So, dynamical systems theory is, not surprisingly, the study of dynamical systems. The term "dynamical system" itself has a few equivalent definition, but the one I shall use is the following:
Definition: Suppose we have a map F: U ⊆ ℝ → ℝ, a time interval I ⊆ U, and an initial condition (t0,x0) ∈ (I,ℝ). Then we have a (1-D, homogenous, continuous) dynamical system given by the solutions x to the differential equation
|x'(t) = F(x) |x( t0) = x0Usually, for convenience, we set t0 = 0.
Why is this important? Because the Law of Accelerating Returns (LAR) is itself a dynamical system, and can be analyzed according to the rules of dynamical systems theory. It is given by the equation
|x'(t) = ax
|x(0) = x0
where a < 0 is some constant representing the rate of decay, x(t) is the cost of information technology at time t, and x0 is the current cost. In ordinary language, it states that the cost of information technology falls at a rate proportional to the cost itself.
This is very important to note: this is all the LAR says! Kurzweil himself will say this. The LAR therefore cannot be used to derive any additional conclusions about the so-called march of technology, at least not without plugging in additional facts or assumptions about society, the economy, etc.
EDIT: It looks like I'm going to have to make this a series after all. The next entry in the series is going to contain examples of how to solve the LAR under certain societal conditions.