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Search: MSC category 34C05 ( Location of integral curves, singular points, limit cycles )

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1. CJM 2002 (vol 54 pp. 1038)

Gavrilov, Lubomir; Iliev, Iliya D.
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 $\mathcal{S}_3$. 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.

Categories:34C07, 34C05, 34C10

2. CJM 1997 (vol 49 pp. 212)

Coll, B.; Gasull, A.; Prohens, R.
Differential equations defined by the sum of two quasi-homogeneous vector fields
In this paper we prove, that under certain hypotheses, the planar differential equation: $\dot x=X_1(x,y)+X_2(x,y)$, $\dot y=Y_1(x,y)+Y_2(x,y)$, where $(X_i,Y_i)$, $i=1$, $2$, are quasi-homogeneous vector fields, has at most two limit cycles. The main tools used in the proof are the generalized polar coordinates, introduced by Lyapunov to study the stability of degenerate critical points, and the analysis of the derivatives of the Poincar\'e return map. Our results generalize those obtained for polynomial systems with homogeneous non-linearities.

Categories:34C05, 58F21

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