SLIDE 1
Background
- Discovered in 1963 by Ed Lorenz
- Simple model of convection in
atmosphere
- First showing of strange attractor and
chaos
- No fixed points
- No periodic orbits
- Solutions do not infinity with time
The Chaotic Waterwheel: Exploring the Lorenz Equations Stephanie - - PowerPoint PPT Presentation
The Chaotic Waterwheel: Exploring the Lorenz Equations Stephanie - - PowerPoint PPT Presentation
The Chaotic Waterwheel: Exploring the Lorenz Equations Stephanie Moyerman Math 164 Final Project Background Discovered in 1963 by Ed Lorenz Simple model of convection in atmosphere First showing of strange attractor and chaos
SLIDE 2
SLIDE 3
Derivation
NO! But…
- Conservation of Mass
- Torque Balance
- Amplitude Equations
I gRa q K b b Ka b a / ) (
1 1 1 1 1 1 1
π νω ω ω ω ω + − = + − − = − = & & &
Intertia
- f
Moment Wheel
- f
Radius (Variable) Gravity Rate Damping Rotational Rate Inflow Rate Leakage
1
= = = = = = I R g q K ν
SLIDE 4
The Waterwheel
SLIDE 5
Animations and Results
SLIDE 6
The Lorenz Equations
Just a change of variables away! Name No Number Rayleigh Number Prandtl = = = b r σ
SLIDE 7
Solution Reliability
SLIDE 8
Solution Reliability
SLIDE 9
Solution Reliability
SLIDE 10
Liapunov Functions
Measure Divergence of Nearby Trajectories with Increasing Time
SLIDE 11
Behaviors
SLIDE 12
SLIDE 13
Chaos and Sensitive Dependence
SLIDE 14
Chaos and Sensitive Dependence
SLIDE 15