Introduction to Dynamical Systems

  • 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 present without proofs notions and facts on the dynamical systems which are necessary to understand this book. We recall the notion of an invariant set and show the most important examples of such sets: fixed and periodic points, \(\omega -\) and \(\alpha -\)limit sets, wandering and nonwandering sets, chain recurrent sets, topologically transitive sets. We discuss the notion of stability of a dynamical system with respect to one of its characteristics, structural stability and \(\varOmega -\)stability in particular. We consider hyperbolic invariant sets, recall the theorem on existence of the stable and the unstable manifold for a point of such a set. We recall Hartman-Grobman theorem that a diffeomorphism in a neighborhood of a hyperbolic periodic point is topologically conjugate to its linearizion. We give the topological classification of hyperbolic fixed points. We present a brief explanation of the results of the “epoch of the hyperbolic revolution” begun in 1960s with the classical works by S. Smale and D. Anosov. We show the relations between the nonwandering set, the chain recurrent set and the limit set of a dynamical system (diffeomorphism). We present Smale’s spectral decomposition theorem which allows us to represent a hyperbolic nonwandering set of a diffeomorphism as the union of the disjoint closed transitive (basic) sets if the nonwandering set is the closure of the periodic points. We recall the criteria of structural and \(\varOmega -\)stability. The concepts of the symbolic dynamics, the reverse limit and the solenoid are presented as the means to describe the restriction of a dynamical system to its invariant set with complex dynamics. For more details see for example the books [17, 24, 28, 33, 43, 46], the surveys [3, 4, 5, 6, 48] and the papers [1, 7, 8, 9, 11, 13, 14, 18, 23, 26, 27, 29, 30, 34, 47, 51].


