Summary

J. Emmanuel JohnsonTakaya Uchida

In these sets of notebooks, we look at the canonical Lorenz systems. We look at the Lorenz-63, the Lorenz-96 and the two level Lorenz-96 ODEs.

Lorenz 63

Demo

dxdt=σ(y−x)dydt=x(ρ−z)−ydzdt=xy−βz\begin{aligned} \frac{dx}{dt} &= \sigma (y - x) \\ \frac{dy}{dt} &= x (\rho - z) - y \\ \frac{dz}{dt} &= xy - \beta z \end{aligned}
(1)

Lorenz 96

dxdt=(xi+1−xi−2)xi−1−xi+F\frac{dx}{dt} = (x_{i+1} - x_{i-2})x_{i-1}-x_i+F
(2)

Demo

Lorenz 96 (2 Level)

dxdt=(xi+1−xi−2)xi−1−xi+F−hcb∑jyjdydt=−bc(yj+2−yj−1)yj+1−cyj−hcbxi\begin{aligned} \frac{dx}{dt} &= (x_{i+1} - x_{i-2})x_{i-1}-x_i + F - \frac{h c}{b} \sum_{j}y_j \\ \frac{dy}{dt} &= -b c (y_{j+2} - y_{j-1})y_{j+1}- c y_j - \frac{h c}{b} x_i \end{aligned}
(3)

Demo

Lorenz 96 - Two Level (TODO)

Special Cases
Lat-Lon Domains
Lorenz
Lorenz 63