A Note on a Result of G. S. Petrov About the Weakened 16th Hilbert Problem

G. S. Petrov [Functional Anal. Appl. 22 (1988), 72-73] and P. Mardesic [Ergodic Theory Dynamical Systems10 (1990), 523-529] have proved that for system ẋ = y + ϵP(x, y), ẏ = 1 − 3x2 + ϵQ(x, y), where P(x, y) and Q(x, y) are polynomials of x, y with degree ≤ N, if the first order Melnikov function M1...

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Published in:Journal of mathematical analysis and applications 1995-03, Vol.190 (2), p.489-516
Main Authors: Li, B.Y., Zhang, Z.F.
Format: Article
Language:English
Subjects:
Exact sciences and technology
Mathematical analysis
Mathematics
Ordinary differential equations
Sciences and techniques of general use
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Summary:G. S. Petrov [Functional Anal. Appl. 22 (1988), 72-73] and P. Mardesic [Ergodic Theory Dynamical Systems10 (1990), 523-529] have proved that for system ẋ = y + ϵP(x, y), ẏ = 1 − 3x2 + ϵQ(x, y), where P(x, y) and Q(x, y) are polynomials of x, y with degree ≤ N, if the first order Melnikov function M1(h) ≢ 0, then the lowest upper bound B(N) of the number of limit cycles of the above system is N − 1. We prove that if M1(h) ≡ 0 and the second order Melnikov function M2(h) ≢ 0, then [formula]
ISSN:0022-247X
1096-0813
DOI:10.1006/jmaa.1995.1088