1. 1 GEM2505M Frederick H. Willeboordse frederik@chaos.nus.edu.sg Taming Chaos

2. 2 In Class Tutorial 1

3. GEM2505M 3 Questions? Homoclinic Points Received by E-mail Btw. this picture is based on an invertible period 1 map. Invertible implies that there is a unique forward and backward iterate. An (un)-stable manifold cannot intersect itself One homoclinic point -> infinitely many homoclinic points Both based on the fact that a point can have one (and only one) forward/backward iterate.

4. GEM2505M 4 Now that we have seen stretch and fold at work, we can get a bit a better understanding of why the homoclinic points lead to chaotic orbits. Homoclinic Points Let us see what happens to a small area near the stable manifold After a few steps it will be near the fixed point. From Lecture 11

5. GEM2505M 5 Homoclinic Points After arriving at the fixed point the rectangle will be stretched and pushed away along the unstable manifold. Eventually, it will be near the starting point again and overlap the original area. original square From Lecture 11

6. GEM2505M 6 Questions? Cantor Set – Endpoints in triadic Received by E-mail 01 0 1 2 000102101112202122 0.002 0.02 0.202 0.22 So we see: Endpoints either end in 2 or 2

7. GEM2505M 7 Questions? Cantor Set Received by E-mail Again: Endpoints either end in 2 or 2 in triadic But: In order for a point to be a member of the Cantor set, the requirement is only that is can be written with 0 and 2. Therefore: Combinations of 0 and 2 not ending in a 2 or 2 are members of the Cantor set but not endpoints!

8. GEM2505M 8 z0 Questions? Mandelbrot & Julia Sets Received by E-mail Julia Set Mandelbrot Set Change c Fix z 0 = 0 Change z 0 Fix c z0z0 c

9. GEM2505M 9 If you have any questions about the first few lectures, please ask them now! Ask them now! ? Questions

10. GEM2505M 10 1.Make a triangle and label the corners A,B,C 2.Choose any point inside the triangle, make a dot 3.Roll a dice, if it’s 1 or 2 move halfway to A, if it’s 3 or 4, move halfway to B and if it’s 5 or 6 move half way to C. Make a dot at the new position 4.Repeat 3 until dots cover the entire triangle! Roll your dice! A B C The Chaos Game 1,2 3,4 5,6

11. GEM2505M 11 Example A B C The Chaos Game 1,2 3,4 5,6 1 st throw: 4 A B C 1,2 3,4 5,6 A B C 1,2 3,4 5,6 A B C 1,2 3,4 5,6 2 nd throw: 2 3 rd throw: 3 4 th throw: 6