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. © 1975 Fondazione Annali di Matematica Pura ed Applicata

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This paper concerns the analysis of transferring stability properties from an invariant manifold to the\ud whole space for an ordinary differential system. In previous papers we already treated this problem\ud in the case of asymptotic and total stability. Here we deal with the case of non-asymptotic stability.\ud We generalize to differential systems depending on time a reduction principle [Kelley in J. Math. Anal. 18:336-344, 1967; Pliss in Izv.Nauk SSSR Mat Ser 28:1297-1324, 1964)] relative to autonomous\ud systems. Our procedure is very different from the fixed point theorem argument used in [Kelley in J. Math. Anal. 18:336-344, 1967],\ud and it is based on the use of a suitable Liapunov function. Some results concerning integral stability\ud are also given

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In the first part of the paper we give a short review of our recent results concerning the relationship between conditional and unconditional stability properties of time dependent sets, under smooth di¤erential systems in R n. More precisely, let M be an ''s-compact'' invariant set in R Â R n and let F be a smooth invariant set in R Â R n containing M. It is assumed that M is uniformly asymptotically stable with respect to the perturbations lying on F. The unconditional stability properties of M depend on the stability properties of F ''near M''. This dependence has been analyzed in general, and, in the periodic case, complete characterizations are obtained. In the second part, the above results have been applied to bifurcation problems for periodic di¤erential systems. Some our previous statements on the matter are revisited and enriched.

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A Bruno Finzi nel sue 70me compleanno.Santo. -Si generalizzano condizioni sufficienti per la stabilitdb uni/ormeme~te (~siutotica e per l'instabilit~ espresse da atcttni teoremi di Matrosov. I risultati ottenuti so~o utiliz. zabili anche nel case che i secondi membri delle equazioni del mote siano funzioni non limitate.