16th Problem Research Articles

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.

Isoparametric Polynomials and Sums of Squares

Abstract Hilbert’s 17th problem asks whether every nonnegative polynomial can be a sum of squares of rational functions. It has been answered affirmatively by Artin. However, the question as to whether a given nonnegative polynomial is a sum of squares of polynomials is still a central question in real algebraic geometry. In this paper, we solve this question completely for the nonnegative polynomials associated with isoparametric polynomials, initiated by E. Cartan, which define the focal submanifolds of the corresponding isoparametric hypersurfaces. Dedicated to Professor Chia-Kuei Peng on his 80th birthday

Convergence rates of Gibbs measures with degenerate minimum

We study convergence rates of Gibbs measures, with density proportional to exp(−f(x)∕t), as t→0 where f: Rd→R admits a unique global minimum at x⋆. We focus on the case where the Hessian is not definite at x⋆. We assume instead that the minimum is strictly polynomial and give a higher order nested expansion of f at x⋆, which depends on every coordinate. We give an algorithm yielding such an expansion if the polynomial order of x⋆ is no more than 8, in connection with Hilbert’s 17th problem. However, we prove that the case where the order is 10 or higher is fundamentally different and that further assumptions are needed. We then give the rate of convergence of Gibbs measures using this expansion. Finally we adapt our results to the multiple well case.

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.

Globally trace-positive noncommutative polynomials and the unbounded tracial moment problem

A noncommutative (nc ) polynomial is called (globally) trace-positive if its evaluation at any tuple of operators in a tracial von Neumann algebra has nonnegative trace. Such polynomials emerge as trace inequalities in several matrix or operator variables, and are widespread in mathematics and physics. This paper delivers the first Positivstellensatz for global trace positivity of nc polynomials. Analogously to Hilbert’s 17th problem in real algebraic geometry, trace-positive nc polynomials are shown to be weakly sums of hermitian squares and commutators of regular nc rational functions. In two variables, this result is strengthened further using a new sum-of-squares certificate with concrete univariate denominators for nonnegative bivariate polynomials. The trace positivity certificates in this paper are obtained by convex duality through solving the so-called unbounded tracial moment problem, which arises from noncommutative integration theory and free probability. Given a linear functional on nc polynomials, the tracial moment problem asks whether it is a joint distribution of integral operators affiliated with a tracial von Neumann algebra. A counterpart to Haviland’s theorem on solvability of the tracial moment problem is established. Moreover, a variant of Carleman’s condition is shown to guarantee the existence of a solution to the tracial moment problem. Together with semidefinite optimization, this is then used to prove that every trace-positive nc polynomial admits an explicit approximation in the 1-norm on its coefficients by sums of hermitian squares and commutators of nc polynomials.

Crossing limit cycles of planar discontinuous piecewise differential systems formed by isochronous centres

These last years an increasing interest appeared in studying the planar discontinuous piecewise differential systems motivated by the rich applications in modelling real phenomena. The understanding of the dynamics of these systems has many difficulties. One of them is the study of their limit cycles. In this paper, we study the maximum number of crossing limit cycles of some classes of planar discontinuous piecewise differential systems separated by a straight line and formed by combinations of linear centres (consequently isochronous) and cubic isochronous centres with homogeneous nonlinearities. For these classes of planar discontinuous piecewise differential systems we solved the extension of the 16th Hilbert problem, i.e. we provide an upper bound for their maximum number of crossing limit cycles.

The extended 16th Hilbert problem for a class of discontinuous piecewise differential systems

The extended 16th Hilbert problem for a class of discontinuous piecewise differential systems

On the cyclicity of quasi-homogeneous polynomial systems

In this paper, we study the weakened 16th Hilbert problem for a class of quasi-homogeneous polynomial systems and obtain the upper bound as well as the lower one for the number of limit cycles bifurcating from the period annulus of such systems.

The criticality of reversible quadratic centers at the outer boundary of its period annulus

