From Osher Doctorow Ph.D. mdoctorow@comcast.net
Indications accumulate that the Riccati Equation is the equation of Expansion-Contraction-Growth (ECG) across biology and physics and other fields, and that this is a type of motion distinct from the usual one-direction-at-a-time Curvilinear Motion. ECG involves motion simultaneously in 2 or more directions (often infinitely many). Since the Universe expands, it seems like an ideal testing case for the Riccati Equation. The general Riccati Equation is: 1) dy/dt = A(t) + B(t)y + C(t)y^2 where dy/dt is the speed of expansion or contraction of y at time t and y^2 is y-squared (y times y) and A(t), B(t), C(t) depend on time t and may or may not be constant real numbers. It turns out that we can prove the following, although the words "certain plausible conditions" are somewhat restrictive. THEOREM. Under certain plausible simple conditions, the Universe goes from a Big Bang in which Rare Event Theory applies and dy/dt attains its maximum ( = 1) to a Big Crunch in which dy/dt = 1/e where e is the exponential constant approximately equal to 2.72 and 1/e is the boundary of exploding Julia sets for the exponential functions c exp(x) (that is, c = 1/e). When c < 1/e, the Julia set is small, and when c > 1/e the Julia set is the whole complex plane (the latter statement can be seen from Robert L. Devaney (1990) Chaos, Fractals, and Dynamics, Addison-Wesley: Menlo Park, CA). PROOF. Since the Riccati Equation and its special cases the Logistic Equation and Simple Exponential Growth/Decay Equations often involve solutions equal to exp(kt) or rational expressions in exp(kt), put exp(t) (k = 1 for simplicity here) into the general Riccati Equation. Since d(exp(t))/dt = exp(t) from calculus, and exp(t) is extremely difficult to "cancel" from an equation usually unless all the terms involve exp(t), let us write A(t) = A1(t)exp(t), C(t) = C1(t)exp(-t). We don't have to change B(t) as will be seen below. Then we obtain: 2) dy/dt = exp(t) = A1(t)exp(t) + B(t)exp(t) + C1(t)exp(-t)exp(2t) since exp(t) squared is exp(2t) from elementary algebra. Since exp(t) divides out in (2) on both sides, we are left with: 3) dy/dt = A1(t)exp(t) + B(t)y + C1(t)exp(-t)y^2 But dividing through by exp(t) on both sides of (2) also leads to: 4) C1(t) = 1 - A1(t) - B(t) This in turn suggests the simplification: 5) B(t) = - A1(t) which also nicely simplifies C1(t) from (4) to C1(t) = 1 Now equation (3) becomes: 6) dy/dt = A1(t)(exp(t) - y) + exp(-t)y^2 When t = 0, dy/dt = A1(0)(1 - y(0)) + (y(0))^2. When t = 1, dy/dt = A1(1)(e - y(1)) + exp(-1)y(1)^2. Since the Big Bang was "explosive" in various senses, we can set dy/dt at t = 0 equal to 1, in which case 1 = A1(0)(1 - y(0)) + (y(0))^2. This is perfectly satisfied if y(0) = 1. Although this does not hold for y(t) as the radius of the Universe, it does hold if y in a neighborhood of 0 is the Rare Event Probable Influence since the latter is 1 at 0. Choose the neighborhood of 0 as [0, .05), that is to say t goes from 0 to less than .05, in accordance with the typical statistical .05 "level of significance". Choose A1(t) = 1 - t + t^2 in the Rare Event neighbor- hood of 0 and A1(t) = 1 - t in [.05, 1]. Then dy/dt at t = 1 equals exp(-1)y^2(1). This already implicates exp(-1) unless y^2(1) = 0 which needn't hold in general. If y(1) = 1, then dy/dt = exp(-1) = 1/e at t = 1. But y(1) = 1 and y(0) = 1 for continuous but non- constant y describes a cyclic universe, and 1/e is the explosive boundary. It should be noted that y(t) and t are between 0 and 1 in the Rare Event scale and that this scale should be restored in the neighborhood of 1 since 1 in the Rare Event scale corresponds to infinite time or to the supremum of finite times, which is a Rare Event (although A1(t) needn't change). Q.E.D. The words "Big Crunch" may not be precisely what we think of as a second "Big Bang", since for example at present the Universe is accelerating in size, but in Rare Events things may not quite be what they seem. Osher Doctorow