Structural stability
In mathematics, structural stability is a fundamental property of a dynamical system which means that the qualitative behavior of the trajectories is unaffected by small perturbations (to be exact C1-small perturbations).
Examples of such qualitative properties are numbers of fixed points and periodic orbits (but not their periods). Unlike Lyapunov stability, which considers perturbations of initial conditions for a fixed system, structural stability deals with perturbations of the system itself. Variants of this notion apply to systems of ordinary differential equations, vector fields on smooth manifolds and flows generated by them, and diffeomorphisms.
Structurally stable systems were introduced by Aleksandr Andronov and Lev Pontryagin in 1937 under the name "systèmes grossiers", or rough systems. They announced a characterization of rough systems in the plane, the Andronov–Pontryagin criterion. In this case, structurally stable systems are typical, they form an open dense set in the space of all systems endowed with appropriate topology. In higher dimensions, this is no longer true, indicating that typical dynamics can be very complex (cf strange attractor). An important class of structurally stable systems in arbitrary dimensions is given by Anosov diffeomorphisms and flows.
Let G be an open domain in Rn with compact closure and smooth (n−1)-dimensional boundary. Consider the space X1(G) consisting of restrictions to G of C1vector fields on Rn that are transversal to the boundary of G and are inward oriented. This space is endowed with the C1metric in the usual fashion. A vector field F ∈ X1(G) is weakly structurally stable if for any sufficiently small perturbation F1, the corresponding flows are topologically equivalent on G: there exists a homeomorphism h: G → G which transforms the oriented trajectories of F into the oriented trajectories of F1. If, moreover, for any ε > 0 the homeomorphism h may be chosen to be C0ε-close to the identity map when F1 belongs to a suitable neighborhood of F depending on ε, then F is called (strongly) structurally stable. These definitions extend in a straightforward way to the case of n-dimensional compact smooth manifolds with boundary. Andronov and Pontryagin originally considered the strong property. Analogous definitions can be given for diffeomorphisms in place of vector fields and flows: in this setting, the homeomorphism h must be a topological conjugacy.
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