The Lorenz EquationsErik Ackermann & Emma Crow-

Willard

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

Lorenz derived his system by simplifying the Navier-Stokes Equation

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

The Lorenz Attractor

Solution curve for:σ = 10, β = 8/3 and ρ = 28Initial Condition: (0, 1, 2)

Existence & Uniqueness

The Lorenz Equations satisfy the E&U Theorem:

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.

These are all continuous for all time.

Classification of Equilibria

)1()1(

1)-(11

0-

The equilibrium points are:

and

The Jacobian Matrix evaluated at these points:

Chaotic Systems

Financial Markets

Weather

Sub-atomic Physics

Chaotic systems are characterized by sensitivity to slight variations in initial conditions.