Lorenz models

• Stochastic Lorenz 96 model (available in Matlab, and Python) . This model has two different versions: (1) the mono-scale model, and (2) the multi-scale model.

  • The multi-scale model has two state variables, $x$ and $z$, of dimensions $d_x$ and $d_z$ respectively. While $x$ is a slow-changing variable, $z$ evolves fast in time. It also has four static parameters $F$, $H$, $C$ and $B$.
  • The mono-scale model is a simpler version of the previous one, ignoring the variable $z$ and the influence it has in $x$. The number of static parameters are also reduced, and there is only $F$.

  The dimension of the model can be tuned, making it interesting to try algorithms and methodologies in higher dimensions.

• Stochastic Lorenz 63 model (available in Matlab and Python). This is a 3-dimensional chaotic model, described by the SDEs:

  $$dx_1 = -S (x_1 - x_2) + \sigma dw_1,$$

  $$dx_2 = R x_1 - x_2 - x_1 x_3 + \sigma dw_2,$$

  $$dx_3 = x_1 x_2 - B x_3 + \sigma dw_3,$$

  where the $w_i$'s are independent 1-dimensional Wiener processes, $\sigma$ is the standard deviation of the state noise, and $S$, $R$, and $B$ are static parameters.

Some links

• Wikipedia page of the Lorenz 63 system, with code in different programming languages
• Integration of Lorenz 63 model with a 4th Runge Kutta method (instead of Euler-Maruyama) in Python
• The original paper of Edward N. Lorenz in the Journal of the Atmospheric Sciences, 1963
• Wikipedia page for the Lorenz 96 system
• Thesis of van Kekem, Dirk Leendert: Dynamics of the Lorenz-96 model. I recommend to have a look at Figure 1.3 to understand the physical meaning of the model. There is also a summary of Lorenz's life and achievements.

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