On the Morse Index of a Functional Arising in Contact Form Geometry

In this paper we study a functional at infinity associated to a contact form on a three dimensional manifold. The Morse index of this functional at infinity can be decomposed into two parts, one along the characteristic manifold and the other along the normal directions. We prove that we can redistribute the negative directions between the two subspaces through a local deformation of the contact form near a critical point at infinity.

Bifurcations of Limit Cycles From Infinity in Quadratic Systems

We investigate the bifurcation of limit cycles in one-parameter unfoldings of quadractic differential systems in the plane having a degenerate critical point at infinity. It is shown that there are three types of quadratic systems possessing an elliptic critical point which bifurcates from infinity together with eventual limit cycles around it. We establish that these limit cycles can be studied by performing a degenerate transformation which brings the system to a small perturbation of certain well-known reversible systems having a center. The corresponding displacement function is then expanded in a Puiseux series with respect to the small parameter and its coefficients are expressed in terms of Abelian integrals. Finally, we investigate in more detail four of the cases, among them the elliptic case (Bogdanov-Takens system) and the isochronous center S3. We show that in each of these cases the corresponding vector space of bifurcation functions has the Chebishev property: the number of the zeros of each function is less than the dimension of the vector space. To prove this we construct the bifurcation diagram of zeros of certain Abelian integrals in a complex domain.