Chaos or nonlinear dynamics can paradoxically be explained by very simple equations, called nonlinear equations, which Briggs and Peat (1989) describe are “like a mathematical version of the twilight zone. Solvers making their way through an apparently normal mathematical landscape can suddenly find themselves in an alternate reality” (p. 23). Hayles (1990) remarks that “chaos theory has in effect opened up, or more precisely brought into view, a third territory that lies between order and disorder" ( p. 15).
Nonlinear equations apply to discontinuous things like explosions, sudden breaks in materials, and natural phenomena of all kinds from high winds to volcanic earthquakes. In these equations, a small change in one variable “can have a disproportional, even catastrophic impact on other variables” (p. 24). Hawkins (1995) in quoting from Michael Crichton's 1990 novel, Jurrasic Park, lets Ian Malcolm, chaos theorist, explain:
"I guess [chaos theory] is one way to look at things," Grant said. "No," Malcolm said, "It’s the only way to look at things. At least, the only way that is true to reality . . . we have soothed ourselves into imagining sudden change as something that happens outside the normal order of things. An accident, like a car crash . . . we do not conceive of a sudden, radical irrational change as built into the very fabric of existence. Yet it is. And chaos theory teaches us," Malcolm said, "that straight linearity, which we have come to take for granted in everything from physics to fiction, simply does not exist. Linearity is an artificial way of viewing the world. Real life isn’t a series of interconnected events occurring one after the other like a string of beads on a necklace. Life is actually a series of encounters in which one event may change those that follow it, in a wholly unpredictable, even devastating way." (quoted in Hawkins, 1995, p. 37)
Nonlinear equations do not have explicit solutions. Instead of linearity being the rule with nonlinearity the exception, as in the Newtonian mechanistic view of the world, chaos theory has revealed that in fact, nonlinearity rules! Virginia Woolf elegantly discusses the naturalness of unpredicitablilty and uncertainty and our resistance to it in her Pensees:
We are floating in a medium of vast extent, always drifting uncertainly, blown to and fro; whenever we think we have a fixed point to which we can cling and make fast, it shifts and leaves us behind; if we follow it, it eludes our grasp, slips away and flees eternally before us. Nothing stands still for us. This is our natural state and yet the state most contrary to our own inclinations. We burn with desire to find a firm footing, an ultimate lasting base on which to build a tower rising up to infinity, but our whole foundation cracks and the earth opens. (Briggs, 1992, p. 99)
Nonlinear equations are unpredictable due in part to the fact that they iterate, which means that they feed back into themselves, creating a feedback loop. This feedback can be either positive or negative. With negative feedback, the change or oscillations are damped out, the feedback acts to regulate the system and tends to settle back down, a good example being a thermostat. Negative feedback was put to good use in the 1940s in the study of cybernetics. With positive feedback, oscillations build upon or amplify each other and pretty soon, you get the ear-popping screech that happens with feedback in a microphone. With positive feedback, you can literally end up in places you would or could never predict, as little differences are magnified and can get blown out of proportion. Iteration or “feedback involving continual reabsorption or enfolding of what has come before—crops up in almost everything: rolling weather systems, artificial intelligence, the cycling replacement of the cells in our bodies” (Briggs & Peat, 1989, p. 66). Iteration comes from the Latin iter meaning journey. Van Eenwyk (1997) reminds us that whereas when we travel, our trips are more linear in nature, going directly to our predetermined destination by plane or in a car with a minimum (one hopes) of delays, detours, and changes of destinations: whereas traveling was a much different in Roman times, for back then, aberrations abounded, and there was not so much certainty as to the outcome of one’s journey; delays and detours were a matter of course:
The journeys that chaotic dynamics take resemble travel in Roman times. So does life, actually. Just as life is unpredictable, so are nonlinear iterative equations. Unlike linear equations, which proceed in an orderly fashion to a predictable outcome, iterative equations determine their own destiny. Their outcome is rarely predictable” (Van Eenwyk, p. 48).
Briggs and Peat (1989) point out that “iteration suggests that stability and change are not opposites but mirror images of each other” and the iterative nature of nonlinear equations “represents the interconnected nature of dynamical systems” (p. 69).