44 research outputs found
Periodic solutions of periodically perturbed planar autonomous systems: A topological approach
Aim of this paper is to investigate the existence of periodic solutions of a
nonlinear planar autonomous system having a limit cycle x_0 of least period
T_0>0 when it is perturbed by a small parameter, T_1-periodic, perturbation. In
the case when T_0/T_1 is a rational number l/k, with l, k prime numbers, we
provide conditions to guarantee, for the parameter perturbation e>0
sufficiently small, the existence of klT_0-periodic solutions x_e of the
perturbed system which converge to the trajectory x_1 of the limit cycle as
e->0. Moreover, we state conditions under which T=klT_0 is the least period of
the periodic solutions x_e. We also suggest a simple criterion which ensures
that these conditions are verified. Finally, in the case when T_0/T_1 is an
irrational number we show the nonexistence, whenever T>0 and e>0, of T-periodic
solutions x_e of the perturbed system converging to x_1. The employed methods
are based on the topological degree theory
A continuation principle for a class of periodically perturbed autonomous systems
In this paper we evaluate the topological index of periodic solutions otained
via the Malkin-Loud bifurcation result. Incidentally, we do not assume that the
perturbation is differentiale.Comment: Accepted to Math. Nachr., Vol. 281, (2008), No.
Periodic bifurcation from families of periodic solutions for semilinear differential equations with Lipschitzian perturbations in Banach spaces
Let A:D(A)\to E be an infinitesimal generator either of an analytic compact
semigroup or of a contractive C_0-semigroup of linear operators acting in a
Banach space E. In this paper we give both necessary and sufficient conditions
for bifurcation of -periodic solutions for the equation x'=Ax+f(t,x)+e
g(t,x,e) from a k-parameterized family of T-periodic solutions of the
unperturbed equation corresponding to e=0. We show that by means of a suitable
modification of the classical Mel'nikov approach we can construct a bifurcation
function and to formulate the conditions for the existence of bifurcation in
terms of the topological index of the bifurcation function. To do this, since
the perturbation term g is only Lipschitzian we need to extend the classical
Lyapunov-Schmidt reduction to the present nonsmooth case.Comment: Submitted to Adv. Nonlinear Stu
Existence of weak solutions to stochastic evolution inclusions
We consider the Cauchy problem for a semilinear stochastic differential
inclusion in a Hilbert space. The linear operator generates a strongly
continuous semigroup and the nonlinear term is multivalued and satisfies a
condition which is more heneral than the Lipschitz condition. We prove the
existence of a mild solution to this problem. This solution is not "strong" in
the probabilistic sense, that is, it is not defined on the underlying
probability space, but on a larger one, which provides a "very good extension"
in the sense of Jacod and Memin. Actually, we construct this solution as a
Young measure, limit of approximated solutions provided by the Euler scheme.
The compactness in the space of Young measures of this sequence of approximated
solutions is obtained by proving that some measure of noncompactness equals
zero