Perturbations of non-Hamiltonian reversible quadratic systems with cubic orbits

✍ Scribed by Yulin Zhao

• Publisher: Elsevier Science
• Year: 2006
• Tongue: English
• Weight: 213 KB
• Volume: 64
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/j.na.2005.08.016

✦ Synopsis

This paper is concerned with degree n polynomial perturbations of a class of planar non-Hamiltonian reversible quadratic integrable system whose almost all orbits are cubics. We give an estimate of the number of limit cycles for such a system. If the first-order Melnikov function (Abelian integral) M 1 (h) is not identically zero, then the perturbed system has at most 5 for n = 3 and 3n -7 for n 4 limit cycles on the finite plane. If M 1 (h) is identically zero but the second Melnikov function is not, then an upper bound for the number of limit cycles on the finite plane is 11 for n = 3 and 6n -13 for n 4, respectively. In the case when the perturbation is quadratic (n = 2), there exists a neighborhood U of the initial non-Hamiltonian polynomial system in the space of all quadratic vector fields such that any system in U has at most two limit cycles on the finite plane. The bound for n = 2 is exact.