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- Title
An Improvement on the Number of Limit Cycles Bifurcating from a Nondegenerate Center of Homogeneous Polynomial Systems.
- Authors
Yu, Pei; Han, Maoan; Li, Jibin
- Abstract
In the two articles in Appl. Math. Comput., J. Giné [2012a, 2012b] studied the number of small limit cycles bifurcating from the origin of the system: ẋ=−y+Pn(x,y), ẏ=x+Qn(x,y), where Pn and Qn are homogeneous polynomials of degree n. It is shown that the maximal number of the small limit cycles, denoted by Mh(n), satisfies Mh(n)≥2n−1 for n=4,5,6,7; and Mh(8)≥13, Mh(9)≥16. It seems that the correct answer for their case n=9 should be Mh(9)≥15. In this paper, we apply Hopf bifurcation theory and normal form computation, and perturb the isolated, nondegenerate center (the origin) to prove that Mh(n)≥2n for n=4,5,6,7; and Mh(n)≥2(n−1) for n=8,9, which improve Giné’s results with one more limit cycle for each case.
- Subjects
MATHEMATICAL models; POLYNOMIALS; HOPF bifurcations; STIFF computation (Differential equations); LIMIT cycles
- Publication
International Journal of Bifurcation & Chaos in Applied Sciences & Engineering, 2018, Vol 28, Issue 6, pN.PAG
- ISSN
0218-1274
- Publication type
Article
- DOI
10.1142/S0218127418500785