Abstract view
Bifurcations of Limit Cycles From Infinity in Quadratic Systems


Published:20021001
Printed: Oct 2002
Lubomir Gavrilov
Iliya D. Iliev
Abstract
We investigate the bifurcation of limit cycles in oneparameter
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 wellknown 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 (BogdanovTakens system) and the isochronous center
$\mathcal{S}_3$. 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.
MSC Classifications: 
34C07, 34C05, 34C10 show english descriptions
Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications) Location of integral curves, singular points, limit cycles Oscillation theory, zeros, disconjugacy and comparison theory
34C07  Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications) 34C05  Location of integral curves, singular points, limit cycles 34C10  Oscillation theory, zeros, disconjugacy and comparison theory
