After linearization near the equilibrium, I get a system that looks like this:
$$ \frac{d\vec{x}}{dt} = -A \vec{x} $$
where $A$ is a $4$ by $4$ matrix. After some simplification, I found that the matrix $A$ has the following four eigenvalues under certain condition:
$$ \lambda_1 = 2 + a+b\\ \lambda_2 = 2 - (a+b)\\ \lambda_{3,4} = \pm (a-b) i $$
I ran some simulations and it seems like the system exhibits a periodic limit cycle when $a+b = 2$. However, when $a + b \neq 2$, amplitude either grows or shrinks. Why does having a zero as an eigenvalue cause this behaviour? Is looking at the eigenvalues not enough to understand this?