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Equation (2) is linear in $M_{i}$, and therefore the resulting magnetization can be written as the solution to the following matrix eigenvalue problem:
$$\frac{d}{dt}(M_{i}) = \sum\limits_{j}A_{ij}M_{j}$$
Note that, similar to Eq. (5), a biased field acting on one of the moments can drive the dynamics in the lab frame.
$$\frac{d}{dt}(M_{i}) = \sum\limits_{j}A_{ij}M_{j} + \frac{\gamma}{\mu_{0}}(B_{i} – M_{i}B_{i})$$
The required matrix is
$$A_{ij} = \frac{1}{J_{0}}\left[\frac{g}{J_{0}}\sum\limits_{\ell\ell’}J_{i}J_{j}\cos\theta_{\ell\ell’}J_{i}J_{j}\cos\theta_{i}^{k}\cos\theta_{j}^{k} + \frac{1}{\mu_{0}}B_{i}B_{j}\right]$$
Numerical Simulations of the Nano-Magnonic Circuits
===================================================
As shown in the main text, the dynamics of a nano-magnetic vortex can be controlled by engineering time-dependent magnetic fields and bias fields. Here, we numerically simulate such controlled scenarios with a minimal numerical model of the circuit, where the dynamics of individual magnetic vortices in
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