Structural stability
This article lacks inline citations. (December 2010) |
In
Examples of such qualitative properties are numbers of
Structurally stable systems were introduced by
Definition
Let G be an
It is important to note that topological equivalence is realized with a loss of smoothness: the map h cannot, in general, be a diffeomorphism. Moreover, although topological equivalence respects the oriented trajectories, unlike topological conjugacy, it is not time-compatible. Thus, the relevant notion of topological equivalence is a considerable weakening of the naïve C1 conjugacy of vector fields. Without these restrictions, no continuous time system with fixed points or periodic orbits could have been structurally stable. Weakly structurally stable systems form an open set in X1(G), but it is unknown whether the same property holds in the strong case.
Examples
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
Structural stability of non-singular smooth vector fields on the
History and significance
Structural stability of the system provides a justification for applying the qualitative theory of dynamical systems to analysis of concrete physical systems. The idea of such qualitative analysis goes back to the work of
When Smale started to develop the theory of hyperbolic dynamical systems, he hoped that structurally stable systems would be "typical". This would have been consistent with the situation in low dimensions: dimension two for flows and dimension one for diffeomorphisms. However, he soon found examples of vector fields on higher-dimensional manifolds that cannot be made structurally stable by an arbitrarily small perturbation (such examples have been later constructed on manifolds of dimension three). This means that in higher dimensions, structurally stable systems are not dense. In addition, a structurally stable system may have transversal homoclinic trajectories of hyperbolic saddle closed orbits and infinitely many periodic orbits, even though the phase space is compact. The closest higher-dimensional analogue of structurally stable systems considered by Andronov and Pontryagin is given by the Morse–Smale systems.
See also
References
- ISSN 0036-1445.
- ISBN 0-387-96649-8.
- D. V. Anosov (2001) [1994], "Rough system", Encyclopedia of Mathematics, EMS Press
- Charles Pugh and Maurício Matos Peixoto (ed.). "Structural stability". Scholarpedia.