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Let P be a not necessarily bounded polycycle of an analytic vector field on an open set of the plane. Suppose that the singularities which appear after desingularization of the vertices of P are formally linearizable. Consider the function T defined by the return time near P. It is shown that the function T and its derivative T′ have asymptotic expansions in {x μ } μ∈ R and {x δ log x} δ∈ R . It is also shown that under some other conditions imposed on the polycycle vertices, the asymptotic expansions of T and T′ converge absolutely and uniformly to these functions, respectively. These results are applied to the polycycles of the analytic vector fields which have a Darboux first integral. In particular, it is obtained that if P is a polycycle of a Hamiltonian vector field with an analytic (polynomial if P is unbounded) Hamiltonian function, T is a nonoscillating function. Another application concerns the nilpotent centers or focus, since the singularities which appear after desingularization of such a singularity have analytic first integrals.

We consider the 1-parameter family of planar quintic systems, $\dot x= y^3-x^3$, $\dot y= -x+my^5$, introduced by A. Bacciotti in 1985. It is known that it has at most one limit cycle and that it can exist only when the parameter $m$ is in $(0.36,0.6)$. In this paper, using the Bendixon-Dulac theorem, we give a new unified proof of all the previous results, we shrink this to $(0.547,0.6)$, and we prove the hyperbolicity of the limit cycle. We also consider the question of the existence of polycycles. The main interest and difficulty for studying this family is that it is not a semi-complete family of rotated vector fields. When the system has a limit cycle, we also determine explicit lower bounds of the basin of attraction of the origin. Finally we answer an open question about the change of stability of the origin for an extension of the above systems., 27 pages, 11 figures

We study the stability of a type of unbounded polycycles which appear in some planar differential equations. Each of these polycycles has hyperbolic corners, but the product of the hyperbolicity ratios of all its corners does not decide its stability. We obtain an explicit convergent integral whose sign gives the stability of the polycycle.