The Lorenz Equations
Erik Ackermann & Emma CrowWillard
Background
Navier-Stokes Equations:
Where v is the flow velocity, ρ is the fluid density, p is the pressure, T is the stress
tensor, and f represents body forces
•Equation to describe the motion of viscous fluids
•Derived from Newton’s second Law
•Unknown if solutions always exist in three dimensions
Edward N. Lorenz
•Worked as a mathematician and meteorologist during WWII for the United
States Army.
•Published “Deterministic nonperiodic flow” (Journal of Atmospheric Sciences) in
1963.
•Died April 16, 2008
Lorenz derived his system by simplifying the Navier-Stokes Equation
The Lorenz Attractor
Solution curve for:
σ = 10, β = 8/3 and ρ = 28
Initial Condition: (0, 1, 2)
Existence & Uniqueness
The Lorenz Equations satisfy the E&U Theorem:
These are all continuous for all time.
Solutions to the Lorenz equation never cross and continue to infinity.
Because of this, the Lorenz curve has fractal properties. The Lorenz
Attractor has Hausdorff dimension of 2.06.
Classification of Equilibria
The equilibrium points are:
and
The Jacobian Matrix evaluated at these points:
 -

1

  (   1)

1
 (   1)


  (  - 1) 


0
Chaotic Systems
Chaotic systems are characterized
by sensitivity to slight variations in
initial conditions.
Weather
Financial Markets
Sub-atomic Physics