Hilbert's 16th Problem Research Articles

On the 16th Hilbert problem for limit cycles on nonsingular algebraic curves

We give an upper bound for the maximum number N of algebraic limit cycles that a planar polynomial vector field of degree n can exhibit if the vector field has exactly k nonsingular irreducible invariant algebraic curves. Additionally we provide sufficient conditions in order that all the algebraic limit cycles are hyperbolic. We also provide lower bounds for N.

AN IMPROVED LOWER BOUND ON THE NUMBER OF LIMIT CYCLES BIFURCATING FROM A HAMILTONIAN PLANAR VECTOR FIELD OF DEGREE 7

The limit cycle bifurcations of a Z2 equivariant planar Hamiltonian vector field of degree 7 under Z2 equivariant degree 7 perturbation is studied. We prove that the given system can have at least 53 limit cycles. This is an improved lower bound for the weak formulation of Hilbert's 16th problem for degree 7, i.e. on the possible number of limit cycles that can bifurcate from a degree 7 planar Hamiltonian system under degree 7 perturbation.

The Procesi–Schacher conjecture and Hilbert's 17th problem for algebras with involution

In 1976 Procesi and Schacher developed an Artin–Schreier type theory for central simple algebras with involution and conjectured that in such an algebra a totally positive element is always a sum of hermitian squares. In this paper elementary counterexamples to this conjecture are constructed and cases are studied where the conjecture does hold. Also, a Positivstellensatz is established for noncommutative polynomials, positive semidefinite on all tuples of matrices of a fixed size.

Bifurcations of limit cycles in a -equivariant planar polynomial vector field of degree 7

The present paper intends to explore the number and distribution of limit cycles for a class of perturbed Z4-equivariant planar Hamiltonian system of degree 7. The bifurcation theory of planar dynamical systems and the method of detection functions are applied to study the given perturbed system. By controlling the perturbation parameters, an example of a special perturbed Z4-equivariant vector field having 41 limit cycles is found. The distribution of the above 41 limit cycles is also given. The results are significant to the study of weaken Hilbert's 16th problem and to the acquaintance of the characteristics of higher-order equivariant vector fields.

A Version of Hilbert's 13th Problem for Infinitely Differentiable Functions

In 1957, Kolmogorov and Arnold gave a solution to the 13th problem which had been formulated by Hilbert in 1900. Actually, it is known that there exist many open problems which can be derived from the original problem. From the function-theoretic point of view, Hilbert's 13th problem can be exactly characterized as the superposition representability problem for continuous functions of several variables. In this paper, the solution to the superposition representability problem for infinitely differentiable functions of several variables is given.

A Restricted Version of Hilbert's 16th Problem for Quadratic Vector Fields

The restricted version of the Hilbert 16th problem for quadratic vector requires an upper estimate of the number of limit cycles through a vector parameter that characterizes the vector considered and the limit cycles to be counted. In this paper we give an upper estimate of the number of limit cycles of quadratic vector $\sigma $--distant from centers and $\ka $-distant from singular quadratic vector fields provided that the limit cycles are $\delta $--distant from singular points and infinity.

On the 16th Hilbert problem for algebraic limit cycles

For a polynomial planar vector field of degree n ⩾ 2 with generic invariant algebraic curves we show that the maximum number of algebraic limit cycles is 1 + ( n − 1 ) ( n − 2 ) / 2 when n is even, and ( n − 1 ) ( n − 2 ) / 2 when n is odd. Furthermore, these upper bounds are reached.

