Polynomial Systems Research Articles

Periodicidad en modelos de depredación del tipo Gause considerando colaboración entre los depredadores

Predation models are a great source of study from both an ecological and a mathematical point of view, especially for the analysis of trophic chains. The determination of the dynamics of the systems that describe them, as well as the verification of the nature of these properties by the interacting species, is a topic that is sometimes not always correlated. It is widely known that the incorporation of some mathematical descriptions of ecological phenomena strongly modifies the properties of many of these models. This implies that the systems describing such models are structurally unstable. In this work, we include collaboration or cooperation between predators, a social behavior that describes the help made to capture their favorite prey. It is described by a power function with an exponent between 0 and 1, to indicate the possible interference between them, despite their collaboration. The exponent is interpreted as the density-dependent aggregation index. We show that this assumption originates a varied behavior of the system, with respect to the associated Kolmogorov-type quadratic polynomial system that does not consider collaboration, including the existence of a stable limit cycle around a positive equilibrium point, among other analytical properties.

Practical complexities of probabilistic algorithms for solving Boolean polynomial systems

Solving a polynomial system over a finite field is an NP-complete problem of fundamental importance in both pure and applied mathematics. In particular, the security of the so-called multivariate public-key cryptosystems, such as HFE of Patarin and UOV of Kipnis et al., is based on the postulated hardness of solving quadratic polynomial systems over a finite field. Lokshtanov et al. (2017) were the first to introduce a probabilistic algorithm that, in the worst-case, solves a Boolean polynomial system in time O∗(2δn), for some δ∈(0,1) depending only on the degree of the system, thus beating the brute-force complexity O∗(2n). Later, Björklund et al. (2019) and then Dinur (2021) improved this method and devised probabilistic algorithms with a smaller exponent coefficient δ.We survey the theory behind these probabilistic algorithms, and we illustrate the results that we obtained by implementing them in C. In particular, for random quadratic Boolean systems, we estimate the practical complexities of the algorithms and their probabilities of success as their parameters change.

On facets of the Newton polytope for the discriminant of the polynomial system

On facets of the Newton polytope for the discriminant of the polynomial system

Bifurcation of limit cycles and isochronous centers on center manifolds for four‐dimensional systems

This paper is concerned with the bifurcation of limit cycles and isochronous centers on center manifolds for four‐dimensional nonlinear dynamic systems, which have a pair of purely imaginary eigenvalues and a pair of eigenvalues with negative real part for the Jacobian matrix corresponding to the singularity situated at the origin. In order to investigate the isochronous centers for four‐dimensional systems, a new algorithm to calculate so‐called isochronous constants is given. Moreover, with the calculation of the isochronous constants, the conditions for the existence of an isochronous center are determined in a way which does not need first computing the center manifolds of the four‐dimensional system. Finally, as an application, we investigate a class of quadratic polynomial systems for a model of airfoil and prove that the system can have five small limit cycles around the origin on the center manifold. Moreover, we apply our new methodology to find the sufficient and necessary conditions for the origin to be an isochronous center restricted to the center manifold, illustrating the reliability and availability of the new algorithm.

Application of The Sine-Gordon Expansion Method on Nonlinear Various Physical Models

In this paper, by utilizing the Sine-Gordan expansion method, soliton solutions of the higher-order improved Boussinesq equation, Kuramoto-Sivashinsky equation, and seventh-order Sawada-Kotera equation are obtained. Given partial differential equations are reduced to ordinary differential equations, by choosing the compatible wave transformation associated with the structure of the equation. Based on the solution of the Sine-Gordan equation, a polynomial system of equations is obtained according to the principle of homogeneous balancing. The solution of the outgoing system gives the parameters which are included by the solution. Plot3d and Plot2d graphics are given in detail. As a result, many different graphic models are obtained from soliton solutions of equations that play a very important role in mathematical physics and engineering.

