PHP Chaos

I really do not believe, as the proverbial story goes, that the fluttering of a butterfly in the Amazon can affect the weather in Europe. But Chaos Theory obviously has some reality as can be understood from this link

When I worked in the Nuclear Industry I wrote Fortran computer programs to calculate temperatures in a counterflow boiler which worked in reality by hot CO₂ (from the top of the reactor) passing down through the boiler while water (from the turbine condenser) entered the boiler tubes at the bottom and was heated as it was pumped to the top. We solved the equations by finite difference and integrated from the bottom. To begin the process we therefore had to guess a value for the CO₂ outlet temperature. If this guess was only a few degrees out we got totally chaotic results with wildly impossible temperatures. This represented no kind of reality. Mother Nature had no problem getting things right and there was no possibilty whatever of a real boiler behaving in this way.

When a friend developed an algebraic solution to similar equations for the counterflow in the reactor, it was clear that they involved the difference between large exponentials and double length arithmetic was necessary to get satisfactory solutions.

I have long been of the view, therefore, that so called chaos arises from the way we represent numbers by a finite number of decimal places (or in a computer hexadecimal places) which introduces discontinuities where in nature there are none. If you read the paper to which I refer you will find the very simple equation:

It is a very simplified model of population growth. X, a number between zero and one,  representing the population and K is some parameter representing things affecting how the population grows. I have programmed in PHP a simple histogram graph of this equation and you can try it with differen values of K and starting values X₀. Obviously the steady state answer to the equation is given by :

The question is: does real population ever go into chaotic behaviour as can be brought about with K=3.8 and X₀=0.8 for which the steady state solution would be 0.736842105 to nine decimal places???

Go and play with the equation for yourself