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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 $f:(a_i),(b_i)inmathbb{R}^{10}rightarrow chi_A(x)in Zapprox mathbb{R}^5$,

where $Z$ is the set of monic real polynomials of degree $5$.

$textbf{Proposition}$. If $Ainmathcal{E},rank(Df_A)=5$, 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.

$psi^{-1}circ fcircphi$ is the projection $(x_i)_{ileq 10}inmathbb{R}^{10}rightarrow (x_i)_{ileq 5}inmathbb{R}^5$.

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 $mathcal{H}$.