Modell System for calculations

Most of the informations we have about the climate change are based on calculations. Here we present an easy calculation system to study the influences of changes in parameters and their results. We want to describe, what is known as the "butterfly-effect". The name comes from the fact, that if you transfer the principles of a atomic world into the bigger surrounding of the real world, than you will find out,that a small wind from a butterfly in the rainforest will cause a big hurrican after a while. As you can easily see, it is nonsense and nobody will believe it,but it is true, if you don´t think about your activity. The real backround is the non-linear,chaotic behavior that can be observed everywhere.

Here is a describtion of the modell system:

My example is much more simpler. It is called the"logical illustration" and is known since 1844. We look for the population of fishes in a lake (or something else, similar). If we change the starting conditions only in a very, very small number, this small mistake will causes dramatic changes, if you recalculate every year by taking the result of the one before. So, you take the Number of Fishes (Xn) as per cent of the maximum of population. Because of that, Xn will be a Number between o and 1. We calculate the amount of fishes in the next year by the easy equation: Xn+1 = R*Xn*(1-Xn). R is a parameter to describe the fertility of the fishes. For R smaller than 1, the population dies expotentiell. For R bigger than 1 it can stabilise on a special level. For R bigger than 3 it will result in periodic oscillation. For R bigger than 3,57 it will be chaotic, although the Population can be calculated by the deterministic equation exactly! So, lets start calculation. For that you best use Exel. Use R=3,9 and calculate two lines. First one with 0,6 for Xn and the second one with 0,60001 as starting population. You see: the different is very,very small. After 17 steps (=17 years) you will find complete different results. You can easily see that in a graph. For me, this behavior of these systems is one the main points to have a critical look for calculation systems in generall. They are easy to manipulate and it is a general problem to find the right start conditions. I hope, you all understand my explanation how to do this calculation.

Here are the calculation lines:

It is easy to see, that after 17 years it beginn to differ. After 20 years, it is no longer possible, to predict the Number of Fishes!

It is remarkable, that this easy system is not to calculate properly for a longer distance.

Here are the other described lines:

So please give me your ideas!