The "three-body problem" is an issue in calculus that Henry Poincare was working on when he made the first observations that would lead to chaos theory. In classical mechanics, the three-body problem occurs when you try to model the initial position of a set of exactly three bodies (that is, planets). Poincare's three-body problems are based in differential calculus; thus, they involve deriving rates of change from tangent curves using the basic formula "m = change in y / change in x."
Edward Lorenz was the mathematician who discovered chaos theory by attempting to model the weather. Lorenz's most important discovery was the "Butterfly Effect," which states that even small changes in initial conditions of a system can make a big difference. (Lorenz's example was that of a butterfly in China affecting the weather in New York.) Lorenz used many calculations to model this phenomenon. An example of a Lorenz equation is "dx / dt = o(y - x)," where "x" is the speed of the convectional rolls, "t" is time, "o" is the Prandtl number and "y" is the temperature difference between "p" and "q."
Bifurcation equations deal with the sensitivity of systems (such as weather or finances) to initial conditions. The most well-known bifurcation equation is the logistic equation, in which each value entered depends on a previous value. The logistic equation reads as follows: X(n + 1) = R X(n)(1 - X(n)), where "R" is the specified parameter and "X(n)" is the variable at the nth iteration. This is the equation that explains exactly how sensitive systems are to initial conditions; Lorenz's equations only establish that such a sensitivity exists.
Chaos is not merely a theoretical concept. To the contrary, it has a precise (but abstract) geometrical nature. The geometrical aspects of chaos theory are based on straightforward Euclidian geometry, the equation being D = log(N)/log(1/r). What this equation says is that a given dimension (D) is equal to the logarithm of N (the number of replacement parts) divided by one over the scaling-down ratio. When this equation is modeled, it creates a snowflake whose number of triangular arms increases exponentially.