Morse-Smale system

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In the theory of dynamic systems , various flows and dynamic systems can be summarized under the term Morse-Smale systems . Morse-Smale systems are structurally stable ; H. their qualitative behavior does not change with slight disturbances of the parameters.

definition

Gradient flow of the height function of an embedded torus . The stable manifold of one saddle point agrees with the unstable manifold of the other saddle point, so the transversality condition is not fulfilled.
Example of a Morse-Smale system: gradient flow of the height function of a slightly tilted torus. The stable and unstable manifolds of the saddle points do not intersect, the transversality condition is fulfilled.

A dynamic system is a Morse-Smale system if it meets the following conditions:

example

The gradient flow of a Morse function is Morse-Smale if all stable and unstable manifolds are transverse to each other. The non-migrating set then consists exclusively of fixed points.

Structural stability

Morse-Smale systems are structurally stable. The flow of a vector field on a surface is structurally stable if and only if it is Morse-Smale. But in higher dimensions there are examples of structurally stable systems that are not Morse-Smale.

literature

  • J. Palis, S. Smale, "Structural stability theorems" S.-S. Chern (ed.) S. Smale (ed.), Global analysis, Proc. Symp. Pure Math., 14, Amer. Math. Soc. (1970) pp. 223-232
  • AA Andronov, EA Leontovich, II Gordon, AG Maier, "Theory of bifurcations of dynamic systems on a plane", Israel Program Sci. Transl. (1971)
  • AA Andronov, EA Leontovich, II Gordon, AG Maier, "Qualitative theory of second-order dynamic systems", Wiley (1973)
  • AG Maier, "A structurally stable map of the circle onto itself" Uchen. Zap. Gor'k. Gos. Inst., 12 (1939) pp. 215-229 (Russian)
  • VA Pliss, "On the structural stability of differential equations on the torus" Vestnik Leningrad. Univ. Ser. Mat., 15: 13 (1960) pp. 15–23 (Russian)
  • VI Arnol'd, "Small denominators I. Mapping the circle onto itself" Transl. Amer. Math. Soc. (2), 46 (1965) pp. 213-284 Izv. Akad. Nauk SSSR Ser. Mat., 25: 1 (1961) pp. 21-86
  • VI Arnol'd, "Correction to" Small denominators, I. Mapping the circle onto itself "" Izv. Akad. Nauk SSSR Ser. Mat., 28: 2 (1964) pp. 479-480 (Russian)
  • S. Smale, "Morse inequalities for dynamical systems" Bull. Amer. Math. Soc. , 66 (1960) pp. 43-49
  • S. Smale, "On gradient dynamical systems" Ann. of Math. (2), 74 (1961) pp. 199–206 MR0133139 Zbl 0136.43702
  • M. Shub, "Morse-Smale diffeomorphisms are unipotent in homology," MM Peixoto (ed.), Dynamical Systems (Proc. Conf. Salvador, 1971), Acad. Press (1973) pp. 489-491
  • M. Shub, D. Sullivan, "Homology theory and dynamical systems" Topology, 14 (1975) pp. 109-132
  • D. Asimov, "Homotopy of non-singular vector fields to structurally stable ones" Ann. of Math., 102: 1 (1975) pp. 55-65
  • M. Peixoto, "Sur la classification des equations différentielles" CR Acad. Sci. Paris, 272 (1971) pp. A262-A265
  • MM Peixoto, "Dynamical systems" MM Peixoto (ed.), Dynamical Systems (Proc. Conf. Salvador, 1971), Acad. Press (1973) pp. 389-419
  • Ya.L. Umanskii, "The scheme of a 3-dimensional Morse-Smale dynamical system without closed trajectories" Soviet Math. Dokl. , 17 (1976) pp. 1479–1482 Docl. Akad. Nauk SSSR, 230: 6 (1976) pp. 1286-1289
  • S.Yu. Pilyugin, "Phase diagrams that determine Morse-Smale systems without periodic trajectories on spheres" Diff. Eq., 14: 2 (1978) pp. 170-177 Diff. Uravnen. , 14: 2 (1978) pp. 245-254
  • D. Neumann, T. O'Brien, "Global structure of continuous flows on 2-manifolds" J. Diff. Eq., 22: 1 (1976) pp. 89-110

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