Table 2 Stability classification of equilibrium points based on the Jacobian matrix.
Condition | Eigenvalue Nature | Equilibrium Type | Stability |
|---|---|---|---|
\(J < 0\) | Real, opposite signs | Saddle Point | Always unstable |
\(J > 0\), \(T^2 - 4J \ge 0\) | Real, same sign | Node | Stable if \(T < 0\), unstable if \(T > 0\) |
\(J > 0\), \(T^2 - 4J < 0\), \(T \ne 0\) | Complex conjugates | Focus | Stable if \(T < 0\), unstable if \(T > 0\) |
\(J > 0\), \(T = 0\) | Pure imaginary | Center | Neutrally stable (closed orbits) |
\(J = 0\), Poincaré index = 0 | Degenerate | Zero Point / Cusp | Indeterminate |
