Summary
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.
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References
G. N. Dubosin, Trudy gos. astron. Inst. Sternberg,14, no. 1 (1940).
L. Landau - E. Lifschitz,Mécanique des fluides, Editions Mir, Moscou (1971).
D. Ruelle -F. Takens, Comm. Math. Phys.,20 (1971), pp. 167–191.
E. Hopf, Math. Phys. Kl. Sachs. Akad. Wiss. Leipzig,94 (1942), pp. 1–22.
D. H. Sattinger, J. Math. and Mech.,19 (1970), pp. 797–817.
D. Ruelle -F. Takens, Comm. Math. Phys.,23 (1971), pp. 343–344.
N. P. Bathia -G. P. Szego,Stability theory of dynamical systems, Springer, Berlin (1970).
F. Marchetti -P. Negrini -L. Salvadori -M. Scalia, Atti del II Congresso AIMETA, vol. I, Napoli (1974), pp. 105–116.
P. Seibert, Proc. Intern. Symp. Non-lin. Diff. Eq. and Non-lin. Mech., Academic Press (1963), pp. 463–473.
S. Gorsin, VMU,6, no. 3 (1951), pp. 15–24.
I. G. Malkin, PMM,6 (1942), pp. 411–448.
T. Yoshizawa,Stability theory by Liapunov's Second Method, The Math. Soc. of Japan (1966).
G. Sansone - R. Conti,Non-linear differential equations, Pergamon Press (1964).
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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.).
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Marchetti, F., Negrini, P., Salvadori, L. et al. Liapunov direct method in approaching bifurcation problems. Annali di Matematica 108, 211–226 (1976). https://doi.org/10.1007/BF02413955
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DOI: https://doi.org/10.1007/BF02413955