Hilbert Number Research Articles

Hilbert Number for a Family of Piecewise Nonautonomous Equations

For family x′=(a0+a1cost+a2sint)|x|+b0+b1cost+b2sint\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x'=(a_0+a_1\\cos t+a_2 \\sin t)|x|+b_0+b_1 \\cos t+b_2 \\sin t$$\\end{document}, we solve three basic problems related with its dynamics. First, we characterize when it has a center (Poincaré center focus problem). Second, we show that each equation has a finite number of limit cycles (finiteness problem), and finally we give a uniform upper bound for the number of limit cycles (Hilbert problem 16).

The New Class $L_{p,\Phi}$ of $s$-Type Operators

In this study, the class of $s$-type $\ell_{p}( \Phi )$ operators is introduced and it is shown that $L_{p,\Phi}$ is a quasi-Banach operator ideal. Also, some other classes are defined by using approximation, Gelfand, Kolmogorov, Weyl, Chang, and Hilbert number sequences. Then, some properties are examined.

L-HILBERT MEAN LABELING OF SOME PATH RELATED GRAPHS

Let be a graph with vertices and edges. The th hilbert number is denoted by and is defined by where A - hilbert mean labeling is an injective function , where that induces a bijection defined by
 for all . A graph which admits such labeling is called a - hilbert mean graph. In this paper, a new type of labeling called - hilbert mean labeling is introduced and the path related graphs is studied.
 AMS Subject Classification – 05C78

New lower bound for the Hilbert number in low degree Kolmogorov systems

Our main goal in this paper is to study the number of small-amplitude isolated periodic orbits, so-called limit cycles, surrounding only one equilibrium point a class of polynomial Kolmogorov systems. We denote by MK(n) the maximum number of limit cycles bifurcating from the equilibrium point via a degenerate Hopf bifurcation for a polynomial Kolmogorov vector field of degree n. In this work, we obtain another example such that MK(3)≥6. In addition, we obtain new lower bounds for MK(n) proving that MK(4)≥13 and MK(5)≥22.

Counting configurations of limit cycles and centers

We present several results on the determination of the number and distribution of limit cycles or centers for planar systems of differential equations. In most cases, the study of a recurrence is one of the key points of our approach. These results include the counting of the number of configurations of stabilities of nested limit cycles, the study of the number of different configurations of a given number of limit cycles, the proof of some quadratic lower bounds for Hilbert numbers and some questions about the number of centers for planar polynomial vector fields.

On the Hilbert number for piecewise linear vector fields with algebraic discontinuity set

The second part of Hilbert’s sixteenth problem consists in determining the upper bound H(n) for the number of limit cycles that planar polynomial vector fields of degree n can have. For n≥2, it is still unknown whether H(n) is finite or not. The main achievements obtained so far establish lower bounds for H(n). Regarding asymptotic behavior, the best result says that H(n) grows as fast as n2log(n). Better lower bounds for small values of n are known in the research literature. In the recent paper “Some open problems in low dimensional dynamical systems”, by A. Gasull, Problem 18 proposes a different Hilbert’s sixteenth type problem, namely improving the lower bounds for L(n), n∈N, which is defined as the maximum number of limit cycles that planar piecewise linear differential systems with two zones separated by a branch of an algebraic curve of degree n can have. So far, L(n)≥[n/2],n∈N, is the best known general lower bound. Again, better lower bounds for small values of n are known in the research literature. Here, by using a recently developed second order Melnikov method for nonsmooth systems with nonlinear discontinuity manifold, it is shown that L(n) grows as fast as n2. This will be achieved by providing lower bounds for L(n), which improve previous estimates for n≥4.

Spin structures and baby universes

We extend a 2d topological model of the gravitational path integral to include sums over spin structure, corresponding to Neveu-Schwarz (NS) or Ramond (R) boundary conditions for fermions. This path integral corresponds to a correlator of boundary creation operators on a non-trivial baby universe Hilbert space, and vanishes when the number of R boundaries is odd. This vanishing implies a non-factorization of the correlator, which necessitates a dual interpretation of the bulk path integral in terms of a product of partition functions (associated to NS boundaries) and Witten indices (associated to R boundaries), averaged over an ensemble of theories with varying Hilbert space dimension and different numbers of bosonic and fermionic states. We also consider a model with End-of-the-World (EOW) branes, for which the dual ensemble then includes a sum over randomly chosen fermionic and bosonic states. We propose two modifications of the bulk path integral which restore an interpretation in a single dual theory: (i) a geometric prescription where we add extra boundaries with a sum over their spin structures, and (ii) an algebraic prescription involving “spacetime D-branes”. We extend our ideas to Jackiw-Teitelboim gravity, and propose a dual description of a single unitary theory with spin structure in a system with eigenbranes.

