Perturbations from an elliptic Hamiltonian of degree four. II: Cuspidal loop. (English) Zbl 1034.34036
The authors study Liénard equations of the form \(\dot{x}=y\), \(\dot{y}=P(x)+y Q(x)\), with \(P\) and \(Q\) polynomials of degree 3 and 2, respectivily. Attention goes to perturbations of the Hamiltonian vector field with an elliptic Hamiltonian of degree 4, exhibiting a cuspidal loop. It is proven that the least upper bound for the number of zeros of the related elliptic integral is four, and this upper bound is a sharp one. This permits to prove the existence of Liénard equations of type \((3,2)\) with at least four limit cycles. The paper also contains a complete result on the corresponding number of “small” and “large” limit cycles.
For part I see ibid. 176, 114–157 (2001; Zbl 1004.34018).
For part I see ibid. 176, 114–157 (2001; Zbl 1004.34018).
Reviewer: Valery A. Gaiko (Minsk)
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 |
| 34D10 | Perturbations of ordinary differential equations |
| 37J45 | Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010) |
Citations:
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