Applied Mathematics and Computation, ISSN 0096-3003, 02/2014, Volume 228, pp. 329 - 335

An explicit upper bound B(n) is derived for the number of zeros of Abelian integrals I(h)=∮Γhg(x,y)dx-f(x,y)dy on the open interval (0,1/4), where Γh is an...

Quartic Hamiltonian | Weakened Hilbert 16th problem | Poincare bifurcation | Abelian integrals | MATHEMATICS, APPLIED | LINEAR ESTIMATE | COMPLEX ZEROS | LIMIT-CYCLES | HILBERT PROBLEM | CENTERS | Intervals | Algebra | Upper bounds | Computation | Integrals | Mathematical models | Polynomials

Quartic Hamiltonian | Weakened Hilbert 16th problem | Poincare bifurcation | Abelian integrals | MATHEMATICS, APPLIED | LINEAR ESTIMATE | COMPLEX ZEROS | LIMIT-CYCLES | HILBERT PROBLEM | CENTERS | Intervals | Algebra | Upper bounds | Computation | Integrals | Mathematical models | Polynomials

Journal Article

Journal of Applied Analysis and Computation, ISSN 2156-907X, 12/2018, Volume 8, Issue 6, pp. 1959 - 1970

In this paper, by using the method of Picard-Fuchs equation and Riccati equation, we study the upper bounds for the associated number of zeros of Abelian...

Quadratic reversible center | Abelian integral | Weakened Hilbert’s 16th problem | weakened Hilbert's 16th problem | MATHEMATICS, APPLIED | quadratic reversible center | BIFURCATIONS | ALMOST-ALL | KIND | LIENARD SYSTEM

Quadratic reversible center | Abelian integral | Weakened Hilbert’s 16th problem | weakened Hilbert's 16th problem | MATHEMATICS, APPLIED | quadratic reversible center | BIFURCATIONS | ALMOST-ALL | KIND | LIENARD SYSTEM

Journal Article

Journal of Differential Equations, ISSN 0022-0396, 11/2017, Volume 263, Issue 9, pp. 5554 - 5581

This paper deals with the limit cycles of a class of cubic Hamiltonian systems under polynomial perturbations. We suppose that the corresponding Hamiltonian...

Hamiltonian system | Chebyshev space | Abelian integral | Picard–Fuchs equation | Weakened Hilbert's 16th problem | NUMBER | LINEAR ESTIMATE | QUADRATIC CENTERS | COMPLEX ZEROS | POLYNOMIAL DIFFERENTIAL-SYSTEMS | MATHEMATICS | Picard-Fuchs equation | ALMOST-ALL | ABELIAN-INTEGRALS | QUARTIC HAMILTONIANS | LIMIT-CYCLES | HOMOGENEOUS NONLINEARITIES

Hamiltonian system | Chebyshev space | Abelian integral | Picard–Fuchs equation | Weakened Hilbert's 16th problem | NUMBER | LINEAR ESTIMATE | QUADRATIC CENTERS | COMPLEX ZEROS | POLYNOMIAL DIFFERENTIAL-SYSTEMS | MATHEMATICS | Picard-Fuchs equation | ALMOST-ALL | ABELIAN-INTEGRALS | QUARTIC HAMILTONIANS | LIMIT-CYCLES | HOMOGENEOUS NONLINEARITIES

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 09/2015, Volume 429, Issue 2, pp. 924 - 941

In this study, we determine the number of zeros for Abelian integrals in four cases of quadratic reversible centers of genus one. Based on the results of Li et...