Asymptotic lower bounds on Hilbert numbers using canard cycles

In this work we give an asymptotic lower bound for the Hilbert number for real planar polynomial differential systems. This lower bound equals, up to leading order, to the best existing one, but the method we provide is new and involves slow-fast systems. The construction strongly relies on generalized Liénard systems.

Melnikov analysis in nonsmooth differential systems with nonlinear switching manifold

We study the family of piecewise linear differential systems in the plane with two pieces separated by a cubic curve. Our main result is that 7 is a lower bound for the Hilbert number of this family. In order to get our main result, we develop the Melnikov functions for a class of nonsmooth differential systems, which generalizes, up to order 2, some previous results in the literature. Whereas the first order Melnikov function for the nonsmooth case remains the same as for the smooth one (i.e. the first order averaged function) the second order Melnikov function for the nonsmooth case is different from the smooth one (i.e. the second order averaged function). We show that, in this case, a new term depending on the jump of discontinuity and on the geometry of the switching manifold is added to the second order averaged function.

Bifurcations of limit cycles in complex networks of Hamiltonian systems and Lloyd's conjecture

This paper is motivated by Hilbert's sixteenth problem, concerning the number and distribution of limit cycles in planar polynomial vector fields. We propose an original method, inspired by the complex systems framework, in order to construct a family of Hamiltonian systems admitting non-degenerate centers located on the vertices of a planar network. We introduce two novel polynomial perturbations adapted to those systems, with the aim to produce a high number of limit cycles; the first one is oriented along the gradient of the unperturbed system, and the second one is again constructed with a complex systems approach. Using the Melnikov integral, we reach, in a small number of particular cases, the lower bound of cubical order for the Hilbert number H(n), conjectured by Lloyd in 1988.

New lower bounds for the Hilbert numbers using reversible centers

In this paper we provide the best lower bounds, that are known up to now, for the Hilbert numbers of polynomial vector fields of degree N, , for small values of N. These limit cycles appear bifurcating from symmetric Darboux reversible centers with very high simultaneous cyclicity. The considered systems have, at least, three centers, one on the reversibility straight line and two symmetric outside it. More concretely, the limit cycles are in a three nests configuration and the total number of limit cycles is at least 2n + m, for some values of n and m. The new lower bounds are obtained using simultaneous degenerate Hopf bifurcations. In particular, and

New lower bound for the Hilbert number in piecewise quadratic differential systems

We study the number of limit cycles bifurcating from a piecewise quadratic system. All the differential systems considered are piecewise in two zones separated by a straight line. We prove the existence of 16 crossing limit cycles in this class of systems. If we denote by Hp(n) the extension of the Hilbert number to degree n piecewise polynomial differential systems, then Hp(2)≥16. As fas as we are concerned, this is the best lower bound for the quadratic class. Moreover, in the studied cases, all limit cycles appear nested bifurcating from a period annulus of a isochronous quadratic center.

Combinatorial properties of integer matrices and integer matrices mod k

We consider matrices over the (nonnegative) integers and the integers mod k, and representations of them as linear combinations of permutation matrices. In both instances we describe and compare several minimal generating sets (bases) of permutation matrices. We investigate a mod k generalization of the Hilbert numbers for rational cones. In this context, we also consider matrices with entries in with real, and not mod k, arithmetic.

Double bifurcation for a cubic Kolmogorov model

Our work focuses on the study of a kind of new bifurcation behavior about limit cycles (namely double bifurcation) for a class of cubic Kolmogorov model with a degenerate critical point (1, 1). The positive critical point (1, 1) is a 3-multiple nilpotent fine focus which can become a fine focus of 5th order. By analyzing the change in stability, we show (1, 1) can yield 4 limit cycles. Moreover, the high-order critical point (1, 1) can be broken into three lower-order critical points including a real critical points and two complex critical points under small parameters’ perturbation. Under the condition letting the multiplicity be same one, obtained real critical point (1, 1) can become a fine focus of 2nd order and yield two small limit cycles. In sum, 6 limit cycle can bifurcate near point (1, 1). This is a kind of new bifurcation behavior via double perturbation; in addition, in terms of the Hilbert number of cubic Kolmogorov model, our results are also new.

