Differential equations | Stability theory | Dynamical systems

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. (Wikipedia).

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From playlist Dr. Structure: Structural Analysis Video Lectures

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From playlist Dr. Structure: Structural Analysis Video Lectures

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Related pages

Lyapunov stability | Jacobian matrix and determinant | Dynamical system | Homeomorphism | Fixed point (mathematics) | Hassler Whitney | Denjoy's theorem on rotation number | Rotation number | Topological conjugacy | Boundary (topology) | Andronov–Pontryagin criterion | Torus | Aleksandr Lyapunov | Peixoto's theorem | Dense set | Hyperbolic equilibrium point | Three-body problem | Mathematics | Ordinary differential equation | Diffeomorphism | Limit cycle | Henri Poincaré | Self-stabilization | Stability theory | Anosov diffeomorphism | Marston Morse | Dmitri Anosov | Morse–Smale system | Arnold's cat map | Flow (mathematics) | Superstabilization | Singularity theory | Circle | Open set | Vector field