Luigi Salvadori

Hopf bifurcation and related stability problems for periodic differential systems

This paper deals with Hopf bifurcation for periodic systems of ordinary differential equations. Denote by m the number of Floquet multipliers of the linearized part of our differential system which lie on the unit circle. The case in which m = 1 with a non-resonance condition imposed on the multipliers has been treated in [1, 2]. When m is greater than 1, this problem presents further difficulties. Here we will consider the case m = 2. In order to make more clear our exposition let us first consider the follow- ing autonomous two-dimensional system

First and invariant integrals in stability problems

Abstract We consider again the problem already treated in [Salvadori (Math. Japon. 49 (1) (1999) 1)] of unconditional stability properties of the solution x=0 of a smooth differential system x =f(t,x) , f(t,0)≡0, for which x=0 is uniformly asymptotically stable for perturbations lying on an appropriate invariant set Φ. Precisely in [Salvadori (1999)] it is assumed that Φ={(t,x) : F(t,x)=0} , where F, F(t,0)≡0, is a smooth first integral. Previously to our research the problem was analyzed for autonomous systems in [Aeyels (Systems and Control Letters, Vol. 19, North-Holland, Amsterdam, 1992)] and for periodic systems in [Peiffer (Rend. Sem. Mat. Univ. Padova, 92 (1994) 165)]. In the present paper we weaken the sufficient condition for the stability of the origin given in [Salvadori (1999)] (Φ-positive definitiveness of F). The sufficiency is preserved but the new condition is also necessary for uniform stability and then necessary and sufficient for stability in the periodic case. The results follow from the connections which have been found between the stability of the origin and the stability of the set Φ (for perturbations close to the origin). Even the asymptotic stability of x=0 appears to be connected to corresponding asymptotic stability properties of Φ. If the differential system is periodic, the origin and Φ have the same stability properties. If the differential system is autonomous the results are extendable to the stability of a compact invariant set M. In particular it is shown that M and Φ have the same stability properties. This statement still holds if F=0 is a t-independent invariant integral, with F not necessarily a first integral. This latter result is illustrated through its application to a bifurcation problem.

Recent results on the stability of time dependent sets and their application to bifurcation problems

— 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. More precisely, let M be an ‘‘s-compact’’ invariant set in R R and let F be a smooth invariant set in R R 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.

Perturbed dynamical systems: Displacement and bifurcation functions☆☆☆

where XE R, yE I?“-‘. We suppose f, h, q, (T are suffkiently smooth functions, 2n-periodic in t, and5 h vanish together with their first derivatives in x, y at x = JJ = 0. We also suppose that A is a constant matrix, and a nonresonance condition is satisfied. For those q, u small in an appropriate topology, the problem of the existence and stability of the 2x-periodic solutions of (*) was considered by deoliveira and Hale [ 1 ] by using the Liapounov-Schmidt method. Precisely, this method reduces the problem of the existence of 2x-periodic solutions to the discussion of the zeros of the bifurcation function G(a, q, o). The bifurcation function also contains the qualitative information on the stability of periodic solutions. Precisely [ 1 ] if the eigenvalues of A have negative real part, then the stability properties of a periodic solution corresponding to a root a* of G(a, q, a) are the same as the stability properties of a* as a solution of the equation ci = G(a, q, a). In other words, these stability properties are dependent on the sign that G(a, q, a) assumes on the rightand left-hand sides of a*. In particular these results are interesting in approaching Hopf bifurcation, since an appropriate

Generalized Hopf Bifurcation in Rn and h-Asymptotic Stability

Publisher Summary This chapter discusses the generalized Hopf bifurcation in R n and h-asymptotic stability. The concept and analysis of structural stability of two dimensional autonomous differential systems was provided by Andronov and Pontryagin. Within the next 15 years, Andronov and his group developed a major part of the theory of two dimensional structurally stable and unstable system of differential equations. Motivated by problems in classical and celestial mechanics, mathematicians and physicists have been very concerned with providing a qualitative description of the trajectories of perturbed systems in a neighborhood of a structurally unstable equilibrium point in R n (n ≥ 2 ).