I'm interested in coupled matrix equations with a magnitude multiplier:
$\frac{d\vec{x}}{dt} = \lvert\lvert \vec{x} \rvert\rvert \mathbf{A}\vec{x} - \vec{d}$
Where $\mathbf{A}$ is a 2 by 2 matrix and $\vec{x}$ a 2 by 1 vector $ = \begin{pmatrix} a \\ b \end{pmatrix}$ so that the magnitude $\lvert\lvert \vec{x} \rvert\rvert = \sqrt{a^2 + b^2}$. And where $\vec{d}$ is a constant vector with nonnegative elements.
My interest is for the particular case where $\mathbf{A} = \begin{pmatrix} -\delta & -\lambda \\ \lambda & -\delta \end{pmatrix}$ and $ \vec{d}= \begin{pmatrix} 0 \\ G \end{pmatrix}$ and all parameters ($\delta , \lambda , G$ are nonnegative).
Questions like his have been queried before, and a closed form solution has not yet been suggested. An intriguing result from numerical simulations of these equations is that the period is (approximately?) invariant to the initial conditions (when $\delta \ll \lambda$), despite the nonlinear form of this equation. Would his suggest a Fourier Series approach (at least for $\delta = 0$)? More broadly, is anyone aware of a closed form solution or have any insight into this type of differential equation?
