1. 1 GEK1530 Frederick H. Willeboordse frederik@chaos.nus.edu.sg Nature’s Monte Carlo Bakery: The Story of Life as a Complex System

2. 2 Your questions answered! In-class Tutorial 1

3. GEK1530 3 Chaos Sensitive or Robust During Lecture 1 it was mentioned that a key property of chaotic systems is sensitive dependence on initial conditions. I was also mentioned, however, that many chaotic systems are robust. This appears to be rather contradictory. What’s going on here?

4. GEK1530 4 Strange Attractor The Lorentz attractor is the final (fractal) object that one obtains when numerically solving the equations that Lorentz originally used when investigating the weather. It attracts outlying points. But inside the attractor there is sensitive dependence on initial conditions due to stretch and fold. Lorentz Attractor

5. GEK1530 5 Sensitive Dependence Let us calculate 2.0*very small numbers many times… 0.0000000000000000000000000000000001 0.0000000000000000000000000000000002 0.0000000000000000000000000000000004 0.0000000000000000000000000000000008 0.0000000000000000000000000000000016 0.0000000000000000000000000000000032 0.0000000000000000000000000000000064 0.0000000000000000000000000000000128

6. GEK1530 6 Let us calculate 2.0*very small numbers many times… 0.0000000000000000000000000000000128 0.0000000000000000000000000000000256 0.0000000000000000000000000000000512 0.0000000000000000000000000000001024 0.0000000000000000000000000000002048 0.0000000000000000000000000000004096 0.0000000000000000000000000000008192 0.0000000000000000000000000000016384 As you can see, it only takes a few doublings to shift the leading digit one to left. Ergo, it won’t take long until the number is very big! Sensitive Dependence

7. GEK1530 7 Chaos Let us investigate the logistic map a bit more by exploring the Java applet. The Logistic Map

8. GEK1530 8 Payoff Matrix Chan-Ken-Pong! Well that’s how it sounds to me. In other words, scissors- paper-stone. An important tool for representing who wins what and when is the payoff-matrix. For this simple game it is: draw red blue red blue red blue

9. GEK1530 9 Payoff Matrix The Game of Life and Death S+S+ S-S- S0S0 S+S+ S-S- S0S0 S+S+ S-S- S0S0 Conform Indifferent Contrary Strategy stable unstable variable indifferent Life Death The strategies are as with regards to an increasing or decreasing population after starting from some given population.