Hilbert's 16th Problem Research Articles

Exploring Limit Cycles of Differential Equations through Information Geometry Unveils the Solution to Hilbert's 16th Problem.

The detection of limit cycles of differential equations poses a challenge due to the type of the nonlinear system, the regime of interest, and the broader context of applicable models. Consequently, attempts to solve Hilbert's sixteenth problem on the maximum number of limit cycles of polynomial differential equations have been uniformly unsuccessful due to failing results and their lack of consistency. Here, the answer to this problem is finally obtained through information geometry, in which the Riemannian metrical structure of the parameter space of differential equations is investigated with the aid of the Fisher information metric and its scalar curvature R. We find that the total number of divergences of |R| to infinity provides the maximum number of limit cycles of differential equations. Additionally, we demonstrate that real polynomial systems of degree n≥2 have the maximum number of 2(n-1)(4(n-1)-2) limit cycles. The research findings highlight the effectiveness of geometric methods in analyzing complex systems and offer valuable insights across information theory, applied mathematics, and nonlinear dynamics. These insights may pave the way for advancements in differential equations, presenting exciting opportunities for future developments.

On the crossing limit cycles created by a discontinuous piecewise differential system formed by three linear Hamiltonian saddles

We study planar discontinuous piecewise differential systems formed by three linear Hamiltonian saddles separated by the non-regular line Σ = { ( x , y ) ∈ R 2 : ( y = 0 ) ∨ ( x = 0 ∧ y ≥ 0 ) } . We prove that when the linear Hamiltonian saddles are homogeneous they have no limit cycles, and when they are non homogeneous they can have at most three limit cycles having exactly one point on each branch of Σ, and at most one limit cycle having four intersection points on Σ. Moreover we show that they can have at most one limit cycle having four intersection points on Σ and three limit cycles having three intersection points on Σ, simultaneously. Thus we have solved the extension of the 16th Hilbert problem to this class of piecewise differential systems. Furthermore we show that for the three types of combinations of the limit cycles here studied in two of them the upper bound is sharp by providing examples of these systems with the maximum number of possible limit cycles. For the remaining type of limit cycles the upper bound is four and we have examples with three limit cycles. So at this moment it is an open problem to know if the sharp upper bound is either four or three.

A family of generalized Liénard differential systems with an explicit limit cycle

One of the important problems in the qualitative theory of real planar differential systems is limit cycles i.e., isolated periodic solutions in the set of all periodic this concept was defined at the end of the 19th century by Poincaré, and with the works of van der Pol and Liénard. The significance of these isolated solutions lies in their key role in understanding the dynamics of a certain differential system. Afterward, Hilbert specified a list of 23 problems for the advancement of mathematical science, and from then it started intensive research on these problems throughout the 20th century. Of the 23 problems, only the so-called 16th Hilbert’s problem and the Riemann conjecture remain open until now. The second part of the 16th Hilbert problem has two parts and asks for an upper bound on the number of possible limit cycles and their positions for a planar polynomial differential system of a given degree. The theory of the limit cycle operating as an indispensable mathematical tool has found broad and important applications in modern physics, chemical, biological, ,and other fields. Then the progress of these fields, in turn, promotes limit cycle research. In this paper, we prove the existence and uniqueness of the limit cycle for a family of generalized polynomial Liénard differential systems, the explicit expression of this limit cycle is given. An example illustrating our results is additionally presented.

At Most Five Limit Cycles in a Class of Discontinuous Piecewise Linear Systems with Three Zones

In this paper, we study the maximum number of limit cycles where planar discontinuous piecewise differential systems can exist formed by three regions separated by a nonregular line. We show that such discontinuous piecewise linear systems can have at most five limit cycles, two of which are of the four intersection points type, the third one is of the two intersection points type and the other two are of three intersection points type. In a setting, we have solved the extended 16th Hilbert problem for these discontinuous piecewise linear differential systems.

