In mathematics most things can be predicted quite simply if enough information is provided. This though doesn’t apply to the chaotic system, as in their names are quite chaotic and are difficult to predict. Some examples of this in real world would be the weather. We have developed systems to be able to predict it to some extent but it can vary from initial prediction, this is the basis of butterfly effect that initial condition can vary the final result greatly. The other simple example is a double pendulum, depending on where it is dropped it would result in much different results.
What we did was look at different attractors that would create different systems. In essence attractors are set of numeric values to which a system tends to evolve from. If these values are to be slightly changed it would result in a complete different and non-chaotic system. To have the system figured out over longer and longer periods it would need to be integrated. The reason to find these attractors is to be able to recognize and to eliminate them to help with models.
The system looked at today is called an Arneodo system. Visually it can be though as two reflexive disk that have an extension in the middle but seeing the visual it self is much better. To get the model has to be referred from thermohaline convection and reduction of its partial differential equations to an ordinary differential equation, this would result in Arneodo.
Initial conditions that are found for the formula are system can be seen in the image. The system it self for the most part can be seen quite simple but does result in quite bizarre shape. And these models can be quite hard to do by hand but are usually performed with computers to help model them but still that is very limited to what they look long term.
As can be seen the chaos is quite complex though we are able to mitigate it a bit with these attractors. In long term it is impossible to predict how these things will end up looking like and we are only able to get some sense with the attractors. There are many more of these chaotic maps, some that are complex, and are one dimension.
Illia Stadnyk
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