Polynomial Systems Research Articles

Randomized methods for computing joint eigenvalues, with applications to multiparameter eigenvalue problems and root finding

AbstractIt is well known that a family of $${n}\times {n}$$ n × n commuting matrices can be simultaneously triangularized by a unitary similarity transformation. The diagonal entries of the triangular matrices define the n joint eigenvalues of the family. In this work, we consider the task of numerically computing approximations to such joint eigenvalues for a family of (nearly) commuting matrices. This task arises, for example, in solvers for multiparameter eigenvalue problems and systems of multivariate polynomials, which are our main motivations. We propose and analyze a simple approach that computes eigenvalues as one-sided or two-sided Rayleigh quotients from eigenvectors of a random linear combination of the matrices in the family. We provide some analysis and numerous numerical examples, showing that such randomized approaches can compute semisimple joint eigenvalues accurately and lead to improved performance of existing solvers.

Unified approach to reciprocal matrices with Kippenhahn curves containing elliptical components

We consider tridiagonal matrices ( a ij ) i , j = 1 n with constant main diagonal and such that a i , i + 1 a i + 1 , i = 1 for i = 1 , … , n − 1 . For these matrices, criteria are established under which their Kippenhahn curves contain elliptical components or even consist completely of such. These criteria are in terms of a system of homogeneous polynomial equations in variables ( | a j , j + 1 | − | a j + 1 , j | ) 2 , and established via a unified approach across arbitrary dimensions. The results are illustrated, and specific numerical examples are provided for n = 7, thus generalizing earlier work in the lower-dimensional setting.

Global Dynamics and Integrability of a Leslie-Gower Predator–Prey Model with Linear Functional Response and Generalist Predator

We deal with a Leslie-Gower predator–prey model with a generalist or alternating food for predator and linear functional response. Using a topological equivalent polynomial system we prove that the system is not Liouvillian (hence also not Darboux) integrable. In order to study the global dynamics of this model, we use the Poincaré compactification of R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^2$$\\end{document} to characterize all phase portraits in the Poincaré disc, obtaining two different topological phase portraits.

An analysis of the Crossbred Algorithm for the MQ Problem

The Crossbred algorithm is currently the state-of-the-art method for solving overdetermined multivariate polynomial systems over 𝔽 2 . Since its publication in 2017, several record breaking implementations have been proposed and demonstrate the power of this hybrid approach. Despite these practical results, the complexity of this algorithm and the choice of optimal parameters for it are difficult open questions. In this paper, we prove a bivariate generating series for potentially admissible parameters of the Crossbred algorithm.

Smooth symmetric systems over a finite field and applications

We study the set of common Fq–rational solutions of “smooth” systems of multivariate symmetric polynomials with coefficients in a finite field Fq. We show that, under certain conditions, the set of common solutions of such polynomial systems over the algebraic closure of Fq has a “good” geometric behavior. This allows us to obtain precise estimates on the corresponding number of common Fq–rational solutions. In the case of hypersurfaces we are able to strengthen the results. We illustrate the interest of these estimates through their application to certain classical combinatorial problems over finite fields.

Reducing overdefined systems of polynomial equations derived from small scale variants of the AES via data mining methods

Reducing overdefined systems of polynomial equations derived from small scale variants of the AES via data mining methods

Symbolic-numeric algorithm for parameter estimation in discrete-time models with exp

Dynamic models describe phenomena across scientific disciplines, yet to make these models useful in application the unknown parameter values of the models must be determined. Discrete-time dynamic models are widely used to model biological processes, but it is often difficult to determine these parameters. In this paper, we propose a symbolic-numeric approach for parameter estimation in discrete-time models that involve univariate non-algebraic (locally) analytic functions such as exp. We illustrate the performance (precision) of our approach by applying our approach to two archetypal discrete-time models in biology (the flour beetle ‘LPA’ model and discrete Lotka-Volterra competition model). Unlike optimization-based methods, our algorithm guarantees to find all solutions of the parameter values up to a specified precision given time-series data for the measured variables provided that there are finitely many parameter values that fit the data and that the used polynomial system solver can find all roots of the associated polynomial system with interval coefficients.