Phase Portrait Periodic Point Unstable Manifold Homoclinic Orbit Topological Invariant 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Adler, R., Konheim, A.G., McAndrew, M.: Topological entropy. Trans. Am. Math. Soc. 114, 309–319 (1965). doi: 10.2307/1994177 MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Andronov, A., Pontryagin, L.: Rough systems. Dokl. Akad. Nauk SSSR 14(5), 247–250 (1937)Google Scholar
  3. 3.
    Anosov, D.: Geodesic flows on closed Riemann manifolds with negative curvature. In: Proceedings of the Steklov Institute of Mathematics, vol. 90. MAIK Nauka/Interperiodica, Pleiades Publishing, Moscow; Springer (1967)Google Scholar
  4. 4.
    Anosov, D.: About one class of invariant sets of smooth dynamical systems. Proc. Int. Conf. Non-linear Oscillation 2, 39–45 (1970)Google Scholar
  5. 5.
    Anosov, D.: Structurally stable systems. Proc. Steklov Inst. Math. 169, 61–95 (1986)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Anosov, D., Solodov, V.: Hyperbolic sets. In: Itogi Nauki Tekhniki; Sovremennye Problemy Matematiki, Fundamental’nye Napravleniya, Dynamical Systems 9, VINITI, Akad. Nauk SSSR, vol. 66, pp. 12–99. Moscow (1991) (Russian), English translation in Encyclopaedia of Mathematical Sciences, Dynamical Systems IX, vol. 66. Springer, Berlin (1995)Google Scholar
  7. 7.
    Bowen, R.: Topological entropy and Axiom A. Proc. Symp. Pure Math. 14, 23–41 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bowen, R.: Periodic points and measures for Axiom A diffeomorphisms. Trans. Am. Math. Soc. 154, 377–397 (1971)Google Scholar
  9. 9.
    Dankner, A.: On Smale’s Axiom A dynamical systems. Ann. Math. 107, 517–553 (1978)Google Scholar
  10. 10.
    Doering, C.: Persistently transitive flows on three-dimensional manifolds. In: Camacho, M.I., Pacifico, M.J., Takens, F. (eds.) Dynamical Systems and Bifurcation Theory, Pitman Research Notes Mathematics Series, vol. 160, pp. 59–89. Pitman, London (1987)Google Scholar
  11. 11.
    Franks, J.: Anosov diffeomorphisms. Proc. Symp. Pure Math. 14, 61–94 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Franks, J.: Necessary conditions for stability of diffeomorphisms. Trans. Am. Math. Soc. 158(2), 301–308 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Grines, V., Zhuzhoma, E.: On structurally stable diffeomorphisms with codimension one expanding attractors. Trans. Am. Math. Soc. 357(2), 617–667 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Grines, V., Zhuzhoma, E.: Dynamical systems with nontrivially recurrent invariant manifolds. Dynamics, Games and Science I, pp. 421–470. Springer, Berlin (2011)Google Scholar
  15. 15.
    Hayashi, S.: Diffeomorphisms in \(^1(M)\) satisfy Axiom A. Ergod. Theory Dyn. Syst. 12, 233–253 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hu, S.: A proof of \(C^1\) stability conjecture for three-dimensional flows. Trans. Am. Math. Soc. 342(2), 753–772 (1994)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge (1995)CrossRefzbMATHGoogle Scholar
  18. 18.
    Kurata, M., et al.: Hyperbolic nonwandering sets without dense periodic points. Nagoya Math. J. 74, 77–86 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Maier, A.: A rough transformation of circle into circle. Uchenye Zap. Univ. 12, 215–229 (1939)Google Scholar
  20. 20.
    Mañé, R.: Contributions to the stability conjecture. Topology 17(4), 383–396 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Mañé, R.: An ergodic closing lemma. Ann. Math. 116, 503–540 (1982)Google Scholar
  22. 22.
    Mañé, R.: A proof of the \(C^1\) stability conjecture. Publications Mathématiques de l’IHÉS 66, 161–210 (1987)CrossRefzbMATHGoogle Scholar
  23. 23.
    Manning, A.: There are no new Anosov diffeomorphisms on tori. Am. J. Math. 96, 422–429 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Mitropol’skii, Y.A. (ed.): Ninth summer math school. Math. Inst. Acad. Sci. Ukraine. Kiev (1972)Google Scholar
  25. 25.
    Moser, J.: On a theorem of Anosov. J. Differ. Equs. 5, 411–440 (1969)Google Scholar
  26. 26.
    Newhouse, S.E.: On codimension one Anosov diffeomorphisms. Am. J. Math. 92(3), 761–770 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Newhouse, S., Palis, J.: Hyperbolic nonwandering sets on two-dimensional manifolds. Dynamical Systems: Proceedings, p. 293. Academic Press, New York (1973)Google Scholar
  28. 28.
    Nitecki, Z.: Differentiable dynamics: An introduction to the orbit structure of diffeomorphisms. MIT Press, Cambridge (1971)Google Scholar
  29. 29.
    Palis, J.: On Morse-Smale dynamical systems. Topology 8(4), 385–404 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Palis, J.: A note on \(\varOmega \)-stability. Proc. Symp. Pure Math. 14, 221–222 (1970)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Palis, J.: \(\varOmega \)-explosões. Bull. Braz. Math. Soc. 1(1), 55–56 (1970)CrossRefGoogle Scholar
  32. 32.
    Palis, J.: \(\varOmega \)-stability and explosions. Lect. Notes Math. 1206, 40–42 (1971)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Palis, J., De Melo, W.: Geometric Theory of Dynamical Systems. Springer, Berlin (1982)Google Scholar
  34. 34.
    Palis, J., Smale, S.: Structural stability theorems. Proc. Symp. Pure Math. 14, 223–231 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Peixoto, M.M.: Structural stability on two-dimensional manifolds. Topology 1(2), 101–120 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Peixoto, M.M.: Structural stability on two-dimensional manifolds: a further remark. Topology 2(1), 179–180 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Pugh, C.C.: An improved closing lemma and a general density theorem. Am. J. Math. 89(4), 1010–1021 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Pugh, C.C., Robinson, C.: The \(C^1\) closing lemma, including hamiltonians. Ergodic Theory Dyn. Syst. 3(02), 261–313 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Robbin, J.W.: A structural stability theorem. Ann. Math. 94, 447–493 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Robinson, C.: \(C^r\) structural stability implies kupka-smale. In: Peixoto, M. (ed.) Dynamical Systems, pp. 443–449. Academic, New York (1973)CrossRefGoogle Scholar
  41. 41.
    Robinson, C.: Structural stability of vector fields. Ann. Math. 99, 154–175 (1974)Google Scholar
  42. 42.
    Robinson, C.: Structural stability of \(C^1\) diffeomorphisms. J. Differ. Equs. 22, 28–73 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Robinson, C.: Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, vol. 28. CRC Press, Boca Raton (1999)zbMATHGoogle Scholar
  44. 44.
    Robinson, R.C.: Structural stability of \(C^1\) flows. Dynamical Systems-Warwick 1974, pp. 262–275. Springer, Berlin (1975)Google Scholar
  45. 45.
    Sannami, A.: The stability theorems for discrete dynamical systems on two-dimensional manifolds. Nagoya Math. J. 90, 1–55 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Shilnikov, L.P., Shilnikov, A.L., Turaev, D.V., Chua, L.O.: Methods of Qualitative Theory in Nonlinear Dynamics, vol. 5. World Scientific, Singapore (2001)Google Scholar
  47. 47.
    Smale, S.: Dynamical systems and the topological conjugacy problem for diffeomorphisms. Proc. Int. Congr. Math. 1962, 490–496 (1963)MathSciNetGoogle Scholar
  48. 48.
    Smale, S.: Differentiable dynamical systems. Bull. Am. Math. Soc. 73(6), 747–817 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Smale, S.: The \(\varOmega \)-stability theorem. Proc. Symp. Pure Math. 14, 289–297 (1970)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Tao, L.S.: On the stability conjecture. Chin. Ann. Math. Ser. B 1(1), 8–30 (1980)MathSciNetGoogle Scholar
  51. 51.
    Williams, R.F.: Expanding attractors. Publications Mathématiques de l’IHES 43(1), 169–203 (1974)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

Personalised recommendations