This paper deals with the period function of the reversible quadratic centersXν=−y(1−x)∂x+(x+Dx2+Fy2)∂y, where ν=(D,F)∈R2. Compactifying the vector field to S2, the boundary of the period annulus has two connected components, the center itself and a polycycle. We call them the inner and outer boundary of the period annulus, respectively. We are interested in the bifurcation of critical periodic orbits from the polycycle Πν at the outer boundary. A critical period is an isolated critical point of the period function. The criticality of the period function at the outer boundary is the maximal number of critical periodic orbits of Xν that tend to Πν0 in the Hausdorff sense as ν→ν0. This notion is akin to the cyclicity in Hilbert's 16th Problem. Our main result (Theorem A) shows that the criticality at the outer boundary is at most 2 for all ν=(D,F)∈R2 outside the segments {−1}×[0,1] and {0}×[0,2]. With regard to the bifurcation from the inner boundary, Chicone and Jacobs proved in their seminal paper on the issue that the upper bound is 2 for all ν∈R2. In this paper the techniques are different because, while the period function extends analytically to the center, it has no smooth extension to the polycycle. We show that the period function has an asymptotic expansion near the polycycle with the remainder being uniformly flat with respect to ν and where the principal part is given in a monomial scale containing a deformation of the logarithm, the so-called Écalle-Roussarie compensator. More precisely, Theorem A follows by obtaining the asymptotic expansion to fourth order and computing its coefficients, which are not polynomial in ν but transcendental. Theorem A covers two of the four quadratic isochrones, which are the most delicate parameters to study because its period function is constant. The criticality at the inner boundary in the isochronous case is bounded by the number of generators of the ideal of all the period constants but there is no such approach for the criticality at the outer boundary. A crucial point to study it in the isochronous case is that the flatness of the remainder in the asymptotic expansion is preserved after the derivation with respect to parameters. We think that this constitutes a novelty that is of particular interest also in the study of similar problems for limit cycles in the context of Hilbert's 16th Problem. Theorem A also reinforces the validity of a long standing conjecture by Chicone claiming that the quadratic centers have at most two critical periodic orbits. A less ambitious goal is to prove the existence of a uniform upper bound for the number of critical periodic orbits in the family of quadratic centers. By a compactness argument this would follow if one can prove that the criticality of the period function at the outer boundary of any quadratic center is finite. Theorem A leaves us very close to this existential result.

Crossing limit cycles for discontinuous piecewise differential systems formed by linear Hamiltonian saddles or linear centers separated by a conic

The extension of the 16th Hilbert problem to discontinuous piecewise linear differential systems asks for an upper bound for the maximum number of crossing limit cycles that such systems can exhibit. The study of this problem is being very active, especially for discontinuous piecewise linear differential systems defined in two zones and separated by one straight line. In the case that the differential systems in these zones are formed either by linear centers or linear Hamiltonian saddles it is known that there are no crossing limit cycles. However it is also known that the number of crossing limit cycles can change if we change the shape of the discontinuity curve. In this paper we study the maximum number of crossing limit cycles of discontinuous piecewise differential systems formed by either linear Hamiltonian saddles or linear centers and separated by a conic which intersect the conic in two points. For this class of discontinuous piecewise differential systems we solve the extended 16th Hilbert problem.

Cyclicity of canard cycles with hyperbolic saddles located away from the critical curve

The goal of our paper is to study limit cycles in smooth planar slow-fast systems, Hausdorff close to canard cycles containing hyperbolic saddles located away from the critical (or slow) curve. Our focus is on a broad class of smooth slow-fast systems with a Hopf breaking mechanism (often called a generic turning point), a jump breaking mechanism or a non-generic turning point. Such canard cycles naturally occur in predator-prey systems with Holling type II and IV response functions and with a small predator's death rate. The study of canard cycles is also relevant for the Hilbert's 16th problem. We primarily focus on the canard cycles with one hyperbolic saddle (located away from the slow curve) and we allow isolated singularities in the slow dynamics.

Functional norms, condition numbers and numerical algorithms in algebraic geometry

Abstract In numerical linear algebra, a well-established practice is to choose a norm that exploits the structure of the problem at hand to optimise accuracy or computational complexity. In numerical polynomial algebra, a single norm (attributed to Weyl) dominates the literature. This article initiates the use of $L_p$ norms for numerical algebraic geometry, with an emphasis on $L_{\infty }$ . This classical idea yields strong improvements in the analysis of the number of steps performed by numerous iterative algorithms. In particular, we exhibit three algorithms where, despite the complexity of computing $L_{\infty }$ -norm, the use of $L_p$ -norms substantially reduces computational complexity: a subdivision-based algorithm in real algebraic geometry for computing the homology of semialgebraic sets, a well-known meshing algorithm in computational geometry and the computation of zeros of systems of complex quadratic polynomials (a particular case of Smale’s 17th problem).

