Equilibrium of an autonomous differential equation

An equilibrium is a state where the autonomous vector field vanishes.

Linear stability analysis classifies a hyperbolic equilibrium from the eigenvalues of the vector field's Jacobian.

A conservative equilibrium with one real pair of linear eigenvalues $\pm \lambda$ and one imaginary pair $\pm i\omega$ has saddle and centre directions. Generic perturbations have an exponentially growing component, even though finely chosen initial conditions can remain in the centre-stable subspace. The local L2 Lagrange point dynamics is an example.

A sideband instability is growth of disturbances with wavenumbers slightly displaced from a carrier pattern's wavenumber. A real physical perturbation uses the displaced mode together with its complex conjugate; a complex amplitude equation can couple those two sidebands through its cubic term. The Eckhaus instability is a longitudinal phase-modulation example.

An oscillatory normal mode whose amplitude grows, corresponding to a complex growth rate with positive real part.

For a real $2\times 2$ Jacobian matrix $J$, both eigenvalue have negative real part exactly whenThe linearization stability theorem then makes a hyperbolic equilibrium locally asymptotically stable.