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 Henri Poincaré on the three-body problem in celestial mechanics. Around the same time, Aleksandr Lyapunov rigorously investigated stability of small perturbations of an individual system. In practice, the evolution law of the system (i.e. the differential equations) is never known exactly, due to the presence of various small interactions. It is, therefore, crucial to know that basic features of the dynamics are the same for any small perturbation of the "model" system, whose evolution is governed by a certain known physical law. Qualitative analysis was further developed by George Birkhoff in the 1920s, but was first formalized with introduction of the concept of rough system by Andronov and Pontryagin in 1937. This was immediately applied to analysis of physical systems with oscillations by Andronov, Witt, and Khaikin. The term "structural stability" is due to Solomon Lefschetz, who oversaw translation of their monograph into English. Ideas of structural stability were taken up by Stephen Smale and his school in the 1960s in the context of hyperbolic dynamics. Earlier, Marston Morse and Hassler Whitney initiated and René Thom developed a parallel theory of stability for differentiable maps, which forms a key part of singularity theory. Thom envisaged applications of this theory to biological systems. Both Smale and Thom worked in direct contact with Maurício Peixoto, who developed Peixoto's theorem in the late 1950's.
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.
Read more about this topic: Structural Stability
Famous quotes containing the words history and/or significance:
“I am not a literary man.... I am a man of science, and I am interested in that branch of Anthropology which deals with the history of human speech.”
—J.A.H. (James Augustus Henry)
“The hypothesis I wish to advance is that ... the language of morality is in ... grave disorder.... What we possess, if this is true, are the fragments of a conceptual scheme, parts of which now lack those contexts from which their significance derived. We possess indeed simulacra of morality, we continue to use many of the key expressions. But we have—very largely if not entirely—lost our comprehension, both theoretical and practical, of morality.”
—Alasdair Chalmers MacIntyre (b. 1929)