Quadratic reversible center | Abelian integral | Weakened 16th Hilbert problem | SYSTEM | MATHEMATICS, APPLIED | GLOBAL BIFURCATION | NUMBER | LINEAR ESTIMATE | DIFFERENTIAL CENTER | ISOCHRONOUS CENTERS | MATHEMATICS | ALMOST-ALL | POLYNOMIAL VECTOR-FIELDS | LIMIT-CYCLES | ZEROS | Data mining | Analysis

Quadratic reversible center | Abelian integral | Weakened 16th Hilbert problem | SYSTEM | MATHEMATICS, APPLIED | GLOBAL BIFURCATION | NUMBER | LINEAR ESTIMATE | DIFFERENTIAL CENTER | ISOCHRONOUS CENTERS | MATHEMATICS | ALMOST-ALL | POLYNOMIAL VECTOR-FIELDS | LIMIT-CYCLES | ZEROS | Data mining | Analysis

Journal Article

International Journal of Bifurcation and Chaos, ISSN 0218-1274, 02/2016, Volume 26, Issue 2, p. 1650020

In this study, we determine the associated number of zeros for Abelian integrals in four classes of quadratic reversible centers of genus one. Based on the...

Quadratic reversible center | Abelian integral | weakened 16th Hilbert problem | SYSTEM | GLOBAL BIFURCATION | CYCLICITY | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | ALMOST-ALL | MULTIDISCIPLINARY SCIENCES | LINEAR ESTIMATE | LIMIT-CYCLES | FAMILY

Quadratic reversible center | Abelian integral | weakened 16th Hilbert problem | SYSTEM | GLOBAL BIFURCATION | CYCLICITY | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | ALMOST-ALL | MULTIDISCIPLINARY SCIENCES | LINEAR ESTIMATE | LIMIT-CYCLES | FAMILY

Journal Article

Nonlinear Analysis: Real World Applications, ISSN 1468-1218, 02/2016, Volume 27, pp. 350 - 365

In this paper, we give the upper bound of the number of zeros of Abelian integral I(h)=∮Γhg(x,y)dy−f(x,y)dx, where Γh is the closed orbit defined by...

Hamiltonian system | Chebyshev space | Abelian integral | Weakened Hilbert’s 16th problem | Picard–Fuchs equation | Picard-Fuchs equation | Weakened Hilbert's 16th problem | MATHEMATICS, APPLIED | NUMBER | LINEAR ESTIMATE | QUADRATIC CENTERS | COMPLEX ZEROS | CYCLICITY | ALMOST-ALL | KIND | LIMIT-CYCLES | BIFURCATION | Intervals | Nonlinearity | Polynomials | Upper bounds | Integrals | Saddles

Hamiltonian system | Chebyshev space | Abelian integral | Weakened Hilbert’s 16th problem | Picard–Fuchs equation | Picard-Fuchs equation | Weakened Hilbert's 16th problem | MATHEMATICS, APPLIED | NUMBER | LINEAR ESTIMATE | QUADRATIC CENTERS | COMPLEX ZEROS | CYCLICITY | ALMOST-ALL | KIND | LIMIT-CYCLES | BIFURCATION | Intervals | Nonlinearity | Polynomials | Upper bounds | Integrals | Saddles

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 2009, Volume 359, Issue 1, pp. 209 - 215

The finite generators of Abelian integral I ( h ) = ∮ Γ h f ( x , y ) d x − g ( x , y ) d y are obtained, where Γ h is a family of closed ovals defined by H (...

Hamiltonian system | Abelian integral | Picard–Fuchs equation | Weakened Hilbert's 16th problem | Picard-Fuchs equation | MATHEMATICS | COMPLETE ELLIPTIC INTEGRALS | MATHEMATICS, APPLIED | NUMBER | N=2 | LINEAR ESTIMATE | COMPLEX ZEROS | HILBERT PROBLEM

Hamiltonian system | Abelian integral | Picard–Fuchs equation | Weakened Hilbert's 16th problem | Picard-Fuchs equation | MATHEMATICS | COMPLETE ELLIPTIC INTEGRALS | MATHEMATICS, APPLIED | NUMBER | N=2 | LINEAR ESTIMATE | COMPLEX ZEROS | HILBERT PROBLEM

Journal Article

Applied Mathematics and Computation, ISSN 0096-3003, 2008, Volume 204, Issue 1, pp. 202 - 209

In this paper, we give the upper bound of the number of zeros of Abelian integral I ( h ) = ∮ Γ h P ( x , y ) d x - Q ( x , y ) d y , where Γ h is the closed...

