From Osher Doctorow Ph.D.
The equations dy/dt = ky, dy/dt = ky(1 - y) or ky(k2 - y), dy/dz = f(z) + g(z)y + h(z)y^2 and the latter similarly for y^3 in an added term are the "standard" growth equations provided that z can be regarded as a non-oscillating function of time. If z were to oscillate so much that it cancelled out the growth efect of y or time, then it would defeat the intuitive idea of growth-expansion- contraction which is not rapidly oscillating (back-and-forth) increase and decrease although that could belong to some different theory. It is possible for things to expand during "youth" and contract during "old age", but oscillating quantities of usual trigonometric type (sin(t), cos(t), etc.) are generally found in science and mathematics to characterize rapid rather than long-term change at least as we intuitively understand that as human beings. Another type of problem, though sometimes interacting with the previous one, occurs when equations change very much during transformations to polar or spherical or cylindrical coordinates (other coordinates won't be considered here). In that case, as long as the above types of equations work in one coordinate system (Cartesian or rectangular being the most common one, which is the ordinary one used in Euclidean Space), that is fine. With "extended" growth equations that involve cardioids or equations of type dy/dt = k/y or dy/dt = sqrt(1 - y) or sqrt(y), this may fail because the main coordinate system may be polar or spherical or cylindrical in which trigonometric oscillating functions like sin(t), cos(t), etc., are the main type in these coordinates. For example, one of the simplest equations of a circle is in polar coordinates: 1) r = asin(u) where u is the polar angle. Unless we restrict u to be between 0 and 2pi, this oscillates too much. Even with such a restriction, there would have to a very good physical or biological and mathematical reason for making that restriction, and this is generally unlikely since there is so much difference between the restricted and the unrestricted cases. The cardioid which is so important to the Mandelbrot set suffers from this difficulty. In polar coordinates it is of form: 2) r = a(1 - cos(u)) In parametric coordinates, it has the form (with sine and cosine possibly changed to cosine and sine respectively, etc.): 3) x = a(t - sin(t)) y = a(1 - cos(t)) which repeats its graph every 2api intervals from 0 onward on the x axis. In my opinion, we are justified in restricting the interval of t or u to be only one of those intervals, which eliminates the oscillation, but only because the Mandelbrot set is critical to fractals and chaos. Even here, there is a question of whether u or t is taken to be time or not. If it is time, then we are asserting that the physical object - for example, the Universe as a whole - is drawing out a cardioid in spacetime, not just in space. This may or may not be what we want to theorize. If u or t is NOT time but some spatial angle only, then we are not even talking about growth as expansion-contraction in time. We would then have to separately hypothesize how the Mandelbrot set or its cardioids change in time, and this is an entirely different question. Osher Doctorow Ph.D.