Introduction to Dynamical Systems

  • Yuri A. Kuznetsov
Part of the Applied Mathematical Sciences book series (AMS, volume 112)


This chapter introduces some basic terminology. First, we define a dynamical system and give several examples, including symbolic dynamics. Then we introduce the notions of orbits, invariant sets, and their stability. As we shall see while analyzing the Smale horseshoe, invariant sets can have very complex structures. This is closely related to the fact discovered in the 1960s that rather simple dynamical systems may behave “randomly,” or “chaotically.” Finally, we discuss how differential equations can define dynamical systems in both finite- and infinite-dimensional spaces.


State Space Phase Portrait Evolution Operator Monodromy Matrix Vertical Strip 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliographical notes

  1. Birkhoff, G. (1966), Dynamical Systems, With an addendum by Jurgen Moser. American Mathematical Society Colloquium Publications, Vol. IX, American Mathematical Society, Providence, RI.Google Scholar
  2. Doedel, E., Keller, H. and Kernévez, J.-P. (1991b), `Numerical analysis and control of bifurcation problems: (II) Bifurcation in infinite dimensions’, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 1, 745–772.MathSciNetzbMATHCrossRefGoogle Scholar
  3. Dulac, M. (1923), `Sur les cycles limites’, Bull. Soc. Math. France 51, 45–188. Dumortier, F. (1978), Singularities of Vector Fields, Mathematical Monographs 32,IMPA, Rio de Janeiro.Google Scholar
  4. Nemytskii, V.V. and Stepanov, V.V. (1949), Qualitative Theory of Differential Equations,GITTL, Moscow-Leningrad. In Russian.Google Scholar
  5. Coddington, E. and Levinson, N. (1955), Theory of Ordinary Differential Equations, McGraw-Hill, New York.zbMATHGoogle Scholar
  6. Hille, E. and Phillips, R. (1957), Functional Analysis and Semigroups, American Mathematical Society, Providence, RI.Google Scholar
  7. Lorenz, E. (1963), `Deterministic non-periodic flow’, J. Atmos. Sci. 20, 130–141.CrossRefGoogle Scholar
  8. Shil’nikov, L.P. (1966), `On the generation of a periodic motion from a trajectory which leaves and re-enters a saddle-saddle state of equilibrium’, Soviet Math. Dokl. 7, 1155–1158.zbMATHGoogle Scholar
  9. Shil’nikov, L.P. (1967a), `The existence of a denumerable set of periodic motions in four-dimensional space in an extended neighborhood of a saddle-focus’, Soviet Math. Dokl. 8, 54–58.zbMATHGoogle Scholar
  10. Turaev, D.V. and Shil’nikov, L.P. (1995), `Blue sky catastrophes’, Dokl. Math. 51, 404–407.zbMATHGoogle Scholar
  11. Fenichel, N. (1971), `Persistence and smoothness of invariant manifolds for flows’, Indiana Univ. Math. J. 21, 193–226.MathSciNetzbMATHCrossRefGoogle Scholar
  12. Gaspard, P. (1983), `Generation of a countable set of homoclinic flows through bifurcation’, Phys. Lett. A 97, 1–4.MathSciNetCrossRefGoogle Scholar
  13. Khibnik, A.I. (1990), LINLBF: A program for continuation and bifurcation analysis of equilibria up to codimension three, in D. Roose, B. De Dier and A. Spence, eds, `Continuation and Bifurcations: Numerical Techniques and Applications (Leuven, 1989)’, Vol. 313 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Kluwer Acad. Publ., Dordrecht, pp. 283–296.Google Scholar
  14. Melnikov, V.K. (1962), Qualitative description of resonance phenomena in nonlinear systems, P-1013, OIJaF, Dubna. In Russian.Google Scholar
  15. Moser, J. (1973), Stable and Random Motions in Dynamical Systems, Princeton University Press, Princeton, NJ.Google Scholar
  16. Palis, J. and Pugh, C. (1975), Fifty problems in dynamical systems, in `Dynamical Systems (Warwick, 1974)’, Vol. 468 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, pp. 345–353.Google Scholar
  17. Sternberg, S. (1957), `Local contractions and a theorem of Poincaré’, Amer. J. Math. 79, 809–824.MathSciNetzbMATHCrossRefGoogle Scholar
  18. Hartman, P. (1964), Ordinary Differential Equations, Wiley, New York.zbMATHGoogle Scholar
  19. Holmes, P. and Rand, D. (1978), `Bifurcations of the forced van der Pol oscillator’, Quart. Appl. Math. 35, 495–509.MathSciNetzbMATHGoogle Scholar
  20. Moss, G., Arneodo, A., Coullet, P. and Tresser, C. (1981), Simple computation of bifurcating invariant circles for mappings, in D. Rand and L.-S. Young, eds, ‘Dynamical Systems and Turbulence’, Vol. 898 of Lecture Notes in Mathematics, Springer-Verlag, New York, pp. 192–211.Google Scholar
  21. Nitecki, Z. (1971), Differentiable Dynamics, MIT Press, Cambridge, MA.Google Scholar
  22. Palis, J. and Takens, F. (1993), Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations: Fractal Dimensions and Infinitely Many Attractors, Vol. 35 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge.Google Scholar
  23. Shil’nikov, A.L., Nicolis, G. Si Nicolis, C. (1995), `Bifurcation and predictability analysis of a low-order atmospheric circulation model’, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 5, 1701–1711.MathSciNetzbMATHCrossRefGoogle Scholar
  24. Temam, R. (1997), Infinite-dimensional Dynamical Systems in Mechanics and Physics, 2nd edition, Springer-Verlag, New York.Google Scholar
  25. Dieci, L. and Friedman, M. (2001), `Continuation of invariant subspaces’, Numer. Linear Algebra Appl. 8, 317–327.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Yuri A. Kuznetsov
    • 1
    • 2
  1. 1.Department of MathematicsUtrecht UniversityUtrechtThe Netherlands
  2. 2.Institute of Mathematical Problems of BiologyRussian Academy of SciencesPushchino, Moscow RegionRussia

Personalised recommendations