Mathematics of Fractals

In my previous blog article, I introduced the concept of fractals in an abstract sense by discussing various examples of fractals occurring in nature. We defined fractals as a simple mathematical concept of patterns repeating at all scales. In the current article, I plan to provide a sneak peek into the math behind fractal geometries. My intention here is not to cover each and every detail about fractal mathematics (we have a lot of books that will do a much better job at it!) nor do I claim to know everything. Rather I plan to provide an appreciation for the subject by discussing a simple example in detail and show how a simple mathematical construct can get convoluted into geometric patterns (fractals).

Population Models:

Population models try to predict the future population numbers (of humans, insects, birds, bacteria, viruses, etc.) using simple mathematical relations (iterative or differential equations). To begin with, let's consider the very basic exponential model (Fig. 1), where the population of the next time step (x_n+1) depends on the population in the current time step (x_n) and the growth rate (constant ‘r’). The problem with a trivial model like this as shown in Fig.1 is that as the number of time steps increases the predicted population would either tend to infinity (r>1) or zero (r<1), which is obviously not realistic. One thing to be noted here is that all the examples quoted in this article have been normalized with respect to the maximum population limit, therefore we would be coming across initial population (x_o) of the scales such as 0.01, which might not physically make sense at the first glance.

Fig.1 Exponential growth model.

To resolve the above issue one could introduce an additional factor of starvation (Fig.2), to make sure the population does not exceed the maximum limit (x_lim=1). This additional term in the equation (1-x_n) simulates the fact that limited resources are available for the population to survive, i.e. death rate would increase if the population is higher than what the available resources could accommodate. This model is known as the logistic map and unlike the exponential model, it tends to converge to an equilibrium population size as shown in Fig. 2.

Fig.2 Logistic map model.

Like any dynamic system, the response of the population model depends on the initial population size (x_0) and the growth rate (r). Fig.3 plots the population trends for three different scenarios with the same growth rate (r = 2) but different initial populations (x_0 = 0.01, 0.2, 0.9). It can be observed in Fig.3 that irrespective of the initial condition the model converges to the same population equilibrium in all three cases. It might be worthwhile for our intuitive understanding to discuss the case of x_0 = 0.9 in further detail. Since the initial population is really high (x_0 = 0.9) and resources are limited the rate of deaths is very high initially (>birth rate), after crossing the equilibrium point the curve starts to correct itself (death rate decreases) and has an inflection point around n=2 (birth rate > death rate).

Fig.3 Effect of initial population size (x_0) on the characteristics of the population curve.

Similarly, Fig.4 inspects the influence of growth rate (r=0.5,2.0,2.5,3.0,3.5) on the population response for a certain initial population size (x_0=0.1). For a growth rate less than 1.0 (r=0.5) the population becomes extinct, while for a growth rate greater than 1.0 the population approaches a steady state. It can be observed that as ‘r’ increases the steady-state level raises and after a certain point we tend to have oscillatory behavior (r=3.0, r=3.5). This signifies that after a certain growth rate value the population size fluctuates between equilibrium points. An interesting observation to be made here is that in the case of r=3.0 the population has a period of ‘2’ (population size repeats it seld after 2 time steps) while in the case r=3.5 a periodicity of ‘4’ is observed. The behavior need not be so neat and oscillatory every time, if we further explore different ‘r’ values we can come across cases that are non-periodic/chaotic in nature.

Fig.4 Effect of growth rate (r) on the characteristics of the population curve.

One can make use of bifurcation diagrams to better visualize the complete dependency of population response on the growth rate (r). The animation provided in Fig.5 plots the time domain population response on the left and the bifurcation diagram on the right. You need not be too startled after hearing the word bifurcation diagram, these are simple diagrams that plot the value a dynamic system would asymptotically approach as a function of a system parameter (in this case the equilibrium population vs the growth rate 'r').

https://video.wixstatic.com/video/050535_3d51efda79144748a6e4522013ddaa7c/720p/mp4/file.mp4

Fig.5 Time domain response and bifurcation diagram animation of a logistic map.

A detailed zoomed-in figure of the bifurcation diagram observed at the end of the above animation is provided in Fig. 6. As discussed previously it can be observed that the population goes extinct for r<1.0, while for 1.0<r<3.0 it has a single stable population equilibrium. At r=3.0 a Pitchfork bifurcation is observed due to which the population tends to oscillate between two equilibrium points. Around r=3.4 each of these individual branches again independently observes bifurcations. Each of such bifurcations physically signifies a period-doubling of the population response. After a certain value of 'r', the bifurcation diagram becomes completely chaotic with some intermittent stable regions (marked by green arrows on top of the zoomed plot).

Fig.6 Bifurcation diagram of a logistic map.

Though to the naked eye the chaotic region might look like a complete mess one can observe fractals here. Fig. 7 provides multiple levels of zoomed-in plots of the bifurcation diagram that are self-similar, signifying the fractal nature of the chaotic region (as bizarre as it might sound, sometimes there can be symmetry even in complete total chaos!!).

Fig.7 Multi-level zoom of logistic map's Bifurcation diagram.

There are several such simple equations from which one could generate complex geometrical facets, a few of the famous examples are; Mandelbrot sets, Julian set, Dragon Curve, etc. Also, the population model discussed in this article (logistic map) is one of the most basic models, there are far detailed modelling techniques such as the 'SIR' models which make use of non-linear differential equations, which could lead to further complex bifurcation diagrams (the internet is full of fascinating fractal videos !!). That being said my intention with this article was to provide a simple example as to how one could develop complicated fractal structures using simple mathematical equations. In my next article, I will be discussing a few fascinating practical applications of fractals.