Perturbation from an elliptic Hamiltonian of degree four. IV: Figure eight-loop. (English) Zbl 1057.34015
From the paper’s abstract: “The paper deals with Liénard equations of the form \(\dot x=y\), \(\dot y = P(x)+yQ(x)\) with \(P\) and \(Q\) polynomials of degree, 3 and 2, respectively. Attention goes to perturbations of the Hamiltonian vector fields with an elliptic Hamiltonian of degree four, exhibiting a figure eight-loop. It is proved that the least upper bound of the number of zeroes of the related elliptic integral is five, and this is a sharp bound, multiplicity taken into account. Moreover, if restricting to the level curves ‘inside’ a saddle loop, or ‘outside’ the figure eight-loop, the sharp upper bound is two or four, respectively; also, the multiplicity of the zeroes is at most four.”
The authors also show the existence of a cubic Liénard system with five limit cycles. This paper is the fourth one of four papers devoted to the study of perturbations of planar Hamiltonian vector fields. In the three previous papers [Part III, see ibid. 188, No. 2, 473-511 (2003; Zbl 1056.34044)], the authors studied the saddle loop, the two saddle cycle, the cuspidal loop and the global centre cases.
The authors also show the existence of a cubic Liénard system with five limit cycles. This paper is the fourth one of four papers devoted to the study of perturbations of planar Hamiltonian vector fields. In the three previous papers [Part III, see ibid. 188, No. 2, 473-511 (2003; Zbl 1056.34044)], the authors studied the saddle loop, the two saddle cycle, the cuspidal loop and the global centre cases.
Reviewer: Marco Sabatini (Trento)
MSC:
| 34C07 | Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations |
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
Citations:
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