Currently, while still exploring what was learned through the Digital Advocacy course and experimenting with how it fits into the NCP wiki and systems map, I have been taking a course in Dynamical Systems and Chaos taught by Prof. David P. Feldman through ComplexityExplorer.

I am now questioning what my most recently created Causal Loop Diagram maps are really telling me. What I present here, as a rough summary, is in consideration of my questions and should not be thought of in any way as an even partial substitute for the course. It is simply an attempt to try to apply someone"q new learning. It gets abstract because it involves some mathematical concepts but the ideas in the course are for a general audience and are presented here as general as possible.

A dynamical system is simply a rule for how something changes in time. The NCP systems maps are also intended to provide this type of information though at a different level of precision. The ComplexityExplorer course deals with two types of dynamical systems, iterative functions and differential equations.

The logistic equation, shown above, is a simple model of population growth. It's an iterated function which might tell us how a population changes from year to year. We do the same thing, apply the same function, this logistic equation with a fixed r value, over and over again, using the output for one year as the input for the next. In the standard form of the logistic equation r is a growth rate parameter, r then is something that could change, and we could then see how the behavior of the equation changes. As an iterated function time is discrete, we are not monitoring the population at every instant. A continuous change would involve a differential equation.

The logistic equation is a second order polynomial, a parabola; a very simple function studied in high school, not an exotic or complicated function. The course offers a couple of simple tools for single iterations and comparisons. It is also pretty simple to create a spreadsheet which can push beyond the parameters set by the web tools. The logistic equation is deterministic. Simply an iterated function, an action repeated again and again which ought to be completely predictable.

The first question is whether the simple circles making up the various loops of a Causal Loop Diagram convey smooth transition rather than the more likely true jumps both positive and negative found with iterative functions, then whether additional loops are sequential or occur more or less simultaneously?

The logistic equation deals with populations, as in a finite collection of items under consideration dealt with as a whole, more specifically for our purposes, a community of living entities in which interbreeding occurs among members and is subject to a growth rate parameter from internal and external forces. There is a natural positive increase in population through procreation that exceeds the forces that work against it by some factor greater than 1. The population is changeable to the point that it can be annihilated, from lack of resources, inability to compete for resources, inability to reproduce and maintain the population, and by destruction from environmental or external forces.

The logistic equation is not a means of measurement, in the same way, say as using a Newtonian Law to determine the rate of cooling of an object. As the course explained, the logistic equation is more of a caricature than a detailed portrait or photo. The NCP maps then are a few brush strokes but as has been said many times before, while wrong in terms of the totality of information provided can still be useful, in some cases more useful in conveying insights or at least a different level of insights.

If this year's population is larger than last year’s, based on r, the growth factor considered alone, is greater than 1 then next year's will be larger still tending if without bound towards infinity. If r is 1 then the population stays fixed. If r is less than 1 but greater than 0, the population diminishes approaching 0. If r, for example, is 0.5 then next year there will be half as many say rabbits and half again the following year.

The idea that populations grow without bound is though unrealistic. There is some limit to the growth. There is some maximum population beyond which the population can't pass. There will always be some limit to the number of rabbits, or whatever it is being studied. A term is then added to the equation, a term known in this case as the "Annihilation" population or "Apocalypse" population; meaning that if the rabbits ever reach this “A” population then the next year there will be no more rabbits. The rabbits eat all their food, so the following year there are no rabbits left. The maximum possible number of rabbits is determined by this function in which x is measured as a fraction of the annihilation parameter so the equation can be display as:

For small populations very far away from annihilation, with P much less than A, we should have the potential for rapid growth. When, however, P gets to be large, enough that the rabbits start running out of food, is when population growth starts slowing down approaching its limit. Once the population gets large, the Annihilation term starts to matter more and population growth slows down. There is an absolute upper limit, at the annihilation or apocalypse number, which if reached the population completely crashes.

Nobody should think, however, that the logistic equation actually controls real rabbit populations or fox populations or moose and wolves populations for that matter. It is simply a thought experiment to interpret as the reality, what's actually happening, and what will happen. A rabbit population once established and under normal circumstances would be unlikely to reach either the maximum of the Annihilation Population or total collapse. If an increase in foxes decreases a rabbit population then there will likely be a subsequent decrease in the fox population from a lack of prey and the rabbit population may rebound, explaining the parabolic shape of the curve of the function.

Arguably, many of the elements of the NCP maps could be considered to be populations serving different even opposing functions within a community. There has to be a certain portion of the community population comprising or utilizing the elements making up these maps. I am not sure how to determine an Annihilation population equivalency but the annihilation of elements is possible fundamentally changing the system. The NCP CLD maps denote positive and negative forces or influences by blue and red connections, respectively. The actual nature of Community Advocacy portrayed by the Kumu map then is a result of the net influences making up the particular configuration of the system at that point in time. Causal Loop Diagrams also repeat but adding in more loops, at likely different rates of growth or influence, can create numerous complex outcomes. With the NCP maps, increased calls for greater transparency and open data in government could result in greater pushback by entrenched government institutions. How much of an equivalency is there then with the logistic equation applied to rabbits? Open Data and transparency are ideas made manifest. Can they be propagated and annihilated the same as rabbits or perhaps viral infection would be a better analogy?

The logistic equation is capable of cyclic behavior that is stable or attracting. Different r values can also give rise to cycles of different periodicities, a cycle of period 2 takes two iterations to complete a cycle. The logistic equation with an attracting cycle of period 4 takes four iterations to cycle back then it repeats. It’s attracting because nearby orbits are pulled towards it. If a population is in such a cycle and gets pushed off, it will return back to that cycle.

While the NCP CLD maps are cyclic in nature, they don’t convey periodicities even though subsequent revolutions along the paths could result in far different outcomes with each completion. InsightMaker would seem to be better at this than Kumu. There is though a great probability for interacting cycles to move towards stable orbits regardless if they are desired, particularly when involved with Entrenched Government Institutions. Entrenched Government Institutions have been dealt with before, most recently with Active Citizens in a Digital Age Embracing Organized Complexity. This might raise the question whether the ideas being considered here apply to Warren Weaver's concept of Organized Complexity in "Science and Complexity". I believe it does as the parameters set by the equation don't depend upon the actions or consequences of any individual member but the community as a whole.

However, for a logistic equation with r=4, and other values, the orbit is aperiodic. The orbit doesn't hit some regular cycle, it just keeps bouncing around all over, never repeating. An incredibly repetitious process which always produces something new. The orbit and never settles into periodic behavior. In other words, it is chaotic, a concept which needs to be explored further.