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

Bifurcations of limit cycles from quintic Hamiltonian systems with a double figure eight loop☆☆The work was supported in part by Australia Research Counsil under the Discovery Projects scheme (grant ID: DP0559111).

AbstractThis paper deals with Liénard equations of the form x˙=y, y˙=P(x)+yQ(x,y), with P and Q polynomials of degree 5 and 4 respectively. Attention goes to perturbations of the Hamiltonian vector fields with an elliptic Hamiltonian of degree six, exhibiting a double figure eight loop. The number of limit cycles and their distributions are given by using the methods of bifurcation theory and qualitative analysis

The Development of Mathematical Methods for Tackling Problems in Non-Perturbative Quantum Field Theory, Cosmology, and Gravity

We have extended two recently developed theoretical methods, the Quantum Finite Elements (QFE) and the Euclidean-signature semi-classical method (ESSCM). The QFE is a technique for constructing lattice field theories (LFTs) on curved Riemannian manifolds. We extended the applicability of the QFE to formulating LFTs on certain three and four dimensional Riemannian manifolds such as $\mathbb{S}^{3}$ and \mathbb{R} \cross \mathbb{S}^{3}. This was done by first constructing a novel simplicial approximation to $\mathbb{S}^{3}$. Then, after correctly computing the weights of the links and vertices that make up this simplicial approximation, we defined a Laplacian on it, whose low lying spectrum was observed to approach the known continuum limit as we further refined our simplicial complex. To facilitate a comparison between the QFE and the bootstrap, we calculated an estimate of the fourth-order Binder cumulant using CFT data extracted from the conformal bootstrap. The ESSCM is a methodology for facilitating the use of already known mathematical theorems/results to approach Lorentzian signature problems in bosonic field theory and quantum gravity in terms of their Euclidean-signature analogs. We further developed this method by applying it in a novel fashion to quantum cosmological models with matter sources. In particular, for the Taub models, we proved for the first time the existence of a countably infinite number of well behaved ‘excited’ state solutions when $\Lambda$ is present. Both methods are promising and have applications for field theory, beyond standard model physics, and quantum gravity

Quantum-induced interactions in the moduli space of degenerate BPS domain walls

In this paper quantum effects are investigated in a very special two-scalar field model having a moduli space of BPS topological defects. In a $(1+1)$-dimensional space-time the defects are classically degenerate in mass kinks, but in $(3+1)$ dimensions the kinks become BPS domain walls, all of them sharing the same surface tension at the classical level. The heat kernel/zeta function regularization method will be used to control the divergences induced by the quantum kink and domain wall fluctuations. A generalization of the Gilkey-DeWitt-Avramidi heat kernel expansion will be developed in order to accommodate the infrared divergences due to zero modes in the spectra of the second-order kink and domain wall fluctuation operators, which are respectively $N\times N$ matrix ordinary or partial differential operators. Use of these tools in the spectral zeta function associated with the Hessian operators paves the way to obtain general formulas for the one-loop kink mass and domain wall tension shifts in any $(1+1)$- or $(3+1)$-dimensional $N$-component scalar field theory model. Application of these formulae to the BPS kinks or domain walls of the $N=2$ model mentioned above reveals the breaking of the classical mass or surface tension degeneracy at the quantum level. Because the main parameter distinguishing each member in the BPS kink or domain wall moduli space is essentially the distance between the centers of two basic kinks or walls, the breaking of the degeneracy amounts to the surge in quantum-induced forces between the two constituent topological defects. The differences in surface tension induced by one-loop fluctuations of BPS walls give rise mainly to attractive forces between the constituent walls except if the two basic walls are very far apart. Repulsive forces between two close walls only arise if the coupling is approaches the critical value from below.Comment: 34 pages, 7 figures, to appear in JHE