TY - JOUR
T1 - Abelian integrals for the one-parameter bogdanov takens system
AU - Zhang, Yongkang
AU - Li, Baoyi
AU - Li, Cuiping
PY - 2011/9
Y1 - 2011/9
N2 - 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.
AB - 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.
KW - Abelian integrals
KW - Bogdanov Takens system
KW - Hilbert's 16th problem
UR - https://www.scopus.com/pages/publications/81055123870
U2 - 10.1142/S0218127411030052
DO - 10.1142/S0218127411030052
M3 - 文章
AN - SCOPUS:81055123870
SN - 0218-1274
VL - 21
SP - 2723
EP - 2727
JO - International Journal of Bifurcation and Chaos
JF - International Journal of Bifurcation and Chaos
IS - 9
ER -