Displaying similar documents to “Abelian integrals of quadratic Hamiltonian vector fields with an invariant straight line.”

Abelian integrals related to Morse polynomials and perturbations of plane hamiltonian vector fields

Lubomir Gavrilov (1999)

Annales de l'institut Fourier

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Let 𝒜 be the real vector space of Abelian integrals I ( h ) = ∫ ∫ { H ≤ h } R ( x , y ) d x ∧ d y , h ∈ [ 0 , h ˜ ] where H ( x , y ) = ( x 2 + y 2 ) / 2 + ... is a fixed real polynomial, R ( x , y ) is an arbitrary real polynomial and { H ≤ h } , h ∈ [ 0 , h ˜ ] , is the interior of the oval of H which surrounds the origin and tends to it as h → 0 . We prove that if H ( x , y ) is a semiweighted homogeneous polynomial with only Morse critical points, then 𝒜 is a free finitely generated module over the ring of real polynomials ℝ [ h ] , and compute its rank. We find the generators of 𝒜 in the case when H is an...

Uniqueness of limit cycles bounded by two invariant parabolas

Eduardo Sáez, Iván Szántó (2012)

Applications of Mathematics

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In this paper we consider a class of cubic polynomial systems with two invariant parabolas and prove in the parameter space the existence of neighborhoods such that in one the system has a unique limit cycle and in the other the system has at most three limit cycles, bounded by the invariant parabolas.

Matchings Extend to Hamiltonian Cycles in 5-Cube

Fan Wang, Weisheng Zhao (2018)

Discussiones Mathematicae Graph Theory

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Ruskey and Savage asked the following question: Does every matching in a hypercube Qn for n ≥ 2 extend to a Hamiltonian cycle of Qn? Fink confirmed that every perfect matching can be extended to a Hamiltonian cycle of Qn, thus solved Kreweras’ conjecture. Also, Fink pointed out that every matching can be extended to a Hamiltonian cycle of Qn for n ∈ {2, 3, 4}. In this paper, we prove that every matching in Q5 can be extended to a Hamiltonian cycle of Q5.