The 16th Hilbert Problem for Discontinuous Piecewise Linear Hamiltonian Saddles and Isochronous Centers Separated by a Straight Line

The 16th Hilbert Problem for Discontinuous Piecewise Linear Hamiltonian Saddles and Isochronous Centers Separated by a Straight Line

Analytic Nullstellensätze and the model theory of valued fields

AbstractWe present a uniform framework for establishing Nullstellensätze for power series rings using quantifier elimination results for valued fields. As an application, we obtain Nullstellensätze for ‐adic power series (both formal and convergent) analogous to Rückert's complex and Risler's real Nullstellensatz, as well as a ‐adic analytic version of Hilbert's 17th Problem. Analogous statements for restricted power series, both real and ‐adic, are also considered.

Limit Cycles of Discontinuous Piecewise Differential Hamiltonian Systems Separated by a Straight Line

In this article, we study the maximum number of limit cycles of discontinuous piecewise differential systems, formed by two Hamiltonians systems separated by a straight line. We consider three cases, when both Hamiltonians systems in each side of the discontinuity line have simultaneously degree one, two or three. We obtain that in these three cases, this maximum number is zero, one and three, respectively. Moreover, we prove that there are discontinuous piecewise differential systems realizing these maximum number of limit cycles. Note that we have solved the extension of the 16th Hilbert problem about the maximum number of limit cycles that these three classes of discontinuous piecewise differential systems separated by one straight line and formed by two Hamiltonian systems with a degree either one, two, or three, which such systems can exhibit.

Limit cycles and homoclinic networks in two-dimensional polynomial systems.

In this paper, the properties of equilibriums in planar polynomial dynamical systems are studied. The homoclinic networks of sources, sinks, and saddles in self-univariate polynomial systems are discussed, and the numbers of sources, sinks, and saddles are determined through a theorem, and the first integral manifolds are determined. The corresponding proof of the theorem is completed, and a few illustrations of networks for source, sinks, and saddles are presented for a better understanding of the homoclinic networks. Such homoclinic networks are without any centers even if the networks are separated by the homoclinic orbits. The homoclinic networks of positive and negative saddles with clockwise and counterclockwise limit cycles in crossing-univariate polynomial systems are studied secondly, and the numbers of saddles and centers are determined through a theorem, and the first integral manifolds are determined through polynomial functions. The corresponding proof of the theorem is given, and a few illustrations of networks of saddles and centers are given to show the corresponding geometric structures. Such homoclinic networks of saddles and centers are without any sources and sinks. Since the maximum equilibriums for such two types of planar polynomial systems with the same degrees are discussed, the maximum centers and saddles should be obtained, and maximum sinks, sources, and saddles should be achieved. This paper may provide a different way to determine limit cycles in the Hilbert 16th problem.

The Solution of the Extended 16th Hilbert Problem for Some Classes of Piecewise Differential Systems

The limit cycles have a main role in understanding the dynamics of planar differential systems, but their study is generally challenging. In the last few years, there has been a growing interest in researching the limit cycles of certain classes of piecewise differential systems due to their wide uses in modeling many natural phenomena. In this paper, we provide the upper bounds for the maximum number of crossing limit cycles of certain classes of discontinuous piecewise differential systems (simply PDS) separated by a straight line and consequently formed by two differential systems. A linear plus cubic polynomial forms six families of Hamiltonian nilpotent centers. First, we study the crossing limit cycles of the PDS formed by a linear center and one arbitrary of the six Hamiltonian nilpotent centers. These six classes of PDS have at most one crossing limit cycle, and there are systems in each class with precisely one limit cycle. Second, we study the crossing limit cycles of the PDS formed by two of the six Hamiltonian nilpotent centers. There are systems in each of these 21 classes of PDS that have exactly four crossing limit cycles.

Hilbert's 13th problem for algebraic groups

The algebraic form of Hilbert's 13th problem asks for the resolvent degree \operatorname{RD}(n) of the general polynomial f(x) = x^n + a_1 x^{n-1} + \cdots + a_n of degree n , where a_1, \ldots, a_n are independent variables. The resolvent degree is the minimal integer d such that every root of f(x) can be obtained in a finite number of steps, starting with \mathbb{C}(a_1, \ldots, a_n) and adjoining algebraic functions in \!{}\leqslant\nobreak d variables at each step. Recently Farb and Wolfson defined the resolvent degree \operatorname{RD}_k(G) for every finite group G and any base field k of characteristic 0 . In this setting \operatorname{RD}(n) = \operatorname{RD}_{\mathbb{C}}(\operatorname{S}_n) , where \operatorname{S}_n denotes the symmetric group. In this paper we extend their definition of \operatorname{RD}_k(G) to an arbitrary algebraic {group} G over an arbitrary field k . We investigate the dependency of this quantity on k and show that \operatorname{RD}_k(G) \leqslant 5 for any field k and any connected group G . The question whether \operatorname{RD}_k(G) can be bigger than 1 for any field k and any algebraic group G over k (not necessarily connected) remains open.

