butterfly's wins in China will cause a hurricane in New England down the road. This is NOT Synchronicity, or the acausal connection, Jung's bridge to the mystical, this is an obscure cause of huge outcomes. A horse getting the stone in its shoe, that cause it to throw the general, who then dies and the battle is lost. In terms of D&D, these are the actions of he players, who beat the huge big guys. The Ring Bearer destroying Mordor. This is highly uncommon in real life, an airplane cleaner leaved the Peto Tube (an opening for the altimeter sensor) covered with scotch tape, and the plane crashes into the mountain side, but is the norm in literature and in story-telling.
SENSITIVITY TO INITIAL CONDITIONS
In order for the Buttlerfly Effect to do its magic, the system has to be vulnerable. Water is safe to swim in, but from the height of 10000 feet, it seems like a solid mass, and if you fall into water from 10000 feet, it will kill you just as surely as the ground will. Space Shuttle Columbia gains a chink in its ceramic armor, and it is vulnerable to hot gases destroying it during the re-entry. In psychical environment, there has to be potential for explosive decompression, starvation, disease, flood or other such calamity that will destroy the system. In social and in literary, and in gaming terms, there has to be danger and tension, where things going wrong can have catastrophic consequences. One mistake, and thieves are caught, if the enemy attacks, the outpost will surely be destroyed, if the King dies, the Queen's loyal guards will be decimated by their enemies.
This is the single most useful analytical tool that came out of the Chaos Theory. Bifurcation is the splitting of the pathways in the development of events. After irreversible incident X, things are going A, and if the incident did not happen, things would have went B, but now, B can never happen and things will never be the same again. Plane was flying to A, but was hijacked to B, and now the passengers' lives will never be the same again. You go into a room, kill the monster, take the treasure. This is not a bifurcation, because this can happen again and again and again. You have a fight with your significant other. You fight every day, week, full moon, the relationship continues. No bifurcation. You say the magic words and this turns into a relationship ending fight. It's over. Things will never be the same again. BIFURCATION. In D&D terms, you kill a bunch of goblins rolled from a random encounter table, No Bifurcation, they can appear again, and gain or loss of 1d6 Goblins armed with spears will not change the course of the adventure. Players capture the Goblin King, when the DM was not planning for it, and they do it early in the game. Now the Goblin defenses are bereft of leadership. They are weaker. Players get the added recognition from the much higher level NPC's as a result of their distinguished captive. Game changer. BIFURCATION. In the real world, the more bifurcations the system has, the greater the chances of a collapse or catastrophic failure. Picture bifurcation as pieces of fuselage getting ripped off the plane expanding the initially tiny puncture during the explosive decompression. A complex system will have a whole bunch of non-irreversible events, modulated through various positive and negative feedback loops, which grants the system stability and capacity for complex systemic behavior, but the BIFURCATION DIAGRAM will be a single line. Presence of many bifurcations trumpets an unstable system on the brink of collapse, but in the game world, many bifurcations mean a fast developing story. A BIFURCATION SYSTEM is especially useful tool for injecting story development into a genuine sandbox.