–> See Oxford course notes, Chaos

Characteristics of chaos

• Countable infinity of periodic orbits
• Uncountable infinity of aperiodic orbits
• Sensitive dependence on initial conditions. See Lyapunov exponents in Nonlinear dynamical systems
• A dense orbit, which comes arbitrarily close to all periodic orbits.

http://mathworld.wolfram.com/Chaos.html (Topological transitivity)

Chaotic maps

Symbolic dynamics can be used to analyze them

• Logistic map
• Binary shift map
• Skinny baker map
• Smale horseshoe map

Chaotic Nonlinear dynamical systems

• Lorenz system
• Rossler system
• Duffing oscillator

Routes to chaos

Period doubling

Sarkovskii's theorem

Period 3 implies chaos

Intermittency

Blue sky catastrophe

Every chaotic dynamical system is a fractal-manufacturing machine

An Introduction to chaotic dynamical systems. Second edition

animation of rossler attractor