LIMIT CYCLE BIFURCATION FOR A NILPOTENT SYSTEM IN <i>Z</i><sub>3</sub>-EQUIVARIANT VECTOR FIELD

Our work is concerned with the problem on limit cycle bifurcation for a class of Z3-equivariant Lyapunov system of five degrees with three third-order nilpotent critical points which lie in a Z3-equivariant vector field. With the help of computer algebra system-MATHEMATICA, the first 5 quasiLyapunov constants are deduced. The fact of existing 12 small amplitude limit cycles created from the three third-order nilpotent critical points is also proved. Our proof is algebraic and symbolic, obtained result is new and interesting in terms of nilpotent critical points' Hilbert number in equivariant vector field.

Behavior of limit cycle bifurcations for a class of quartic Kolmogorov models in a symmetrical vector field

In this study, we consider the limit cycle bifurcation problem for a class of quartic Kolmogorov models with five positive singular points, i.e., (1,1), (1,2), (2,1), (1,3), and (3,1), which lie in a symmetrical vector field relative to the line y=x. We classify these singular points. We show that points (1,2) and (2,1) can bifurcate into three small limit cycles by simultaneous Hopf bifurcation, and that points (1,3) and (3,1) can bifurcate into three small limit cycles by simultaneous Hopf bifurcation. In addition, we construct limit cycles for this model and we show that four positive singular points, i.e., (1,1), (1,2), (2,1), and (1,3), can bifurcate into eight limit cycles in total, among which six cycles may be stable. Few previous studies have considered a symmetrical Kolmogorov model with several positive singular points. Our results are good in terms of the Hilbert number for the Kolmogorov model.

Parallelization of the Lyapunov constants and cyclicity for centers of planar polynomial vector fields

Christopher in 2006 proved that under some assumptions the linear parts of the Lyapunov constants with respect to the parameters give the cyclicity of an elementary center. This paper is devoted to establish a new approach, namely parallelization, to compute the linear parts of the Lyapunov constants. More concretely, it is shown that parallelization computes these linear parts in a shorter quantity of time than other traditional mechanisms.To show the power of this approach, we study the cyclicity of the holomorphic center z˙=iz+z2+z3+⋯+zn under general polynomial perturbations of degree n, for n≤13. We also exhibit that, from the point of view of computation, among the Hamiltonian, time-reversible, and Darboux centers, the holomorphic center is the best candidate to obtain high cyclicity examples of any degree. For n=4,5,…,13, we prove that the cyclicity of the holomorphic center is at least n2+n−2. This result gives the highest lower bound for M(6),M(7),…,M(13) among the existing results, where M(n) is the maximum number of limit cycles bifurcating from an elementary monodromic singularity of polynomial systems of degree n. As a direct corollary we also obtain the highest lower bound for the Hilbert numbers H(6)≥40, H(8)≥70, and H(10)≥108, because until now the best result was H(6)≥39, H(8)≥67, and H(10)≥100.

Bifurcation of the separatrix skeleton in some 1-parameter families of planar vector fields

This article deals with the bifurcation of polycycles and limit cycles within the 1-parameter families of planar vector fields Xmk, defined by x˙=y3−x2k+1,y˙=−x+my4k+1, where m is a real parameter and k≥1 is an integer. The bifurcation diagram for the separatrix skeleton of Xmk in function of m is determined and the one for the global phase portraits of (Xm1)m∈R is completed. Furthermore for arbitrary k≥1 some bifurcation and finiteness problems of periodic orbits are solved. Among others, the number of periodic orbits of Xmk is found to be uniformly bounded independently of m∈R and the Hilbert number for (Xmk)m∈R, that thus is finite, is found to be at least one.

THE HILBERT NUMBER OF A CLASS OF DIFFERENTIAL EQUATIONS

The notion of Hilbert number from polynomial differential systems in the plane of degree $n$ can be extended to the differential equations of the form \[\dfrac{dr}{d\theta}=\dfrac{a(\theta)}{\displaystyle \sum_{j=0}^{n}a_{j}(\theta)r^{j}} \eqno(*)\] defined in the region of the cylinder $(\tt,r)\in \Ss^1\times \R$ where the denominator of $(*)$ does not vanish. Here $a, a_0, a_1, \ldots, a_n$ are analytic $2\pi$--periodic functions, and the Hilbert number $\HHH(n)$ is the supremum of the number of limit cycles that any differential equation $(*)$ on the cylinder of degree $n$ in the variable $r$ can have. We prove that $\HHH(n)= \infty$ for all $n\ge 1$.

New lower bounds for the Hilbert number of polynomial systems of Liénard type

In this paper, we study the maximal number of limit cycles of some kinds of polynomial Liénard systems with arbitrary degree and obtain some new lower bounds for the Hilbert number of the systems, which improve truly the certain existing results.