Finite Singular Points Research Articles

Analytic integrability of cubic–linear planar polynomial differential systems

For the cubic–linear polynomial planar differential systems with a finite singular point, we classify the ones which have a local analytic first integral around the origin and the ones that have a global analytic first integral.

Characterization of self-adjoint domains for differential operators with interior singular points

We characterize the self-adjoint domains of general even order linear ordinary differential operators which have finite interior singular points in terms of real-parameter solutions of the differential equation. For the purpose we constructed a direct sum space. By the theory of direct sum space and the decomposition of the corresponding maximal domain, we give this complete and analytic characterization in terms of limit-circle solutions. This is for endpoints which are regular or singular and for arbitrary deficiency index.

Diagonal Ising susceptibility: elliptic integrals, modular forms and Calabi–Yau equations

We give the exact expressions of the partial susceptibilities χ(3)d and χ(4)d for the diagonal susceptibility of the Ising model in terms of modular forms and Calabi–Yau ODEs, and more specifically, 3F2([1/3, 2/3, 3/2], [1, 1]; z) and 4F3([1/2, 1/2, 1/2, 1/2], [1, 1, 1]; z) hypergeometric functions. By solving the connection problems we analytically compute the behavior at all finite singular points for χ(3)d and χ(4)d. We also give new results for χ(5)d. We see, in particular, the emergence of a remarkable order-6 operator, which is such that its symmetric square has a rational solution. These new exact results indicate that the linear differential operators occurring in the n-fold integrals of the Ising model are not only ‘derived from geometry’ (globally nilpotent), but actually correspond to ‘special geometry’ (homomorphic to their formal adjoint). This raises the question of seeing if these ‘special geometry’ Ising operators are ‘special’ ones, reducing, in fact systematically, to (selected, k-balanced, ...) q + 1Fq hypergeometric functions, or correspond to the more general solutions of Calabi–Yau equations.

Analytic integrability of quadratic–linear polynomial differential systems

AbstractFor the quadratic–linear polynomial differential systems with a finite singular point, we classify the ones which have a global analytic first integral, and provide the explicit expression of their first integrals.

A quadratic system with two parallel straight-line-isoclines

In this paper, a quadratic system with two parallel straight-line-isoclines is considered. This system corresponds to the system of class II in the classification of Ye Yanqian [Ye Yanqian et al., Theory of Limit Cycles, Transl. Math. Monogr., vol. 66, American Mathematical Society, Providence, RI, 1986]. Using the field rotation parameters of the constructed canonical system and geometric properties of the spirals filling the interior and exterior domains of its limit cycles, we prove that the maximum number of limit cycles in a quadratic system with two parallel straight-line-isoclines and two finite singular points is equal to two. Besides, we obtain the same result in a different way: applying the Wintner–Perko termination principle for multiple limit cycles and using the methods of global bifurcation theory developed in [V.A. Gaiko, Global Bifurcation Theory and Hilbert’s Sixteenth Problem, Kluwer, Boston, 2003].

Inverse problems for invariant algebraic curves: explicit computations

Given an algebraic curve in the complex affine plane, we describe how to determine all planar polynomial vector fields which leave this curve invariant. If all (finite) singular points of the curve are non-degenerate, we give an explicit expression for these vector fields. In the general setting we provide an algorithmic approach, and as an alternative we discuss sigma processes.

A laurent expansion and residue theorems of k-regular functions in clifford analysis

In this context, we mainly study the behavior in the neighborhood of finite singular points for k-regular functions in 1 R1n with values in R0,n. We get a Laurent expansion of them in an open set, prove its uniqueness, give the definitions of k-poles, isolated and essential singular points and removable singularity, discuss some properties, and further obtain the residue theorems.

SINGULAR POINTS OF QUADRATIC SYSTEMS: A COMPLETE CLASSIFICATION IN THE COEFFICIENT SPACE ℝ12

Planar quadratic differential systems occur in many areas of applied mathematics. Although more than one thousand papers were written on these systems, a complete understanding of this class is still missing. Classical problems, and in particular, Hilbert's 16th problem [Hilbert,1900], are still open for this class. Even when not dealing with limit cycles, still some problems have remained unsolved like a complete classification of different phase portraits without limit cycles.For some time it was thought (see [Coppel, 1966]) there could exist a set of algebraic functions whose signs would completely determine the phase portrait of a quadratic system. Nowadays we already know that this is not so, and that there are some analytical, nonalgebraic functions that also play a role when dealing with limit cycles and separatrix connections.However, it is possible to find out a set of algebraic functions whose signs determine the characteristics of all finite and infinite singular points. Most of the work up to now has dealt with this problem studying it using different normal forms adapted to some subclasses of quadratic systems. A general work useful for any quadratic system regardless of affine changes has only been done for the study of infinite singular points [Schlomiuk et al., 2005].In this paper, we give a complete global classification of quadratic differential systems according to their topological behavior in the vicinity of the finite singular points. Our classification Main Theorem gives us a complete dictionary describing the local behavior of finite singular points using algebraic invariants and comitants which are a powerful tool for algebraic computations. Linking the result of this paper with the main one of [Schlomiuk et al., 2005] which uses the same algebraic invariants, it is possible to complete the algebraic classification of singular points (finite and infinite) for quadratic differential systems.

