From Osher Doctorow Ph.D.
We get closer to the relationship between chaos/fractals and physics and biology by considering the linked logistic system: 1) dv1/dt = k1v1(1 - v1) + k2v2(1 - v2) dv2/dt = k3v1(1 - v1) + k3v2(1 - v2) which is a modification of the predator-pret Lotka predator-prey equations in which two speeds (2-population growth speeds) v1 and v2 are interlinked logistically with each other. James E. Bruno and I and Chris Kappner in 1981 did some publications on this and related systems in population growth/migration, but we didn't explore the physics picture or even the equations beyond the particular empirical problem studied there. The equation (1) has arisen again, this time when I introduced it today in stardrive1@yahoogroups.com to model a 2-velocity spacetime warp star drive of the Alcubierre-related type. What I want to emphasize here is one of the strangest outcomes of the use of equations (1) in physics. Here is the basic result: PRINCIPLE OF SPEED 1/2. The speed 1/2, on a scale from speed 0 to the speed of light taken as 1 (here I'm not considering possible superluminal speeds), is not only the average between these two extremes but is also the speed at which Force becomes very highly proportional to the time rate of change of "logistic momentum" which has the form mv(1 - v). We know since Sir Isaac Newton's time in the late 1700s that force is the time rate of change of momentum, but "logistic momentum" is more realistically the growth-related analog of momentum. For those who know calculus (for others, just notice that dv/dt cancels in the second equation below if it isn't 0 and then solve algebraically for v), force F = kd[mv(1 - v)]/dt and approximating m by a constant and incorporating it into k leads to (using the same constant k) dv/dt = kdv/dt - 2kvdv/dt which has the solution v = (1/2) - [1/(2k)] and for k very large this is close to (1/2) and gets closer the larger k is. Now if we approximate v1 in (1) by (1/2), it becomes a simple matter to solve system (1) by the same method as solving the logistic equation, that is to say by partial fractions noting that 1[v(1 - v)] = 1/v + 1/(1 - v). The logistic system of course is central to chaos and fractals and environment- limited growth in its discretized (and non-discretized) version and so on. Osher Doctorow Ph.D.