I am provided a system of ODEs $$\dot{x} = 5x + xy$$ $$\dot{y} = y + 2x(y-x)$$ for $(x,y) \in \mathbb{R}^2.$
I know the Jacobian Matrix, $Df$, works out to be: $$Df = \begin{bmatrix} 5+y & x\\ 2y-4x & 1+2x \end{bmatrix} $$ and $$Df(0) = \begin{bmatrix} 5 & 0\\ 0 & 1 \end{bmatrix} $$ Therefore, the eigenvalues are: $\lambda_1 = 5, \lambda_2 = 1 $
However, I am uncertain whether $Df(0)$ is resonant, given the definition:
We say $Df(0)$ is resonant if there are non-negative integers $m_i$ with $ \sum m_k \geq 2$ such that
$$(m,\lambda) := \sum^{d}_{k=1}m_{k}\lambda_{k} = \lambda_{s} $$
for some $s \in \{1,...,d\}$.
$|m|$ is the order of the resonance.
Given your definition, $Df(0)$ is indeed resonant.
Take $m_1 = 0, m_2 = 5$. As both are non-negative, these are valid values for them. Then, we get \begin{equation} \sum^{d}_{k=1}m_{k}\lambda_{k} = 0\cdot5 + 5\cdot1 = 5 = \lambda_1 \end{equation}