Bifurcations of limit cycles for a perturbed quintic Hamiltonian system with four infinite singular points

By using the theory of bifurcations of dynamical systems and the method of detection function to investigate the bifurcation of limit cycles of a perturbed quintic Hamiltonian system with 25 finite singular points and four infinite singular points. From the detection functions for the perturbed system, we prove that under different determined parameter condition, the given system has at least 22 and 20 limit cycles and the configurations of compound eyes are also obtained.

Critical Points of Functions, sl2 Representations, and Fuchsian Differential Equations with only Univalued Solutions

Let a second order Fuchsian differential equation with only univalued solutions have finite singular points at z_1, ..., z_n with exponents (a_1,b_1), ..., (a_n,b_n). Let the exponents at infinity be (A,B). Then for fixed generic z_1,...,z_n, the number of such Fuchsian equations is equal to the multiplicity of the irreducible sl_2 representation of dimension |A-B| in the tensor product of irreducible sl_2 representations of dimensions |a_1-b_1|, >..., |a_n-b_n|. To show this we count the number of critical points of a suitable function which plays the crucial role in constructions of the hypergeometric solutions of the sl_2 KZ equation and of the Bethe vectors in the sl_2 Gaudin model. As a byproduct of this study we conclude that the Bethe vectors form a basis in the space of states for the sl_2 inhomogeneous Gaudin model.

Integrable Fredholm Operators and Dual Isomonodromic Deformations

The Fredholm determinants of a special class of integral operators K supported on the union of m curve segments in the complex plane are shown to be the tau-functions of an isomonodromic family of meromorphic covariant derivative operators D_l. These have regular singular points at the 2m endpoints of the curve segments and a singular point of Poincare index 1 at infinity. The rank r of the vector bundle over the Riemann sphere on which they act equals the number of distinct terms in the exponential sums entering in the numerator of the integral kernels. The deformation equations may be viewed as nonautonomous Hamiltonian systems on an auxiliary symplectic vector space M, whose Poisson quotient, under a parametric family of Hamiltonian group actions, is identified with a Poisson submanifold of the loop algebra Lgl_R(r) with respect to the rational R-matrix structure. The matrix Riemann-Hilbert problem method is used to identify the auxiliary space M with the data defining the integral kernel of the resolvent operator at the endpoints of the curve segments. A second associated isomonodromic family of covariant derivative operators D_z is derived, having rank n=2m, and r finite regular singular points at the values of the exponents defining the kernel of K. This family is similarly embedded into the algebra Lgl_R(n) through a dual parametric family of Poisson quotients of M. The operators D_z are shown to be analogously associated to the integral operator obtained from K through a Fourier-Laplace transform.

Detection function method and its application to a perturbed quintic Hamiltonian system

We employ the methods of both qualitative analysis and numerical exploration to investigate the bifurcation of limit cycles of a perturbed quintic Hamiltonian system. With the help of the detection functions for the perturbed Hamiltonian system, we study the perturbation of a quintic Hamiltonian system with 25 finite singular points and 4 infinite singular points. We first classify the phase portraits of the unperturbed system and categorize the closed orbits, then obtain the detection functions for the perturbed system, from which the detection curves and the number and the distribution of limit cycles are obtained. As application examples, we numerically compute the detection curves and draw the distribution of limit cycles. It seems that our methods are very effective on the study of the bifurcation of limit cycles of quintic Hamiltonian systems.

SOME BIFURCATION DIAGRAMS FOR LIMIT CYCLES OF QUADRATIC DIFFERENTIAL SYSTEMS

Concrete numerical examples of quadratic differential systems having three limit cycles surrounding one singular point are shown. In case another finite singular point also exists, a (3, 1) distribution of limit cycles is also obtained. This is the highest number of limit cycles known to occur in a quadratic differential system so far. Representative bifurcation diagrams are drawn for realistic parameter values.

