Redundant Picard-Fuchs system for Abelian integrals

We derive an explicit system of Picard-Fuchs differential equations satisfied by Abelian integrals of monomial forms and majorize its coefficients. A peculiar feature of this construction is that the system admitting such explicit majorants, appears only in dimension approximately two times greater than the standard Picard-Fuchs system. The result is used to obtain a partial solution to the tangential Hilbert 16th problem. We establish upper bounds for the number of zeros of arbitrary Abelian integrals on a positive distance from the critical locus. Under the additional assumption that the critical values of the Hamiltonian are distant from each other (after a proper normalization), we were able to majorize the number of all (real and complex) zeros. In the second part of the paper an equivariant formulation of the above problem is discussed and relationships between spread of critical values and non-homogeneity of uni- and bivariate complex polynomials are studied.Comment: 31 page, LaTeX2e (amsart

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

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

On the number of zeros of Abelian integrals for a kind of quadratic reversible centers

Hilbert$'$s 16th problem is extensively studied in mathematics and its applications. Arnold proposed a weakened version focusing on differential equations. While significant progress has been made for Hamiltonian systems, less attention has been given to integrable non-Hamiltonian systems. In recent years, investigating quadratic reversible systems in integrable non-Hamiltonian systems has gained widespread attention and shown promising advancements. In this academic context, our study is based on qualitative analysis theory. It explores the upper bound of the number of zeros of Abelian integrals for a specific class of quadratic reversible systems under perturbations with polynomial degrees of n. The Picard-Fuchs equation method and the Riccati equation method are employed in our investigation. The research findings indicate that when the degree of the perturbing polynomial is n ($n\geq5$), the upper bound for the number of zeros of Abelian integrals is determined to be $7n-12$. To achieve this, we first numerically transform the Hamiltonian function of the quadratic reversible system into a standard form. By applying a combination of the Picard-Fuchs equation method and the Riccati equation method, we derive the representation of the Abelian integrals. Using relevant theorems, we estimate the upper bound for the number of zeros of the Abelian integrals, which consequently provides an upper bound for the number of limit cycles in the system. The research results demonstrate that when the perturbation polynomial degree is high or equal to n, the Picard-Fuchs equation method and the Riccati equation method can be applied to estimate the upper bound of the number of zeros of the Abelian integrals

Abelian Integral Method and its Application

Oscillation is a common natural phenomenon in real world problems. The most efficient mathematical models to describe these cyclic phenomena are based on dynamical systems. Exploring the periodic solutions is an important task in theoretical and practical studies of dynamical systems. Abelian integral is an integral of a polynomial differential 1-form over the real ovals of a polynomial Hamiltonian, which is a basic tool in complex algebraic geometry. In dynamical system theory, it is generalized to be a continuous function as a tool to study the periodic solutions in planar dynamical systems. The zeros of Abelian integral and their distributions provide the number of limit cycles and their locations. In this thesis, we apply the Abelian integral method to study the limit cycles bifurcating from the periodic annuli for some hyperelliptic Hamiltonian systems. For two kinds of quartic hyperelliptic Hamiltonian systems, the periodic annulus is bounded by either a homoclinic loop connecting a nilpotent saddle, or a heteroclinic loop connecting a nilpotent cusp to a hyperbolic saddle. For a quintic hyperelliptic Hamiltonian system, the periodic annulus is bounded by a more degenerate heteroclinic loop, which connects a nilpotent saddle to a hyperbolic saddle. We bound the number of zeros of the three associated Abelian integrals constructed on the periodic structure by employing the combination technique developed in this thesis and Chebyshev criteria. The exact bound for each system is obtained, which is three. Our results give answers to the open questions whether the sharp bound is three or four. We also study a quintic hyperelliptic Hamiltonian system with two periodic annuli bounded by a double homoclinic loop to a hyperbolic saddle, one of the periodic annuli surrounds a nilpotent center. On this type periodic annulus, the exact number of limit cycles via Poincar{\\u27e} bifurcation, which is one, is obtained by analyzing the monotonicity of the related Abelian integral ratios with the help of techniques in polynomial boundary theory. Our results give positive answers to the conjecture in a previous work. We also extend the methods of Abelian integrals to study the traveling waves in two weakly dissipative partial differential equations, which are a perturbed, generalized BBM equation and a cubic-quintic nonlinear, dissipative Schr\ {o}dinger equation. The dissipative PDEs are reduced to singularly perturbed ODE systems. On the associated critical manifold, the Abelian integrals are constructed globally on the periodic structure of the related Hamiltonians. The existence of solitary, kink and periodic waves and their coexistence are established by tracking the vanishment of the Abelian integrals along the homoclinic loop, heteroclinic loop and periodic orbits. Our method is novel and easily applied to solve real problems compared to the variational analysis

Alien limit cycles in Liénard equations

Agraïments: The second author thanks the Universitat de les Illes Balears (UIB) for its support as invited professor during the period November to December, 2011.This paper aims at providing an example of a family of polynomial Liénard equations exhibiting an alien limit cycle. This limit cycle is perturbed from a 2-saddle cycle in the boundary of an annulus of periodic orbits given by a Hamiltonian vector field. The Hamiltonian represents a truncated pendulum of degree 4. In comparison to a former polynomial example, not only the equations are simpler but a lot of tedious calculations can be avoided, making the example also interesting with respect to simplicity in treatment

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

It is provedin this paper that the maximum number of limit cycles of system [formula] is equal to two in the finite plane, where [formula]. This is partial answer to the seventh question in [2], posed by Arnold