Where a flow becomes unstable
Linearise the equations you already wrote about a base state and ask for their eigenmodes, with the residual of each (left eigenvectors on request). Follow the least stable one as a parameter changes and find the critical point where it starts to grow. Ask for a drift check and the modes that do not survive a finer resolution are removed, and you are told.
Tutorial 1: A first eigenvalue problem: Orr–Sommerfeld · Tutorial 8: Polar geometry: a drum, a pipe, two rotating cylinders
Symmetry subspaces
Declare a reflection or a rotation and solve in the symmetric subspace: the same eigenvalues for a fraction of the cost, after a check that the equations respect the symmetry.
Tutorial 3: Symmetries and symmetry subspaces
Bifurcations, followed
Follow a branch of steady solutions as a parameter moves, through folds, by pseudo-arclength continuation. Pitchforks and Hopf points are detected and located on the way; switch branches, follow the cycles born at a Hopf point, continue a fold or a Hopf point in two parameters, and read the normal form that says what happens next.
Tutorial 4: Nonlinear boundary-value problems and continuation: Bratu · Tutorial 10: Bifurcations: branch switching, cycles, codimension 2, normal forms
Equilibria, travelling waves and periodic orbits
Find the invariant solutions that organise a turbulent or chaotic flow, from a guess or from the state of a running simulation, by Newton–Krylov with a hookstep, and compute their stability. Each solution is reported with its residual.
Tutorial 5: Invariant solutions: Kuramoto–Sivashinsky, L = 22
Time-periodic flows: Floquet analysis
When the base flow oscillates, as in a pulsating channel or pipe, stability is a question about periodic motion. Conatus answers it with a Floquet–Hill method (Fourier in time, spectral in space), for any time-periodic linear problem written in the same language: give the base flow as a function of time and its period, and ask for its Floquet exponents or its optimal growth.
Transient growth
Stable flows can still amplify a disturbance enormously before it decays. Find the optimal perturbation and its growth, about a steady or a time-periodic base flow, in the energy norm you declare.
Chaos: Lyapunov exponents and ensembles
Lyapunov spectra, covariant vectors, the Kaplan–Yorke dimension, Poincaré maps and basins of attraction, from ensembles of trajectories integrated together.
Tutorial 12: Chaos and ensembles: Lyapunov spectrum, covariant vectors, Poincaré map, basins
Adjoints, sensitivity and data
The physical adjoint of an operator in the inner product you declare (never a bare transpose), direct and adjoint global modes, the wavemaker and base-flow sensitivity; the exact adjoint of the time stepper for gradients, in a channel or a spherical shell, with the nonlinear optimal perturbation that grows most at a given energy. DMD and SPOD of data in the same norm.
Tutorial 13: Where a global mode is born, and what the data say about it