Linear estimate of the number of zeros of Abelian integrals for quadratic centers having almost all their orbits formed by cubics

Bifurcation of limit cycles of the nongeneric quadratic reversible system with discontinuous perturbations

“…The list of quadratic center at (0,0), almost all the orbits of which are cubic, looks as follows [9,21]: with the integrating factor µ(x, y) = x −4 . In the present paper, by using the Picard-Fuchs equation and the property of Chebyshev space, we investigate the number of limit cycles of system (1.2) under discontinuous polynomial perturbations of degree n. The system (1.2) has a center (1,0) and h = −1 corresponds to the center (1,0).…”

“…(iii) If h ∈ (−1, 0), a + i,j = a − i,j and b + i,j = b − i,j , then Zhao et al [21] obtained that the number of limit cycles of system (1.4) bifurcating from the period annulus is not more than 3n − 4 for n ≥ 4; 8 for n = 3; 5 for n = 2 (counting multiplicity).…”

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By using the Picard-Fuchs equation and the property of Chebyshev space to the discontinuous differential system, we obtain an upper bound of the number of limit cycles for the nongeneric quadratic reversible system when it is perturbed inside all discontinuous polynomials with degree n.The Hamiltonian triangle:ż = −iz +z 2 .The reversible system:ż = −iz + (2b + 1)z 2 + 2|z| 2 + bz 2 , b = −1.The generic Lotka-Volterra system:Under the perturbations of continuous polynomials of degree n, Horozov and Iliev [6] proved that the number of limit cycles for Q H 3 and Hamiltonian triangle does not exceed 5n + 15, and Zhao et al [21] proved that the number of limit cycles for reversible and generic Lotka-Volterra systems does not exceed 7n.Let z = x + iy and by a linear transformation, the reversible system can be written [21]:

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On the Number of Limit Cycles in Perturbations of a Quadratic Reversible Center

“…The proof of Lemma 5.1 is similar to that of Proposition 2.1 in [12], so we omit it. By similar arguments to Lemmas 2.1 and 2.2, we obtain the following lemmas.…”

The period function for quadratic integrable systems with cubic orbits

“…In the paper [52], we have shown that J −1 (h), J 0 (h), associated with system (1.8), satisfy the following Picard-Fuchs equation, provided A = −2:…”