Explain the term ‘parameter structural stability’?

Structural stability is a natural concept and demands broader influence. Here is a Bourbaki-style definition into which the dynamics structural stability notion fits nicely.

Definition. If a set is equipped with a topology and an equivalence relation then its structurally stable elements are those interior to the equivalence classes. The "structure" is whatever is preserved by the equivalence relation; its structure remains the same when a structurally stable element is perturbed.

For discrete dynamical systems the set is D=Diffeo(M) , equipped with the C1 topology, and the equivalence relation is topological conjugacy. For flows the space is X and the equivalence relation is orbit equivalence. Discrete dynamical systems and flows are actions by the groups Z and R . For actions of more general groups the equivalence relation is similar: orbits are sent to orbits by a homeomorphism.

Structural stability behaves non-trivially on restrictions or extensions of the space and equivalence relation. If X is the rotation vector field on the 2-sphere (its orbits are the latitudes and the poles) then X is structurally stable when considered in the subspace of X consisting of divergence free vector fields, but it is not structurally stable in X . Likewise, if the equivalence relation of topological conjugacy in D is changed to smooth conjugacy then the set of structurally stable diffeomorphisms is empty.