The eigenvalues \lambda satisfy the characteristic equation
det(IJ)=0.\det(\lambda I - J) \;=\; 0.det(IJ)=0.
Expanding gives a cubic polynomial:
3+c12+c2+c3=0,\lambda^3 + c_1 \lambda^2 + c_2 \lambda + c_3 = 0,3+c12+c2+c3=0,
with the standard coefficients expressed in terms of traces and minors of JJJ:
c1=tr(J)=(J11+J22+J33)c_1 = -\operatorname{tr}(J) = -(J_{11}+J_{22}+J_{33})c1=tr(J)=(J11+J22+J33),
c2=c_2 =c2= sum of principal 222\times222 minors
 c2=J11J22+J11J33+J22J33J12J21J13J31J23J32,c_2 = J_{11}J_{22} + J_{11}J_{33} + J_{22}J_{33} - J_{12}J_{21} - J_{13}J_{31} - J_{23}J_{32},c2=J11J22+J11J33+J22J33J12J21J13J31J23J32,
c3=det(J).c_3 = -\det(J).c3=det(J).
(Conventions sign: here polynomial written 3+c12+c2+c3\lambda^3 + c_1\lambda^2 + c_2\lambda + c_33+c12+c2+c3.)
Linear stability (all eigenvalues have negative real parts) is characterized by the Routh--Hurwitz conditions for a cubic:
c1>0,c3>0,c1c2>c3.\boxed{c_1 > 0,\qquad c_3 > 0,\qquad c_1 c_2 > c_3.}c1>0,c3>0,c1c2>c3.
If any condition fails, the equilibrium loses stability.
4. Bifurcation conditions: saddle-node and Hopf
