Liapunov direct method in approaching bifurcation problems

  • 39 Accesses

  • 20 Citations


A concept of total stability for continuous or discrete dynamical systems and a generalized definition of bifurcation are given: it is possible to show the link between an abrupt change of the asymptotic behaviour of a family of flows and the arising of new invariant sets, with determined asymptotic properties. The theoretical results are a contribution to the study of the behaviour of flows near an invariant compact set. They are obtained by means of an extension of Liapunov's direct method.


  1. [1]

    G. N. Dubosin, Trudy gos. astron. Inst. Sternberg,14, no. 1 (1940).

  2. [2]

    L. Landau - E. Lifschitz,Mécanique des fluides, Editions Mir, Moscou (1971).

  3. [3]

    D. Ruelle -F. Takens, Comm. Math. Phys.,20 (1971), pp. 167–191.

  4. [4]

    E. Hopf, Math. Phys. Kl. Sachs. Akad. Wiss. Leipzig,94 (1942), pp. 1–22.

  5. [5]

    D. H. Sattinger, J. Math. and Mech.,19 (1970), pp. 797–817.

  6. [6]

    D. Ruelle -F. Takens, Comm. Math. Phys.,23 (1971), pp. 343–344.

  7. [7]

    N. P. Bathia -G. P. Szego,Stability theory of dynamical systems, Springer, Berlin (1970).

  8. [8]

    F. Marchetti -P. Negrini -L. Salvadori -M. Scalia, Atti del II Congresso AIMETA, vol. I, Napoli (1974), pp. 105–116.

  9. [9]

    P. Seibert, Proc. Intern. Symp. Non-lin. Diff. Eq. and Non-lin. Mech., Academic Press (1963), pp. 463–473.

  10. [10]

    S. Gorsin, VMU,6, no. 3 (1951), pp. 15–24.

  11. [11]

    I. G. Malkin, PMM,6 (1942), pp. 411–448.

  12. [12]

    T. Yoshizawa,Stability theory by Liapunov's Second Method, The Math. Soc. of Japan (1966).

  13. [13]

    G. Sansone - R. Conti,Non-linear differential equations, Pergamon Press (1964).

Download references

Author information

Additional information

A Dario Graffi nel suo 70° compleanno

Entrata in Redazione il 9 febbraio 1975.

Work performed under the auspices of the Italian Council of Research (C.N.R.).

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Marchetti, F., Negrini, P., Salvadori, L. et al. Liapunov direct method in approaching bifurcation problems. Annali di Matematica 108, 211–226 (1976).

Download citation


  • Dynamical System
  • Asymptotic Behaviour
  • Theoretical Result
  • Abrupt Change
  • Asymptotic Property