The Dirichlet problem in the plane with semianalytic raw data, quasi analyticity, and o-minimal structure

We investigate the Dirichlet solution for a semianalytic continuous function on the boundary of a semianalytic bounded domain in the plane. We show that the germ of the Dirichlet solution at a boundary point with angle greater than zero lies in a certain quasi-analytic class used by Ilyashenko [21]–[23] in his work on Hilbert's 16th problem. With this result we can prove that the Dirichlet solution is definable in an o-minimal structure if the angles at the singular boundary points of the domain are irrational multiples of π

Study of the cyclicity of some degenerate graphics inside quadratic systems

In this paper we make essential steps in proving the finitecyclicity of degenerate graphics in quadratic systems, having a lineof singular points in the finite plane. In particular we considerthe graphics $(DF_{1 a})$, $(DF_{2 a})$ of the program of [8]to prove the finiteness part of Hilbert's 16th problem for quadraticvector fields. We make a complete treatment except for one veryspecific problem that we clearly identify.

Topological Classification of Morse Functions and Generalisations of Hilbert's 16-th Problem

The topological structures of the generic smooth functions on a smooth manifold belong to the small quantity of the most fundamental objects of study both in pure and applied mathematics. The problem of their study has been formulated by A. Cayley in 1868, who required the classification of the possible configurations of the horizontal lines on the topographical maps of mountain regions, and created the first elements of what is called today ‘Morse Theory’ and ‘Catastrophes Theory’. In the paper we describe this problem, and in particular describe the classification of Morse functions on the 2 sphere and on the torus.

Finite cyclicity of nilpotent graphics of pp-type surrounding a center

This paper is part of the DRR-program of [4] to prove the finiteness part of Hilbert's 16th problem for quadratic vector fields by showing the finite cyclicity of 121 graphics. In this paper we prove the finite cyclicity of 4 graphics passing through a triple nilpotent point of elliptic type surrounding a center, namely the graphics $(H_7^1)$, $(F_{7a}^1)$, $(H_{11}^3)$ and $(I_{6a}^1)$. These four graphics are of pp-type, in the sense that they join two parabolic sectors of the nilpotent point. The exact cyclicity is 2 for $(H_7^1)$ and $(H_{11}^3)$. The graphics $(F_{7a}^1)$ and $(I_{6a}^1)$ occur in continuous families. Their exact cyclicity is 2 except for a discrete subset of such graphics. The method can be applied to most other graphics of the DRR-program [4] through a triple nilpotent point and surrounding a center.

The Riemann mapping theorem for semianalytic domains and o -minimality

We consider the Riemann mapping theorem in the case of a bounded simply connected and semianalytic domain. We show that the germ at 0 of the Riemann map (that is, biholomorphic map) from the upper half plane to such a domain can be realized in a certain quasianalytic class if the angle of the boundary at the point to which 0 is mapped is greater than 0. This quasianalytic class was introduced and used by Ilyashenko in his work on Hilbert's 16th problem. With this result, we can prove that the Riemann map from a bounded simply connected semianalytic domain onto the unit ball is definable in an o-minimal structure, provided that at singular boundary points the angles of the boundary are irrational multiples of π.

Limit cycles near hyperbolas in quadratic systems

In this paper we introduce the notion of infinity strip and strip of hyperbolas as organizing centers of limit cycles in polynomial differential systems on the plane. We study a strip of hyperbolas occurring in some quadratic systems. We deal with the cyclicity of the degenerate graphics D I 2 a from the programme, set up in [F. Dumortier, R. Roussarie, C. Rousseau, Hilbert's 16th problem for quadratic vector fields, J. Differential Equations 110 (1994) 86–133], to solve the finiteness part of Hilbert's 16th problem for quadratic systems. Techniques from geometric singular perturbation theory are combined with the use of the Bautin ideal. We also rely on the theory of Darboux integrability.

ON THE NUMBER AND DISTRIBUTIONS OF LIMIT CYCLES IN A QUINTIC PLANAR VECTOR FIELD

This paper is concerned with the number and distributions of limit cycles in a Z2-equivariant quintic planar vector field. By applying qualitative analysis method of differential equation, we find that 28 limit cycles with four different configurations appear in this special planar polynomial system. It is concluded that H(5) ≥ 28 = 52+ 3, where H(5) is the Hilbert number for quintic polynomial systems. The results obtained are useful to the study of the second part of 16th Hilbert problem.

