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- Title
ABELIAN INTEGRALS FOR THE ONE-PARAMETER BOGDANOV-TAKENS SYSTEM.
- Authors
ZHANG, YONGKANG; LI, BAOYI; LI, CUIPING
- Abstract
An explicit upper bound Z(2, n) ≤ n + m - 1 is derived for the number of zeros of Abelian integrals M1(h) = ∮γ(h) P(x, y) dy - Q(x, y) dx on the open interval (0, 1/6), where γ(h) is an oval lying on the algebraic curve Hλ = (1/2)x2 + (1/2)y2 - (1/3)x3 - λy3 = h, P(x, y), Q(x, y) are polynomials of x and y, and max{deg P(x, y), deg Q(x, y)} = n. The proof exploits the expansion of the first order Melnikov function M1(h, λ) near λ = 0 and assume (∂m/∂λm)M1(h, λ)|λ = 0 not vanish identically.
- Subjects
ABELIAN functions; PARAMETER estimation; ALGEBRAIC curves; POLYNOMIALS; OVALS; MATHEMATICAL formulas; BIFURCATION theory
- Publication
International Journal of Bifurcation & Chaos in Applied Sciences & Engineering, 2011, Vol 21, Issue 9, p2723
- ISSN
0218-1274
- Publication type
Article
- DOI
10.1142/S0218127411030052