Examples
Structural stability of C1 vector fields on the unit disk D that are transversal to the boundary and on the two-sphere S2 has 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. Furthermore, the non-wandering set of the system is precisely the union of singular points and periodic orbits. In particular, structurally stable vector fields in two dimensions cannot have homoclinic trajectories, which could 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. Using the Poincaré recurrence map, the question is reduced to determining structural stability of diffeomorphisms of the circle. As a consequence of the Denjoy theorem, an orientation preserving C2 diffeomorphism ƒ of the circle is structurally stable if and only if its rotation number is rational, ρ(ƒ) = p/q, and the periodic trajectories, which all have period q, are non-degenerate: the Jacobian of ƒq at the periodic points is different from 1, cf Circle map.
Dmitri Anosov discovered that hyperbolic automorphisms of the torus, such as the Arnold's cat map, are structurally stable. He then generalized this statement to a wider class of systems, which have since been called Anosov diffeomorphisms and Anosov flows. One celebrated example of Anosov flow is given by the geodesic flow on a surface of constant negative curvature, cf Hadamard billiards.
Read more about this topic: Structural Stability
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