Analytic continuation and fixed points of the Poincaré mapping for a polynomial Abel equation

We consider an Abel differential equation y'=p(x)y2+q(x)y3 with p(x) , q(x) polynomials in x . For two given points a and b in \mathbb C , the “Poincaré mapping” of the above equation transforms the values of its solutions at a into their values at b . In this paper we study global analytic properties of the Poincaré mapping, in particular, its analytic continuation, its singularities and its fixed points (which correspond to the “periodic solutions” such that y(a)=y(b) ). On the one hand, we give a general description of singularities of the Poincaré mapping, and of its analytic continuation. On the other hand, we study in detail the structure of the Poincaré mapping for a local model near a simple fixed singularity, where an explicit solution can be written. Yet, the global analytic structure (in particular, the ramification) of the solutions and of the Poincaré mapping in this case is fairly complicated, and, in our view, highly instructive. For a given degree of the coefficients we produce examples with an infinite number of complex “periodic solutions” and analyze their mutual position and branching. Let us recall that Pugh's problem, which is closely related to the classical Hilbert's 16th problem, asks for the existence of a bound to the number of real isolated “periodic solutions”. New findings reported here lead us to propose new insights on the Poincaré mapping. If the “complexity” of the path in the x -plane between a and b is a priori bounded, the number of fixed points should be uniformly bounded. We think that, in some sense, this is close to the complex version of Khovansky's fewnomial theory.

Geometry of Hermitian Algebraic Functions. Quotients of Squared Norms

Hermitian algebraic functions were introduced by D. Catlin and J. D'Angelo under the name of "globalizable metrics". Catlin and D'Angelo proved that any Hermitian algebraic function without nontrivial zeros is a quotient of squared norms, thus giving an answer to a Hermitian analogue of Hilbert's 17th problem in the nondegenerate case. The result was independently proved somewhat earlier by D. Quillen in a special case, and using different methods. In this paper, we characterize all Hermitian algebraic functions that are quotients of squared norms.

Bifurcations of limit cycles in a perturbed quintic Hamiltonian system with six double homoclinic loops

This paper concerns with the number and distributions of limit cycles of a quintic subject to a seven-degree perturbation. With the aid of numeric integral computation provided by Mathematica 4.1, at least 45 limit cycles are found in the above system by applying the method of double homoclinic loops bifurcation, Hopf bifurcation and qualitative analysis. The four configurations of 45 limit cycles of the system are also shown. The results obtained are useful to the study of the weakened 16th Hilbert Problem.

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.

Hilbert's 16th problem for classical Liénard equations of even degree

Classical Liénard equations are two-dimensional vector fields, on the phase plane or on the Liénard plane, related to scalar differential equations x ¨ + f ( x ) x ˙ + x = 0 . In this paper, we consider f to be a polynomial of degree 2 l − 1 , with l a fixed but arbitrary natural number. The related Liénard equation is of degree 2 l. We prove that the number of limit cycles of such an equation is uniformly bounded, if we restrict f to some compact set of polynomials of degree exactly 2 l − 1 . The main problem consists in studying the large amplitude limit cycles, of which we show that there are at most l.

On the positive extension property and Hilbert's 17th problem for real analytic sets

In this work we study the Positive Extension property and Hilbert's 17th problem for real analytic germs and sets. A real analytic germ X0 of has the property if every positive semidefinite analytic function germ on X0 has a positive semidefinite analytic extension to ; analogously one states the property for a global real analytic set X in an open set Ω of ℝn. These properties are natural variations of Hilbert's 17th problem. Here, we prove that: (1) A real analytic germ has the property if and only if every positive semidefinite analytic function germ on X0 is a sum of squares of analytic function germs on X0; and (2) a global real analytic set X of dimension ≦ 2 and local embedding dimension ≦ 3 has the property if and only if it is coherent and all its germs have the property. If that is the case, every positive semidefinite analytic function on X is a sum of squares of analytic functions on X. Moreover, we classify the singularities with the property.