Variants of this notion apply to systems of ordinary differential equations, vector fields on smooth manifolds and flows generated by them, and diffeomorphisms.
Weakly structurally stable systems form an open set in X1(G), but it is unknown whether the same property holds in the strong case.
Necessary and sufficient conditions for the structural stability of C1 vector fields on the unit disk D that are transversal to the boundary and on the two-sphere S2 have been determined in the foundational paper of Andronov and Pontryagin.
According to the Andronov–Pontryagin criterion, such fields are structurally stable if and only if they have only finitely many singular points (equilibrium states) and periodic trajectories (limit cycles), which are all non-degenerate (hyperbolic), and do not have saddle-to-saddle connections.
In particular, structurally stable vector fields in two dimensions cannot have homoclinic trajectories, which enormously complicate the dynamics, as discovered by Henri Poincaré.
Structural stability of non-singular smooth vector fields on the torus can be investigated using the theory developed by Poincaré and Arnaud Denjoy.
The idea of such qualitative analysis goes back to the work of Henri Poincaré on the three-body problem in celestial mechanics.
In practice, the evolution law of the system (i.e. the differential equations) is never known exactly, due to the presence of various small interactions.
It is, therefore, crucial to know that basic features of the dynamics are the same for any small perturbation of the "model" system, whose evolution is governed by a certain known physical law.
Qualitative analysis was further developed by George Birkhoff in the 1920s, but was first formalized with introduction of the concept of rough system by Andronov and Pontryagin in 1937.