The Extended 16th Hilbert Problem for Discontinuous Piecewise Systems Formed by Linear Centers and Linear Hamiltonian Saddles Separated by a Nonregular Line

We study discontinuous piecewise linear differential systems formed by linear centers and/or linear Hamiltonian saddles and separated by a nonregular straight line. There are two classes of limit cycles: the ones that intersect the separation line at two points and the ones that intersect the separation line in four points, named limit cycles of type [Formula: see text] and limit cycles of type [Formula: see text], respectively. We prove that the maximum numbers of limit cycles of types [Formula: see text] and [Formula: see text] are two and one, respectively. We show that all these upper bounds are reached providing explicit examples.

An extension of the 16th Hilbert problem for continuous piecewise linear–quadratic centers separated by a non-regular line

In the last few decades, there has been much interest in studying piecewise differential systems. This is mainly due to the fact that these differential systems allow us to modelize many natural phenomena. In order to describe the dynamics of a differential system, we need to control its periodic orbits and, especially, its limit cycles. In particular, providing an upper bound for the maximum number of limit cycles that such differential systems can exhibit would be desirable, that is solving the extended 16th Hilbert problem. In general, this is an unsolved problem. In this paper, we give an upper bound for the maximum number of limit cycles that a class of continuous piecewise differential systems formed by an arbitrary linear center and an arbitrary quadratic center separated by a non-regular line can exhibit. So for this class of continuous piecewise differential systems, we have solved the extended 16th Hilbert problem, and the upper bound found is seven. The question whether this upper bound is sharp remains open.

The 16th Hilbert Problem for Discontinuous Piecewise Linear Differential Systems Separated by the Algebraic Curve $$y=x^{n}$$

The 16th Hilbert Problem for Discontinuous Piecewise Linear Differential Systems Separated by the Algebraic Curve $$y=x^{n}$$

Some families of quadratic systems with at most one limit cycle

The work of Chicone and Shafer published in 1982 together with the work of Bamon published in 1986 proved that any polynomial differential system of degree two has finitely many limit cycles. But the problem remains open of providing a uniform upper bound for the maximum number of limit cycles that a polynomial differential system of degree two can have, i.e. the second part of the 16th Hilbert problem restricted to the polynomial differential systems of degree two remains open. Here we present six subclasses of polynomial differential systems of degree two for which we can prove that an upper bound for their maximum number of limit cycles is one.

A refinement of Reznick's Positivstellensatz with applications to quantum information theory

In his solution of Hilbert's 17th problem Artin showed that any positive definite polynomial in several variables can be written as the quotient of two sums of squares. Later Reznick showed that the denominator in Artin's result can always be chosen as an N-th power of the squared norm of the variables and gave explicit bounds on N. By using concepts from quantum information theory (such as partial traces, optimal cloning maps, and an identity due to Chiribella) we give simpler proofs and minor improvements of both real and complex versions of this result. Moreover, we discuss constructions of Hilbert identities using Gaussian integrals and we review an elementary method to construct complex spherical designs. Finally, we apply our results to give improved bounds for exponential quantum de Finetti theorems in the real and in the complex setting.