Limit cycles of discontinuous piecewise differential systems formed by linear and cubic isochronous centers

One of the main problems in the qualitative theory of the planar differential systems is to control the existence and the number of their limit cycles. There are many researchers who tried to solve this problem for special classes of planar differential systems, see for instance [7, 16]. In this paper, we study the maximum number of limit cycles for discontinuous planar piecewise differential systems formed by four classes of isochronous cubic centers separated by irregular straight line. We provide a sharp upper bound for the maximum number of crossing limit cycles that these classes of discontinuous piecewise differential systems can exhibit. Therefore, we will solve the extended of the 16th Hilbert problem for these classes. Keywords: limit cycles; cubic isochronous centers; linear differential center.

Limit Cycles of Continuous Piecewise Differential Systems Formed by Linear and Quadratic Isochronous Centers I

First, we study the planar continuous piecewise differential systems separated by the straight line [Formula: see text] formed by a linear isochronous center in [Formula: see text] and an isochronous quadratic center in [Formula: see text]. We prove that these piecewise differential systems cannot have crossing periodic orbits, and consequently they do not have crossing limit cycles. Second, we study the crossing periodic orbits and limit cycles of the planar continuous piecewise differential systems separated by the straight line [Formula: see text] having in [Formula: see text] the general quadratic isochronous center [Formula: see text], [Formula: see text] after an affine transformation, and in [Formula: see text] an arbitrary quadratic isochronous center. For these kind of continuous piecewise differential systems the maximum number of crossing limit cycles is one, and there are examples having one crossing limit cycles. In short for these families of continuous piecewise differential systems we have solved the extension of the 16th Hilbert problem.

The Extended 16th Hilbert Problem for Discontinuous Piecewise Linear Centers Separated by a Nonregular Line

The study of the piecewise linear differential systems goes back to Andronov, Vitt and Khaikin in 1920’s, and nowadays such systems still continue to receive the attention of many researchers mainly due to their applications. We study the discontinuous piecewise differential systems formed by two linear centers separated by a nonregular straight line. We provide upper bounds for the maximum number of limit cycles that these discontinuous piecewise differential systems can exhibit and we show that these upper bounds are reached. Hence, we solve the extended 16th Hilbert problem for this class of piecewise differential systems.

First-order perturbation for multi-parameter center families

In the weak 16th Hilbert problem, the Poincaré-Pontryagin-Melnikov function, M1(h), is used for obtaining isolated periodic orbits bifurcating from centers up to a first-order analysis. This problem becomes more difficult when a family of centers is considered. In this work we provide a compact expression for the first-order Taylor series of the function M1(h,a) with respect to a, being a the multi-parameter in the unperturbed center family. More concretely, when the center family has an explicit first integral or inverse integrating factor depending on a. We use this new bifurcation mechanism to increase the number of limit cycles appearing up to a first-order analysis without the difficulties that higher-order studies present. We show its effectiveness by applying it to some classical examples.

Cyclicity of a Class of Hamiltonian Systems under Perturbations of Piecewise Smooth Polynomials

In the study of the weakened Hilbert’s 16th problem, there usually appears period annulus surrounding more than one singular point, implying that there are often more than three generate functions in the Abelian integrals (or the first order Melnikov functions). It is also known that there are usually more than three generate functions in the study of the bifurcation of limit cycles in piecewise smooth Hamiltonian systems. This paper aims to explore a method so that we can estimate the upper bound of limit cycles bifurcating from the period annulus if the unperturbed system has some good properties such as Picard–Fuchs equations. This method is based on [Gavrilov & Iliev, 2009] and is useful for smooth and piecewise smooth polynomial vector fields of a general degree. Finally, we present an example to illustrate an application of the theoretical results.

Smale 17th Problem: Advances and Open Directions

We give an overview of Smale's 17th problem describing the context in whichSmale proposed it, the ideas that led to its solution, and the extensionsand subsequent progress after this solution.

The quest for Diophantine finite-fold-ness

The Davis-Putnam-Robinson theorem showed that every partially computable $m$-ary function f(a1, ..., am) = c on the natural numbers can be specified by means of an exponential Diophantine formula involving, along with parameters a1, ... am, c, some number k of existentially quantified variables. Yuri Matiyasevich improved this theorem in two ways: on the one hand, he proved that the same goal can be achieved with no recourse to exponentiation and, thereby, he provided a negative answer to Hilbert's 10th problem; on the other hand, he showed how to construct an exponential Diophantine equation specifying f which, once a1, ... am have been fixed, is solved by at most one tuple of values for the remaining variables. This latter property is called single-foldness. Whether there exists a single- (or, at worst, finite-) fold polynomial Diophantine representation of any partially computable function on the natural numbers is as yet an open problem. This work surveys relevant results on this subject and tries to draw a route towards a hoped-for positive answer to the finite-fold-ness issue.