I'm trying to prove stability of the following dynamic system but I think my Mathematics knowledge is not deep enough.
My dynamic system consists of a state vector $x \in \mathbb{R}^n$. The system changes the value of $x_i$ as in the following equation,
x'_i = (x_i + KT(\sum_{j \ne i}^n{\frac{1}{T - |\Delta_{i,j}|}}- \sum_{j \ne i}^n{\frac{1}{|\Delta_{i,j}|}})) \mod{T},
where $\Delta_{i,j} = x_i - x_j$, $K$ and $T$ are constants, and I assume |$\Delta_{i,j}| \in (0, T)$ .
The system changes the vector $x$ in circular order. For example, $x_1$ is changed first. Then, $x_2$ is changed and so on. Until $x_n$ is changed, the system re-starts with $x_1$ again.
I've ever proved only the stability of a linear dynamic system by using eigenvalues. However, with this equation form, I think eigenvalues do not work because it's non-linear dynamic system.
Any help?