Bifurcation from Periodic Solutions in Functional Differential Equations.
The paper deals with the bifurcation phenomena of heteroclinic orbits for diffeomorphisms. The existence of a Melnikov-like function for the two-dimensional case is shown. Simple possibilities of the set of heteroclinic points are described for higherdimensional cases.
Chaos generated by the existence of Smale horseshoe is the well-known phenomenon in the theory of dynamical systems. The Poincaré-Andronov-Melnikov periodic and subharmonic bifurcations are also classical results in this theory. The purpose of this note is to extend those results to ordinary differential equations with multivalued perturbations. We present several examples based on our recent achievements in this direction. Singularly perturbed problems are studied as well. Applications are given...
Ordinary differential inclusions depending on small parameters are considered such that the unperturbed inclusions are ordinary differential equations possessing manifolds of periodic solutions. Sufficient conditions are determined for the persistence of some of these periodic solutions after multivalued perturbations. Applications are given to dry friction problems.
Variational inequalities are studied, where is a closed convex cone in , , is a matrix, is a small perturbation, a real parameter. The assumptions guaranteeing a Hopf bifurcation at some for the corresponding equation are considered and it is proved that then, in some situations, also a bifurcation of periodic solutions to our inequality occurs at some . Bifurcating solutions are obtained by the limiting process along branches of solutions to penalty problems starting at constructed...
It is proved in this paper that the maximum number of limit cycles of system⎧ dx/dt = y⎨⎩ dy/dt = kx - (k + 1)x2 + x3 + ε(α + βx + γx2)yis equal to two in the finite plane, where k > (11 + √33) / 4 , 0 < |ε| << 1, |α| + |β| + |γ| ≠ 0. This is partial answer to the seventh question in [2], posed by Arnold.