Dynamics

Everything Conatus does with a model once it is written: stability, growth, invariant solutions, their branches, their symmetries and their sensitivities.

The model you simulate is the model you analyse. A native model (cn.Model) gives its eigenvalue problem with m.eigenvalues(...) or m.linearize(about=...), its time stepper with m.simulate(...) (the flow map that every search for an invariant solution uses), and its compiled problem with m.compile(N).problem for the tools of conatus.expert. Every result carries the residual that validates it and an explain() that lists the choices made.

Page

Question it answers

Eigenvalues and stability

Is this state stable? Where does it become unstable?

Onset of convection

The critical Rayleigh number of convection and its mode, from a ready model or from the equations of a normal mode

Non-modal analysis

How much can a stable flow amplify a perturbation or a forcing?

Invariant solutions

Which equilibria, travelling waves and periodic orbits exist?

Continuation and bifurcations

How do they change with a parameter? Where are the folds and bifurcations?

Floquet analysis

Is a time-periodic flow stable?

Eigenvalues in a region

Which eigenvalues lie in this region of the complex plane, and are there any others?

Chaos and ensembles

How chaotic is the flow? Where do trajectories end up?

Invariant manifolds and slices

How are invariant solutions connected? How to remove a continuous symmetry?

Symmetry subspaces

How to compute in a symmetry subspace, and what that hides

Adjoints and sensitivity

Gradients, adjoint modes, the wavemaker and base-flow sensitivity

Tangent, adjoint and optimal perturbations

Which initial perturbation grows most, linearly or not (Cartesian and spherical shell)?

Data-driven analysis

What do snapshots of a flow say? DMD and SPOD in the exact norm

The snippets on these pages are executed as they are shown, at small resolutions so that they run in seconds; the tutorials carry each method to a published value.

Dynamics