Exact semiclassical and non-perturbative BPS spectra in N = 2 susy QCD

These notes are based on work done in collaboration with Frank Ferrari. We show how to determine the spectra of stable BPS states in N = 2 supersymmetric SU (2) Yang-Mills theories that are asymptotically free, i.e. without and with N f = 1, 2, 3 quark hypermultiplets. In all cases: • • There is a curve of marginal stability diffeomorphic to a circle and going through all (finite) singular points of moduli space. • • The BPS spectra are discontinuous across the curves. • • The strong-coupling spectra (inside the curves) contain only those BPS states that can become massless and are responsible for the singularities. Except for N f = 3, they form a multiplet (with different masses) of the broken global discrete symmetry. • • All other semi-classical BPS states must and do decay consistently when crossing the curves. • • The weak-coupling, i.e. semi-classical BPS spectra, contain no magnetic charges larger than one for N f = 0, 1, 2 and no magnetic charges larger than two for N f = 3.

Discontinuous BPS spectra in N = 2 susy QCD

These notes are based on work done in collaboration with Frank Ferrari. We show how to determine the spectra of stable BPS states in $N=2$ supersymmetric $SU(2)$ Yang-Mills theories that are asymptotically free, i.e. without and with $N_f=1,2,3$ quark hypermultiplets. In all cases: $\bullet$ There is a curve of marginal stability diffeomorphic to a circle and going through all (finite) singular points of moduli space. $\bullet$ The BPS spectra are discontinuous across these curves. $\bullet$ The strong-coupling spectra (inside the curves) contain only those BPS states that can become massless and are responsible for the singularities. Except for $N_f=3$, they form a multiplet (with different masses) of the broken global discrete symmetry. $\bullet$ All other semi-classical BPS states must and do decay consistently when crossing the curves. $\bullet$ The weak-coupling, i.e. semi-classical BPS spectra, contain no magnetic charges larger than one for $N_f=0,1,2$ and no magnetic charges larger than two for $N_f=3$.

Affine invariant conditions for the topological distinction of quadratic systems with a critical point of the 4th multiplicity

The affine invariant partition of the set of quadratic systems with one finite singular point of the 4th multiplicity with respect to different topological classes is accomplished. The conditions corresponding to this partition are semi-algebraic, i.e. they are expressed as equalities or inequalities between polynomials.

On series representation in laguerre polynomials

This paper is devoted to study the existence of singularities of series expansions in Laguerre polynomials of holomorphic functions on the boundaries of its natural region of convergence. An Equiconvergence theorem is proved by which the convergence problem of Laguerre series can be reduced to the simpler discussion of the convergence of associated with them trigonometric series. Based on this the existence of Laguerre series having no finite singular points is established. A theorem is also proved which makes us sure that as inthe case for power series, for the analytic continuation of functions defined by series in Laguerre polynomials it can be judged by the properties of the series coefficients. The final section of the paper contains some proposals as a result of applying the classical method of constructing non-continuable series by means of gaps.

Properties of the solutions of multilinear elliptical systems of nondivergence type

Results on the existence of finite and infinite singular points are obtained for multilinear elliptical systems that satisfy an analogue of the Cordes condition.

Quadratic systems with a weak focus

In this paper we study the number and the relative position of the limit cycles of a plane quadratic system with a weak focus. In particular, we prove the limit cycles of such a system can never have (2, 2)-distribution, and that there is at most one limit cycle not surrounding this weak focus under any one of the following conditions:(i) the system has at least 2 saddles in the finite plane,(ii) the system has more than 2 finite singular points and more than 1 singular point at infinity,(iii) the system has exactly 2 finite singular points, more than 1 singular point at infinity, and the weak focus is itself surrounded by at least one limit cycle.

The Asymptotic Behavior of a Variation of Polarized Hodge Structure

0.1. The purpose of this paper is to give the asymptotic behavior of variation of polarized Hodge structures in the several-dimensional case. We do not discuss here why and how the notion of variation of Hodge structures arises and it is developed by P. A. Griffiths, P. Deligne, W. Schmid and others. What motivates us is to generalize Zucker's result to the several-dimensional case. His result is as follows: the cohomology groups of a variation of Hodge structure on the compact curve with finite singular points have also a Hodge structure,, He proceeds his proof as follows. As an analytic tool, he uses the harmonic analysis (Hodge-Kodaira theory) and as a geometric tool he uses W. Schmid's result that we discuss later. By using the Kahler metric on the curve which behaves with the special property at singular points and the Hermitian metric of the vector bundle which arises from the polarization of Hodge structure, he succeeds to express the cohomology groups of the variation as the L-cohomology groups. Since the L-cohomology group is isomorphic to the space of harmonic forms, by decomposing harmonic forms into (/?, q} -forms, he obtains the Hodge decomposition of the cohomology group of Hodge structure. However, in order to prove the first step—to express the cohomology group by L-cohomology group—he is obliged to use the result of W. Schmid on the asymptotic behavior of variations of Hodge structures at singularity. In this paper, we generalize W. Schmid's result to the several-