General Properties of the Morse–Smale Diffeomorphisms

  • Viacheslav Z. GrinesEmail author
  • Timur V. Medvedev
  • Olga V. Pochinka
Part of the Developments in Mathematics book series (DEVM, volume 46)


In this chapter we consider an important class of discrete structurally stable systems which adequately describe processes with regular dynamics, Morse-Smale diffeomorphisms. We present with proofs the properties of Morse-Smale diffeomorphisms which are necessary for the topological classification. The asymptotic behavior and the embedding into the ambient manifold (the phase space) of the stable and the unstable manifolds of the saddle periodic points plays the key role in understanding of the dynamics of such diffeomorphisms. To describe the topological invariants which reflect these properties we consider the space of wandering orbits which belong to some specially chosen invariant sets of the diffeomorphism. We describe the important (for the subsequent results) construction of the sequence of the “attractor-repeller” pairs suggested by C. Conley. This construction is based on introduction of an order on the set of the periodic orbits which satisfies the Smale partial relation. The proof of existence of a trapping neighborhood of an attractor (a repeller) relies on the local Morse-Lyapunov function constructed in this chapter. All the proofs are presented for the class \(MS(M^n)\) of the orientation preserving Morse-Smale diffeomorphisms \(f:M^n\rightarrow M^n\) on an orientable manifold \(M^n\). The results are partly announced and proved in the surveys [1, 2, 3, 9] and the papers [4, 5, 6, 7, 8].


Periodic Orbit Phase Portrait Periodic Point Invariant Manifold Unstable Manifold 
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  1. 1.
    Bonatti, C., Grines, V., Langevin, R.: Dynamical systems in dimension 2 and 3: conjugacy invariants and classification. The geometry of differential equations and dynamical systems. Comput. Appl. Math. 20(1–2), 11–50 (2001)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Grines, V., Pochinka, O.: On topological classification of Morse-Smale diffeomorphisms. Dynamics, Games and Science II, pp. 403–427. Springer, Berlin (2011)Google Scholar
  3. 3.
    Grines, V., Pochinka, O.: Morse-Smale cascades on 3-manifolds. Russ. Math. Surv. 68(1), 117–173 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Grines, V., Laudenbach, F., Pochinka, O., et al.: Self-indexing energy function for Morse-Smale diffeomorphisms on 3-manifolds. Mosc. Math. J. 9(4), 801–821 (2009)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Grines, V.Z., Zhuzhoma, E.V., Medvedev, V.S., Pochinka, O.V.: Global attractor and repeller of Morse-Smale diffeomorphisms. Proc. Steklov Inst. Math. 271(1), 103–124 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Grines, V., Laudenbach, F., Pochinka, O.: Dynamically ordered energy function for Morse-Smale diffeomorphisms on 3-manifolds. Proc. Steklov Inst. Math. 278, 34–48 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Palis, J.: On Morse-Smale dynamical systems. Topology 8(4), 385–404 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Palis, J., Smale, S.: Structural stability theorems. Proc. Symp. Pure Math. 14, 223–231 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Smale, S.: Differentiable dynamical systems. Bull. Am. Math. Soc. 73(6), 747–817 (1967)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Viacheslav Z. Grines
    • 1
    Email author
  • Timur V. Medvedev
    • 2
    • 3
  • Olga V. Pochinka
    • 1
  1. 1.Department of Fundamental MathematicsNational Research University Higher School of EconomicsNizhny NovgorodRussia
  2. 2.Department of Differential Equations, Mathematical and Numerical AnalysisNizhny Novgorod State UniversityNizhny NovgorodRussia
  3. 3.Laboratory of Algorithms and Technologies for Networks AnalysisNational Research University Higher School of EconomicsNizhny NovgorodRussia

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