Let us parametrize $M_5(\mathbb R)$ by 10 parameters $(a_1, \dots a_5, b_1, \dots, b_5)$ in following way \begin{align} \tag{$\star$} A = \begin{pmatrix} 0 & -a_1 & 0 & 0 & -b_1 \\ 1 & -a_2 & 0 & 0 & -b_2 \\ 0 & -a_3 & 0 & 0 & -b_3 \\ 0 & -a_4 & 1 & 0 & -b_4 \\ 0 & -a_5 & 0 &1 & -b_5 \end{pmatrix}. \end{align} Let us define a set $\mathcal H$ and $\mathcal E$ by \begin{align*} &\mathcal H = \{A \in (\star): \max_i \text{Re}(\lambda_i(A)) < 0\},\\ &\mathcal E = \{A \in (\star): \max_i \text{Re}(\lambda_i(A)) = 0\}, \end{align*} i.e., matrices parametrized as $\star$ having the largest real part of all eigenvalues lying on the left open half plane and imaginary axis respectively. Naively, I would think the boundary of $H$ is given by $\partial H = \mathcal E$. But I have not been able to prove it. Obviously $\partial H \subset \mathcal E$.
Let \begin{align*} \mathcal F = \{A \in \mathcal E: A \text{ has distinct eigenvalues}\}. \end{align*} I have shown $\mathcal H$ is connected and for every $M \in \mathcal F$, I can show $M \in \partial H$. But no further. I tried to show $\mathcal F$ is dense in $\mathcal E$ but according to an answer here Are matrices diagonalizable dense in a specific parametrization of $M_5(\mathbb R)$?, it is not an easy question to answer.
A partial answer.
Let $A=((a_i),(b_i))\in \star$.
Let $f:(a_i),(b_i)\in\mathbb{R}^{10}\rightarrow \chi_A(x)\in Z\approx \mathbb{R}^5$,
where $Z$ is the set of monic real polynomials of degree $5$.
$\textbf{Proposition}$. If $A\in\mathcal{E},rank(Df_A)=5$, then there is a sequence in $\mathcal{H}$ that tends to $A$.
$\textbf{Proof}$. There are a neighborhood $U$ of $(a_i,b_j)$ and a neighborhood $V$ of $\chi_A$ -a polynomial that admits roots $u_1,\cdots,u_k$ with $<0$ real part and roots $v_1,\cdots,v_{5-k}$ with $0$ real part- and charts $\psi,\phi$ of $U,V$ s.t.
$\psi^{-1}\circ f\circ\phi$ is the projection $(x_i)_{i\leq 10}\in\mathbb{R}^{10}\rightarrow (x_i)_{i\leq 5}\in\mathbb{R}^5$.
In $V$, there is a polynomial $q$ that admits the roots $u_1,\cdots,u_k,v'_1,\cdots,v'_{5-k}$ where the $v'_i$ have $<0$ real part. It suffices to consider a small perturbation of $\psi^{-1}(\phi(q),0_5)$ (in order to obtain distinct roots). $\square$
For a generic $A$, $rank(Df_A)=5$ and we may use the above proposition. Unfortunately, in general, $rank(Df_A)\geq 3$ and, in particular, $rank(Df_A)=3$ when the $(a_i),(b_j)$ are $0$.
In this last case, despite many random tests, I did not find small variations that send $A$ to an element of $\mathcal{H}$.