Table 4 Eigenvalues of eight equilibrium points.
From: Fiscal centralization versus decentralization of transport infrastructure operation and maintenance
Equilibrium points | Eigenvalues | ||
|---|---|---|---|
\({k}_{1}\) | \({k}_{2}\) | \({k}_{3}\) | |
\({E}_{1}\left(\mathrm{0,0,0}\right)\) | \((1-{\gamma )F}_{A}+\left(1-\delta \right){F}_{B}\) \(-{\left(1-\theta \right)C}_{1}-{C}_{3}\) | \(\gamma {F}_{A}{+\alpha T}_{{rA}}+\left(1-\varepsilon \right)\sigma {R}_{A}+b{Q}_{B}\) \(-(1-{\mu }_{A}){E}_{A}^{2}/2-{C}_{3A}\) | \(+{\delta F}_{B}{+\beta T}_{{rB}}+\left(1-\nu \right)\tau {R}_{B}+a{Q}_{A}\) \(-(1-{\mu }_{B}){E}_{B}^{2}/2-{C}_{3B}\) |
\({E}_{2}\left(\mathrm{1,0,0}\right)\) | \(-[(1-{\gamma )F}_{A}+\left(1-\delta \right){F}_{B}\) \(-{\left(1-\theta \right)C}_{1}-{C}_{3}]\) | \({T}_{{rA}}+{F}_{A}+\left(1-\varepsilon \right)\sigma {R}_{A}+b{Q}_{B}\) \(-(1-{\mu }_{A}){E}_{A}^{2}/2-{C}_{3A}\) | \({T}_{{rB}}{+F}_{B}+\left(1-\nu \right)\tau {R}_{B}+a{Q}_{A}\) \(-(1-{\mu }_{B}){E}_{B}^{2}/2-{C}_{3B}\) |
\({E}_{3}\left(\mathrm{0,1,0}\right)\) | \(\left(\alpha -1\right){T}_{{rA}}+\left(1-\delta \right){F}_{B}\) \(-{\left(1-\theta \right)C}_{1}-{C}_{3}\) | \(-[\gamma {F}_{A}{+\alpha T}_{{rA}}+(1-\varepsilon )\sigma {R}_{A}\) \(+b{Q}_{B}-(1-{\mu }_{A}){E}_{A}^{2}/2-{C}_{3A}]\) | \({Q}_{B}+{\delta F}_{B}{+\beta T}_{{rB}}+\left(1-\nu \right)\tau {R}_{B}\) \(-(1-{\mu }_{B}){E}_{B}^{2}/2-{C}_{3B}\) |
\({E}_{4}\left(\mathrm{0,0,1}\right)\) | \(\left(\beta -1\right){T}_{{rB}}+(1-{\gamma )F}_{A}\) \(-{\left(1-\theta \right)C}_{1}-{C}_{3}\) | \({Q}_{A}+\gamma {F}_{A}{+\alpha T}_{{rA}}+\left(1-\varepsilon \right)\sigma {R}_{A}\) \(-(1-{\mu }_{A}){E}_{A}^{2}/2-{C}_{3A}\) | \(-[{\delta F}_{B}{+\beta T}_{{rB}}+\left(1-\nu \right)\tau {R}_{B}\) \(+a{Q}_{A}-(1-{\mu }_{B}){E}_{B}^{2}/2-{C}_{3B}]\) |
\({E}_{5}\left(\mathrm{1,1,0}\right)\) | \(-[\left(\alpha -1\right){T}_{{rA}}+\left(1-\delta \right){F}_{B}\) \(-{\left(1-\theta \right)C}_{1}-{C}_{3}]\) | \(-[{T}_{{rA}}+{F}_{A}+(1-\varepsilon )\sigma {R}_{A}+b{Q}_{B}\) \(-(1-{\mu }_{A}){E}_{A}^{2}/2-{C}_{3A}]\) | \({T}_{{rB}}+{F}_{B}+\left(1-\nu \right)\tau {R}_{B}+{Q}_{B}\) \(-(1-{\mu }_{B}){E}_{B}^{2}/2-{C}_{3B}\) |
\({E}_{6}\left(\mathrm{1,0,1}\right)\) | \(-[\left(\beta -1\right){T}_{{rB}}+(1-{\gamma )F}_{A}\) \(-{\left(1-\theta \right)C}_{1}-{C}_{3}]\) | \({T}_{{rA}}+{F}_{A}+\left(1-\varepsilon \right)\sigma {R}_{A}+{Q}_{A}\) \(-(1-{\mu }_{A}){E}_{A}^{2}/2-{C}_{3A}\) | \(-[{T}_{{rB}}+{F}_{B}+\left(1-\nu \right)\tau {R}_{B}+a{Q}_{A}\) \(-(1-{\mu }_{B}){E}_{B}^{2}/2-{C}_{3B}]\) |
\({E}_{7}\left(\mathrm{0,1,1}\right)\) | \(\left(\alpha -1\right){T}_{{rA}}+\left(\beta -1\right){T}_{{rB}}\) \(-{\left(1-\theta \right)C}_{1}-{C}_{3}\) | \(-[{Q}_{A}+\gamma {F}_{A}{+\alpha T}_{{rA}}+(1-\varepsilon )\sigma {R}_{A}\) \(-(1-{\mu }_{A}){E}_{A}^{2}/2-{C}_{3A}]\) | \(-[{Q}_{B}+{\delta F}_{B}{+\beta T}_{{rB}}+\left(1-\nu \right)\tau {R}_{B}\) \(-(1-{\mu }_{B}){E}_{B}^{2}/2-{C}_{3B}]\) |
\({E}_{8}\left(\mathrm{1,1,1}\right)\) | \(-[\left(\alpha -1\right){T}_{{rA}}+\left(\beta -1\right){T}_{{rB}}\) \(-{\left(1-\theta \right)C}_{1}-{C}_{3}]\) | \(-[{T}_{{rA}}+{F}_{A}+{Q}_{A}+(1-\varepsilon )\sigma {R}_{A}\) \(-(1-{\mu }_{A}){E}_{A}^{2}/2-{C}_{3A}]\) | \(-[{T}_{{rB}}+{F}_{B}+{Q}_{B}+\left(1-\nu \right)\tau {R}_{B}\) \(-(1-{\mu }_{B}){E}_{B}^{2}/2-{C}_{3B}]\) |
