Table 5 Eigenvalues of the Jacobian matrix
Equilibrium points | Eigenvalue \(\lambda_{1}\),\(\lambda_{2}\),\(\lambda_{3}\) | Sign | Stability | Stability conditions |
|---|---|---|---|---|
\(E_{1} \left( {0,0,0} \right)\) | \(- C_{i}\), \(- \frac{1}{2}cg_{1}^{2}\),\(F_{e} + L_{s} - C_{r}\) | (–, –, *) | ESS | \(F_{e} + L_{s}< C_{r}\) |
\(E_{2} \left( {1,0,0} \right)\) | \(C_{i}\), \(R_{e} - \frac{1}{2}cg_{1}^{2}\),\(F_{e} + L_{r} + L_{s} + R_{r} - R_{i} - C_{r}\) | (+ , *, *) | Unstable | – |
\(E_{3} \left( {0,1,0} \right)\) | \(- C_{i}\),\(\frac{1}{2}cg_{1}^{2}\),\(- C_{r}\) | (–, + , –) | Unstable | – |
\(E_{4} \left( {0,0,1} \right)\) | \(L_{e} + L_{i} + R_{i} - C_{i}\), \(F_{e} - \frac{1}{2}cg_{1}^{2}\),\(C_{r} - F_{e} - L_{s}\) | (*, *, *) | ESS | \(L_{e} + L_{i} + R_{i} <C_{i}\) \(F_{e} <\frac{1}{2}cg_{1}^{2}\) \(C_{r}< F_{e} + L_{s}\) |
\(E_{5} \left( {1,1,0} \right)\) | \(C_{i}\), \(- R_{e} + \frac{1}{2}cg_{1}^{2}\),\(R_{r} - R_{i} - C_{r}\) | (+ , *, *) | Unstable | – |
\(E_{6} \left( {1,0,1} \right)\) | \(C_{i} - L_{e} - L_{i} - R_{i}, \, R_{e} + F_{e} + L_{e} - \frac{1}{2}cg_{1}^{2},\)\(R_{i} + C_{r} - F_{e} - L_{r} - L_{s} - R_{r}\) | (*, *, *) | ESS | \(C_{i}< L_{e} + L_{i} + R_{i}\)\(R_{e} + F_{e} + L_{e} <\frac{1}{2}cg_{1}^{2}\)\(R_{i} + C_{r}< F_{e} + L_{r} + L_{s} + R_{r}\) |
\(E_{7} \left( {0,1,1} \right)\) | \(R_{i} - C_{i}\), \(\frac{1}{2}cg_{1}^{2} - F_{e}\),\(C_{r}\) | (*, *, +) | Unstable | – |
\(E_{8} \left( {1,1,1} \right)\) | \(C_{i} - R_{i}\), \(\frac{1}{2}cg_{1}^{2} - R_{e} - F_{e} - L_{e}\),\(R_{i} + C_{r} - R_{r}\) | (*, *, *) | ESS | \(C_{i}< R_{i}\) \(\frac{1}{2}cg_{1}^{2} <R_{e} + F_{e} + L_{e}\)\(R_{i} + C_{r} <R_{r}\) |