Global-in-time existence of unbounded separatrices and blow-up time of solutions for Liénard systems whose restoring force is an odd power function

The existence, uniqueness, and number of limit cycles in Liénard dynamics have been intensely studied since the 1940s because they relate to Hilbert's 16th problem. Limit cycles sharply distinguish the nature of adjacent orbits. Moreover, they are essential for understanding dynamical systems. In contrast, the importance of unbounded separatrices is less well recognized even though they play an identical role. Therefore, this study focuses on the Liénard dynamical system and investigates whether all orbits, including unbounded separatrices, exist globally in time. The blow-up times of solutions that cannot be extended up to negative or positive infinity are also discussed. The result allows classification of all solutions by the time for which they exist. For example, for the van der Pol oscillator known to have the sole limit cycle, all solutions that pass through any point in the region bounded by that limit cycle and the two solutions that correspond to the unbounded separatrices continue to exist for all time, whereas the other solutions have positive or negative blow-up times. This fact cannot be judged solely from the shapes of the orbits obtained by simulation. Phase plane analysis is applied to prove the main theorem. Furthermore, several examples and their plane figures are provided to facilitate the understanding of the proof.

Limit cycles of continuous piecewise differential systems separated by a parabola and formed by a linear center and a quadratic center

<p style='text-indent:20px;'>Due to their applications to many physical phenomena during these last decades the interest for studying the continuous or discontinuous piecewise differential systems has increased strongly. The limit cycles play a main role in the study of any planar differential system. Up to now the major part of papers which study the limit cycles of the planar piecewise differential systems have considered systems formed by two pieces separated by one straight line.</p><p style='text-indent:20px;'>Here we consider planar continuous piecewise differential systems separated by a parabola.</p><p style='text-indent:20px;'>We prove that the planar continuous piecewise differential systems separated by a parabola and formed by a linear center and a quadratic center have at most one limit cycle. Moreover there are systems in this class exhibiting one limit cycle. So in particular we have solved the extension of the 16th Hilbert problem to this class of differential systems.</p>

On the Limit Cycles of a Class of Discontinuous Piecewise Differential Systems Formed by Two Rigid Centers Governed by Odd Degree Polynomials

We provide an upper bound for the maximum number of limit cycles for the class of discontinuous piecewise differential systems formed by two differential systems separated by the straight line [Formula: see text], one of which is a linear rigid center while the other is a rigid center formed of a linear part plus a homogeneous polynomial of odd degree. We solve the extended 16th Hilbert problem for this class of discontinuous piecewise differential systems.

Limit cycles in piecewise polynomial Hamiltonian systems allowing nonlinear switching boundaries

This paper aims to study the limit cycles of planar piecewise polynomial Hamiltonian systems of degree n with the switching boundary y=xm, where m and n are positive integers. We answer a version of the 16th Hilbert problem for such systems, providing an upper bound for the maximum number of limit cycles in the function of m and n. We also are devoted to giving a lower bound via perturbing piecewise linear Hamiltonian systems having at the origin a global center. For this, a complete classification of the center conditions is established. In pursuit of a better lower bound, we require that this global center is nonlinear induced by the piecewise linearity, instead of the normal linear differential center considered in most of the existing articles. This renders that the traveling time of the unperturbed periodic orbits in each smooth zone is not calculable explicitly. Thus it is difficult to use the known Melnikov functions and averaged functions to study the bifurcating limit cycles. To overcome this difficulty, we develop an arbitrary order Melnikov-like function, which does not depend on the traveling time, for general d-dimensional piecewise smooth integrable systems allowing nonlinear switching boundaries. Finally, employing the new Melnikov-like function to the considered perturbation problem, we obtain a lower bound and an upper bound for the maximum number of limit cycles bifurcating from the unperturbed periodic orbits up to any order.

Density of composite places in function fields and applications to real holomorphy rings

AbstractGiven an algebraic function field and a place ℘ on K, we prove that the places that are composite with extensions of ℘ to finite extensions of K lie dense in the space of all places of F, in a strong sense. We apply the result to the case of any real closed field and the fixed place on R being its natural (finest) real place. This leads to a new description of the real holomorphy ring of F, which can be seen as an analog to a certain refinement of Artin's solution of Hilbert's 17th problem. We also determine the relation between the topological space of all ‐places of F (places with residue field contained in ), its subspace of all ‐places of F that are composite with the natural ‐place of R, and the topological space of all R‐rational places. Further results about these spaces as well as various classes of relative real holomorphy rings are proven. At the conclusion of the paper, the theory of real spectra of rings will be applied to interpret basic concepts from that angle and to show that the space has only finitely many topological components.