Mathematical exploration on chaos theory, trigonometry or probability I am actually in need to do a mathematical exploration for my International Baccalaureate program. There's no specific topic but I've chosen to explore chaos theory since I found it to be very interesting. However, I don't know what mathematical aspect can be calculated or investigated from this very topic. It would be really helpful if you can suggest one, even if it's out of chaos theory. Other topics that I would want to do are trigonometry and probability.
 A: Those three things are dramatically different, and I'd suggest picking one more in line with what you're learning in your IB program.  I'd choose either trig or probability.  If you choose trig, you can look into interesting things like ancient Greek perspectives on it, including the sophistication of it as a field of study even thousands of years ago.  
If that doesn't entertain you, a more modern treatment of probability that is accessible to IB students would be to compare and contrast the various schools of thought of probability and the philosophy of them.  For example, is probability defined only in terms of how frequently events occur, or is there an underlying framework that describes "probabilistic variables" more fundamentally?  What are the different methods of representing, working with, and using probability?   How can probabilistic models be applied to the real world?  How do you know if your model is correct, if it relies on probabilistic variables that you can only measure once or twice?  What is a good measurement, and why? (And...the list goes on).
With chaos theory you typically have to know a bit about dynamic systems, which isn't exactly inaccessible to you but I'd stick to the above two topics instead, as the mathematics required for a reasonable treatment of them would be well within most HS IB programs.