Hamiltonian system | Abelian integral | Weakened Hilbert’s 16th problem | Picard–Fuchs equation | Picard-Fuchs equation | Weakened Hilbert's 16th problem | weakened Hilbert's 16th problem | COMPLETE ELLIPTIC INTEGRALS | MATHEMATICS, APPLIED | N=2 | LINEAR ESTIMATE | COMPLEX ZEROS | 16TH PROBLEM

Hamiltonian system | Abelian integral | Weakened Hilbert’s 16th problem | Picard–Fuchs equation | Picard-Fuchs equation | Weakened Hilbert's 16th problem | weakened Hilbert's 16th problem | COMPLETE ELLIPTIC INTEGRALS | MATHEMATICS, APPLIED | N=2 | LINEAR ESTIMATE | COMPLEX ZEROS | 16TH PROBLEM

Journal Article

International Journal of Bifurcation and Chaos, ISSN 0218-1274, 03/2013, Volume 23, Issue 3, pp. 1350047 - 1350018

In this paper, we provide a complete study of the zeros of Abelian integrals obtained by integrating the 1-form (α + βx + x2)ydx over the compact level curves...

degenerated polycycle | Abelian integral | hyperelliptic Hamiltonian | weakened Hilbert 16th problem | BIFURCATIONS | MULTIDISCIPLINARY SCIENCES | CUSPIDAL LOOP | SADDLE | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | NON-OSCILLATION | DEGREE-4 | LIMIT-CYCLES | ELLIPTIC INTEGRALS | HETEROCLINIC LOOPS | Asymptotic expansions | Cusps | Upper bounds | Integrals | Saddles | Chebyshev approximation | Bifurcations | Perturbation

degenerated polycycle | Abelian integral | hyperelliptic Hamiltonian | weakened Hilbert 16th problem | BIFURCATIONS | MULTIDISCIPLINARY SCIENCES | CUSPIDAL LOOP | SADDLE | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | NON-OSCILLATION | DEGREE-4 | LIMIT-CYCLES | ELLIPTIC INTEGRALS | HETEROCLINIC LOOPS | Asymptotic expansions | Cusps | Upper bounds | Integrals | Saddles | Chebyshev approximation | Bifurcations | Perturbation

Journal Article

International Journal of Bifurcation and Chaos, ISSN 0218-1274, 05/2004, Volume 14, Issue 5, pp. 1853 - 1862

In this paper, a one-parameter Hamiltonian system under cubic perturbations is investigated and the upper bound of the number of zeros of the Abelian integral...

Bifurcation of limit cycles | Weakened Hubert 16th problem | Hamiltonian system | Abelian integrals | NUMBER | BIFURCATIONS | weakened Hilbert 16th problem | 16TH HILBERT PROBLEM | MULTIDISCIPLINARY SCIENCES | bifurcation of limit cycles | CYCLICITY | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | HOMOCLINIC LOOPS | LIMIT-CYCLES | ZEROS

Bifurcation of limit cycles | Weakened Hubert 16th problem | Hamiltonian system | Abelian integrals | NUMBER | BIFURCATIONS | weakened Hilbert 16th problem | 16TH HILBERT PROBLEM | MULTIDISCIPLINARY SCIENCES | bifurcation of limit cycles | CYCLICITY | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | HOMOCLINIC LOOPS | LIMIT-CYCLES | ZEROS

Journal Article

Qualitative Theory of Dynamical Systems, ISSN 1575-5460, 12/2019, Volume 18, Issue 3, pp. 947 - 967

The present paper is devoted to study the number of zeros of Abelian integral for the near-Hamilton system $$\begin{aligned} {\left\{ \begin{array}{ll} \dot{x}...

Chebyshev space | Abelian integral | Weakened Hilbert’s 16th problem | Picard–Fuchs equation | Difference and Functional Equations | Mathematics, general | Mathematics | Hamilton system | Dynamical Systems and Ergodic Theory

Chebyshev space | Abelian integral | Weakened Hilbert’s 16th problem | Picard–Fuchs equation | Difference and Functional Equations | Mathematics, general | Mathematics | Hamilton system | Dynamical Systems and Ergodic Theory

Journal Article

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