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The equations of motion for a Hamiltonian Lorenz model are

$$\frac{\mathrm{d} X_n}{\mathrm{d} t} = f(X_n) \sum_{k=1}^K \xi_k \left(X_{n+k}f(X_{n+k}) - X_{n-k}f(X_{n-k}) \right),$$

for $n=0,\ldots,N-1$ (with periodic boundary conditions $X_{n+N}=X_n$)

Here are the main parameters of the model:

• N : number of spatial nodes in the Hamiltonian Lorenz model
• K : length of spatial correlation (integer)
• $\xi$ : coefficients associated with the spatial correlation (list of K floats)
• $f$ : function defining the model

The model is defined on an energy surface $E=\sum_{n=0}^N \frac{X_n^2}{2}$. It has a number of Casimir invariants (depending on the values of $N$, $K$ and $\xi$); among them is $C=\sum_{n=0}^N \phi(X_n)$ where $\phi$ is such that $f(x) \phi\prime(x)=1$.

Reference

For a full mathematical formulation and analysis of these models, see:

Fedele, Chandre, Horvat, and Žagar Hamiltonian Lorenz-like models, Physica D, Vol. 472, 134494 (2025). https://doi.org/10.1016/j.physd.2024.134494

@article{HamLorenz,
    title = {Hamiltonian Lorenz-like models},
    author = {Francesco Fedele and Cristel Chandre and Martin Horvat and Nedjeljka Žagar},
    journal = {Physica D: Nonlinear Phenomena},
    volume = {472},
    pages = {134494},
    year = {2025},
    doi = {https://doi.org/10.1016/j.physd.2024.134494},
}