Algorithm for Finding Solutions of Some Overdetermined Systems of Polynomial Equations

Algorithm for Finding Solutions of Some Overdetermined Systems of Polynomial Equations

Polynomial dynamical systems associated with the KdV hierarchy

In 1974, S.P. Novikov introduced the stationary n-equations of the Korteweg–de Vries hierarchy, namely the n-Novikov equations. These are associated with integrable polynomial dynamical systems, with polynomial 2n integrals, in ℂ3n. In this paper, we construct an infinite-dimensional polynomial dynamical system that is universal for all dynamical systems corresponding to the n-Novikov equations. Thus, we solve the well-known problem of the relationship between the n-Novikov equations for different n.

Predictive Controller for Large-Scale Fuzzy Polynomial Systems

This paper presents an innovative method to tackle the predictive-control challenges associated with large-scale fuzzy polynomial systems comprising interconnected polynomial fuzzy systems. This study models large-scale nonlinear fuzzy systems in a polynomial framework, which can reduce the number of fuzzy rules. We derive the conditions for controller synthesis in the main theorem using the Lyapunov theory and sum-of-squares technique. Simulation results confirm the validity and efficiency of this approach.

Numerical analysis of fractional‐order Euler–Bernoulli beam model under composite model

The primary objective of this study is to develop a new constitutive model by combining a fractional‐order Kelvin–Voigt model with an Abel dashpot element in parallel. Subsequently, this new model will be incorporated into the Euler–Bernoulli beam's governing equation, utilizing shifted Legendre polynomials as basis functions, a classical orthogonal polynomial system, to solve the fractional‐order partial differential equations. By comparing the numerical solutions with the analytical solutions, we aim to evaluate the applicability of shifted Legendre polynomials in solving such problems and the accuracy of the obtained numerical solutions. Furthermore, we will investigate the performance of viscoelastic HDPE beams under different loading conditions and conduct a comparative analysis of the displacements of HDPE beams under the new constitutive model and the traditional fractional‐order Kelvin–Voigt model. Through this research, we hope to gain a deeper understanding of the characteristics of fractional‐order phenomena and provide more accurate and efficient numerical simulation and analysis methods for the field of structural mechanics, promoting the development of related engineering applications.

Explicit solutions of the generalized Kudryashov’s equation with truncated M-fractional derivative

The main purpose of this article is to study the generalized Kudryashov’s equation with truncated M-fractional derivative, which is commonly used to describe the propagation of wide pulses in nonlinear optical fibers. By employing the complete discriminant system of fourth-order polynomials, various types of explicit solutions are systematically classified, which include periodic solutions, the trigonometric functions, the double-period solutions, and the elliptic function solutions. Additionally, a series of 2D, 3D, and contour plots are generated to visually depict the spatial distribution and evolution of various solutions. This not only advances the development of nonlinear equations in theory but also provides valuable guidance in practical applications.

Lessons on Datasets and Paradigms in Machine Learning for Symbolic Computation: A Case Study on CAD

Symbolic Computation algorithms and their implementation in computer algebra systems often contain choices which do not affect the correctness of the output but can significantly impact the resources required: such choices can benefit from having them made separately for each problem via a machine learning model. This study reports lessons on such use of machine learning in symbolic computation, in particular on the importance of analysing datasets prior to machine learning and on the different machine learning paradigms that may be utilised. We present results for a particular case study, the selection of variable ordering for cylindrical algebraic decomposition, but expect that the lessons learned are applicable to other decisions in symbolic computation. We utilise an existing dataset of examples derived from applications which was found to be imbalanced with respect to the variable ordering decision. We introduce an augmentation technique for polynomial systems problems that allows us to balance and further augment the dataset, improving the machine learning results by 28% and 38% on average, respectively. We then demonstrate how the existing machine learning methodology used for the problem—classification—might be recast into the regression paradigm. While this does not have a radical change on the performance, it does widen the scope in which the methodology can be applied to make choices.

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.

Limit Cycles Bifurcating from Planar Polynomial Quasi-Homogeneous Centers of Weight-Degree 3 with Nonsmooth Perturbations

In this paper, we estimate the number of limit cycles bifurcating from the periodic orbits of the weight-homogeneous polynomial centers of weight-degree 3, when they are perturbed inside the class of piecewise smooth polynomial systems of degree n. By using the first-order averaging method, we give the upper and lower bounds on the maximal number of limit cycles which can bifurcate from the period annuluses. Our results indicate that the nonsmooth systems can have more limit cycles than the smooth ones.

