CHAOS

1
CHAOS

• Daniel Charles Massey Martinez

2
A Brief History of Chaos

• 19th Century - 3 kinds of motion
• 1890 - Henri Poincaré - the 3 body problem
• 1960 - Edward Lorenz - weather model

3
Dynamical Systems

• What is a dynamical system?
• Equilibrium Points
• Attractors

4
Lyapunov Exponent
5
The Logistic Map
Xn1 ?xn (1 xn)

• Simple ecological model e.g. insect population
• By varying ? and with xn e 0,1 the map becomes
  chaotic

6
? 2
? 2.5
? 3
? 3.5
? 4
7
Lyapunov Exponent of the Logistic Map

• Graph showing the variation of the Lyapunov
  Exponent when varying ?.