Consider the vector field $V:\mathbb{R}^4\rightarrow\mathbb{R}^4$, defined by
\begin{equation}
 V(x,v,M\_0,M\_1)=(v,\kappa^{-1}(\beta M\_0-v-kx),-M\_0+v M\_1,-M\_1+1-vM\_0),
\end{equation}


such that $\beta,\kappa,k$ are constants. The only equilibrium point occurs at $P^\*=(0,0,0,1)$ and the Jacobian matrix of $V$ at $P^\*$ is
\begin{equation}
 JV\_{P^\*}=
 \begin{pmatrix}
 0 & 1 & 0 & 0 \\
 -\frac{k}{\kappa} & -\frac{1}{\kappa} & \frac{\beta}{\kappa} & 0 \\
 0 & 1 & -1 & 0 \\
 0 & 0 & 0 & -1 
 \end{pmatrix}.
\end{equation}


So, we have $\lambda\_0=-1$ and using Cardano's formula
\begin{equation\*}
 \begin{split}
 \Delta\_0&=\left(\frac{\kappa+1}{\kappa}\right)^2-\frac{3(k+1-\beta)}{\kappa},\\
 \Delta\_1&=2\left(\frac{\kappa+1}{\kappa}\right)^3-\frac{9(k+1-\beta)(\kappa+1)}{\kappa^2}+\frac{27k}{\kappa},\\
 \mathcal{C}&=\sqrt[3]{\frac{\Delta\_1\pm \sqrt{\Delta\_1^2-4\Delta\_0^3}}{2}},\\
 \lambda\_i&=-\frac{1}{3}\left(\frac{\kappa+1}{\kappa}+\xi^{i-1} \mathcal{C}+\frac{\Delta\_0}{\xi^{i-1}\mathcal{C}}\right).
 \end{split}
\end{equation\*}


The motivation is to find a solution to the linearised system $$\vec{\dot{x}}=JV\_{P^\*}\text{ }\vec{x},$$


whereby $\vec{x}=C\_1 e^{\lambda\_1 t}\vec{v}\_1+...+C\_4 e^{\lambda\_4 t}\vec{v}\_4$ iff $JV\_{P^\*}$ is diagonalisable. Due to the Hartman–Grobman Theorem, we rely on the property that $\Re(\lambda\_j)\neq0$, $\forall j$. Is it possible then to isolate the real part of $\lambda\_{1,2,3}$?



Finding the eigenvalues and eigenvectors of Jacobian at equilibrium point of nonlinear ODEs
The characteristic polynomial is
$$P(\lambda) = (\lambda+1)\left(\lambda^3 + \frac{\kappa+1}{\kappa} \lambda^2 + \frac{k-\beta+1}{\kappa} \lambda + \frac{k}{\kappa}\right) $$
Since $\kappa \lambda^3 + (\kappa+1) \lambda^2 + (k-\beta+1) \lambda + k$ is irreducible over the rationals, there's no further factorization possible: if you want explicit expressions for the roots, you will indeed have to use the formulas for roots of a
cubic. Of course, for numerical values of the constants you can use numerical methods to get approximate eigenvalues, or you might try series expansions.