The Lorenz system as a gradient-like system

We formulate, for continuous-time dynamical systems, a sufficient condition to be a gradient-like system, i.e. that all bounded trajectories approach stationary points and therefore that periodic orbits, chaotic attractors, etc do not exist. This condition is based upon the existence of an auxiliary function defined over the state space of the system, in a way analogous to a Lyapunov function for the stability of an equilibrium. For polynomial systems, Lyapunov functions can be found computationally by using sum-of-squares optimisation. We demonstrate this method by finding such an auxiliary function for the Lorenz system. We are able to show that the system is gradient-like for 0⩽ρ⩽12 when σ = 10 and β=8/3 , significantly extending previous results. The results are rigorously validated by a novel procedure: First, an approximate numerical solution is found using finite-precision floating-point sum-of-squares optimisation. We then prove that there exists an exact solution close to this using interval arithmetic.

Symbolic computation in algebra, geometry, and differential equations

In this survey article we describe how symbolic computation in algebra and geometry leads to symbolic, i.e., formula solutions of algebraic differential equations. Symbolic solutions of algebraic differential equations can be derived from parametrizations of corresponding algebraic varieties. Such parametrizations in turn can be computed by elimination methods, i.e., methods for solving systems of polynomial equations.

Fault reconstruction‐based robust tracking control design for periodic piecewise polynomial systems with disturbances

AbstractIn this article, we address the topics of fault reconstruction and robust fault‐tolerant tracking control for periodic piecewise polynomial uncertain systems with time‐varying delays, faults and external disturbances. In detail, the periodic piecewise polynomial learning observer system is configured with the intent to concurrently reconstruct the faults and immeasurable states of the system that is being considered. Precisely, the key intent of this study centers around in devising a robust fault‐tolerant tracking control to achieve the tracking performance of the system as intended with mixed and passivity performance. To this end, a robust fault‐tolerant tracking control is designed by incorporating the reconstructed faults and error feedback, which aids in vouching for the outcomes that were desired. Besides, by building the periodic piecewise polynomial Lyapunov–Krasovskii functional and making use of the matrix polynomial lemma and Wirtinger's inequality, certain adequate conditions are procured in the frame of linear matrix inequalities. Subsequently, the required gain values for the devised controller and the configured observer are found by using the established linear matrix inequalities. Ultimately, in order to attest the controller's intrinsic potential and relevance, numerical example along with the simulation results is supplied.

A Polynomial System of Degree Four with an Invariant Square Containing At Least Five Limit Cycles

We consider a class of polynomial systems of degree four with four real invariant straight lines that form a square, called this an invariant square, and also that contains in its interior at least five small amplitude limit cycles for a certain choice of the parameters. Moreover, we will obtain the necessary and sufficient conditions for the critical point inside the square to be a center.

The variational approach to study the mixed convection boundary layer flow over a permeable Riga plate

AbstractThe physical problem of steady state, laminar, mixed convection (Ri) with double‐diffusive (N) in an electrically low conducting fluid past a semi‐infinite electromagnetic () influenced flat plate with internal uniform heat generation (Q) in the presence of suction/injection (H) by considering viscous dissipation (Ec), thermophoresis (Nt) and thermal diffusion effects (Sr) is mathematically modeled as a simultaneous system of nonlinear partial differential equations. To achieve the solution of the problem numerically, Gyarmati's variational principle known as the “Governing Principle of Dissipative Processes” on the basis of nonequilibrium thermodynamic processes in the theory of continua, is adopted. This research work correlates the phenomenon of fluid around submersibles/space vehicles and provides related insights. To estimate the transportation fluid fields within the boundary layer, the appropriate trial polynomials have been employed, and functionals for the integral variational principle are determined. Next, the Euler–Lagrange equations of the functionals are obtained as a system of polynomial equations involving boundary layer thicknesses of momentum, temperature, and concentration. The expressions of local shear stress, local Nusselt, and local Sherwood numbers have been derived and the effects of various physical factors involved in the problem are explored. A comparison with the previously published results in the literature is provided to confirm the validity of the solution procedure. The results depict that injection () and opposing buoyancy () decrease the skin friction about 38% in sea water and 11% in ionized air when compared to impermeable plate for . The aiding buoyancy () plays a dominant role in heat and mass transfers, respectively, with the massive gradients of 340% and 763% in magnitude for the heavier fluid sea water flow while 47% and 3% for the lighter fluid ionized air flow when . The buoyancy parameters (Ri, N) decrease the heat transfer, but increase the mass transfer.