Displaying similar documents to “Abelian integrals related to Morse polynomials and perturbations of plane hamiltonian vector fields”

Simple exponential estimate for the number of real zeros of complete abelian integrals

Dmitri Novikov, Sergei Yakovenko (1995)

Annales de l'institut Fourier

Similarity:

We show that for a generic polynomial H = H ( x , y ) and an arbitrary differential 1-form ω = P ( x , y ) d x + Q ( x , y ) d y with polynomial coefficients of degree ≤ d , the number of ovals of the foliation H = const , which yield the zero value of the complete Abelian integral I ( t ) = ∮ H = t ω , grows at most as exp O H ( d ) as d → ∞ , where O H ( d ) depends only on H . The main result of the paper is derived from the following more general theorem on bounds for isolated zeros occurring in polynomial envelopes of linear differential equations. Let f 1 ( t ) , ⋯ , f n ( t ) , t ∈ K ⋐ ℝ , be a fundamental system of...

On the number of zeros of Melnikov functions

Sergey Benditkis, Dmitry Novikov (2011)

Annales de la faculté des sciences de Toulouse Mathématiques

Similarity:

We provide an effective uniform upper bound for the number of zeros of the first non-vanishing Melnikov function of a polynomial perturbations of a planar polynomial Hamiltonian vector field. The bound depends on degrees of the field and of the perturbation, and on the order k of the Melnikov function. The generic case k = 1 was considered by Binyamini, Novikov and Yakovenko [BNY10]. The bound follows from an effective construction of the Gauss-Manin connection for iterated integrals. ...

Estimate for the Number of Zeros of Abelian Integrals on Elliptic Curves

Mihajlova, Ana (2004)

Serdica Mathematical Journal

Similarity:

2000 Mathematics Subject Classification: Primary 34C07, secondary 34C08. We obtain an upper bound for the number of zeros of the Abelian integral. The work was partially supported by contract No 15/09.05.2002 with the Shoumen University “K. Preslavski”, Shoumen, Bulgaria.

New sufficient conditions for a center and global phase portraits for polynomial systems.

Hector Giacomini, Malick Ndiaye (1996)

Publicacions Matemàtiques

Similarity:

In this paper we consider cubic polynomial systems of the form: x' = y + P(x, y), y' = −x + Q(x, y), where P and Q are polynomials of degree 3 without linear part. If M(x, y) is an integrating factor of the system, we propose its reciprocal V (x, y) = 1 / M(x,y) as a linear function of certain coefficients of the system. We find in this way several new sets of sufficient conditions for a center. The resulting integrating factors are of Darboux type and the first integrals are in the...

Bifurcations of limit cycles from cubic Hamiltonian systems with a center and a homoclinic saddle-loop.

Yulin Zhao, Zhifen Zhang (2000)

Publicacions Matemàtiques

Similarity:

It is proved in this paper that the maximum number of limit cycles of system ⎧ dx/dt = y ⎨ ⎩ dy/dt = kx - (k + 1)x2 + x3 + ε(α + βx + γx2)y is equal to two in the finite plane, where k > (11 + √33) / 4 , 0 < |ε| << 1, |α| + |β| + |γ| ≠ 0. This is partial answer to the seventh question in [2], posed by Arnold.