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- #1

\mathcal{J} = \begin{pmatrix}

-\sigma & \sigma & 0\\

1 & -1 & -\sqrt{b(r - 1)}\\

\sqrt{b(r - 1)} & \sqrt{b(r - 1)} & - b

\end{pmatrix}

$$

From a quick try and error, I was able to find that when $r = 1.3456171$ we will have 3 negative eigenvalues.

But when $r = 1.3456172$, there will be a complex-conjugate pair of eigenvalues.

Is there a mathematically more elegant way to determine this r value?

$b = \frac{8}{3}$ and $\sigma = 10$