On Series of Orthogonal Polynomials and Systems of Classical Type Polynomials

On Series of Orthogonal Polynomials and Systems of Classical Type Polynomials

Limit Cycle Bifurcations in a Class of Piecewise Smooth Polynomial Systems

In this paper, we consider the bifurcation problem of limit cycles for a class of piecewise smooth cubic systems separated by the straight line [Formula: see text]. Using the first order Melnikov function, we prove that at least [Formula: see text] limit cycles can bifurcate from an isochronous cubic center at the origin under perturbations of piecewise polynomials of degree [Formula: see text]. Further, the maximum number of limit cycles bifurcating from the center of the unperturbed system is at least [Formula: see text] if the origin is the unique singular point under perturbations.

P-Adic valuations associated to fibers and images of polynomial functions

The Chevalley-Warning and the Ax-Katz theorems, which provide the p-adic valuations for the number of solutions of the system of polynomials in finite fields, are stated in terms of the degrees of the associated polynomials. Recently, Clark et al. reformulated the theorems in terms of the cardinalities of fibers and images of the associated polynomial functions, and raised a question in this setting. In this paper, we give a negative answer to that question and improve their results in general case using p-adic lifting via the Teichmüller character.

The Investigation of Nonlinear Polynomial Control Systems

The paper considers methods for estimating stability using Lyapunov functions, which are used for nonlinear polynomial control systems. The apparatus of the Gro¨bner basis method is used to assess the stability of a dynamical system. A description of the Gro¨bner basis method is given. To apply the method, the canonical relations of the nonlinear system are approximated by polynomials of the components of the state and control vectors. To calculate the Gro¨bner basis, the Buchberger algorithm is used, which is implemented in symbolic computation programs for solving systems of nonlinear polynomial equations. The use of the Gro¨bner basis for finding solutions of a nonlinear system of polynomial equations is considered, similar to the application of the Gauss method for solving a system of linear equations. The equilibrium states of a nonlinear polynomial system are determined as solutions of a nonlinear system of polynomial equations. An example of determining the equilibrium states of a nonlinear polynomial system using the Gro¨bner basis method is given. An example of finding the critical points of a nonlinear polynomial system using the Gro¨bner basis method and the Wolfram Mathematica application software is given. The Wolfram Mathematica program uses the function of determining the reduced Gro¨bner basis. The application of the Gro¨bner basis method for estimating the attraction domain of a nonlinear dynamic system with respect to the equilibrium point is considered. To determine the scalar potential, the vector field of the dynamic system is decomposed into gradient and vortex components. For the gradient component, the scalar potential and the Lyapunov function in polynomial form are determined by applying the homotopy operator. The use of Gro¨bner bases in the gradient method for finding the Lyapunov function of a nonlinear dynamical system is considered. The coordination of input-output signals of the system based on the construction of Gro¨bner bases is considered.

The Songling system has exactly four limit cycles

Determining how many limit cycles a planar polynomial system of differential equations can have is a remarkably hard problem. One of the main difficulties is that the limit cycles can reside within areas of vastly different scales. This makes numerical explorations very hard to perform, requiring high precision computations, where the necessary precision is not known in advance. Using rigorous computations, we can dynamically determine the required precision, and localize all limit cycles of a given system. We prove that the Songling system of planar, quadratic polynomial differential equations has exactly four limit cycles. Furthermore, we give precise bounds for the positions of these limit cycles using rigorous computational methods based on interval arithmetic. The techniques presented here are applicable to the much wider class of real-analytic planar differential equations.

Bivariate systems of polynomial equations with roots of high multiplicity

Given a bivariate system of polynomial equations with fixed support sets [Formula: see text] it is natural to ask which multiplicities its solutions can have. We prove that there exists a system with a solution of multiplicity [Formula: see text] for all [Formula: see text] in the range [Formula: see text], where [Formula: see text] is the set of all integral vectors that shift B to a subset of [Formula: see text]. As an application, we classify all pairs [Formula: see text] such that the system supported at [Formula: see text] does not have a solution of multiplicity higher than [Formula: see text].

New complexity estimation on the Rainbow-Band-Separation attack

Multivariate public key cryptography is a candidate for post-quantum cryptography, and it allows generating particularly short signatures and fast verification. The Rainbow signature scheme proposed by Ding and Schmidt is such a multivariate cryptosystem, and it is considered secure against all known attacks. The Rainbow-Band-Separation attack recovers a secret key of Rainbow by solving certain systems of quadratic equations, and its complexity is estimated by the well-known theoretical value called the degree of regularity. However, the degree of regularity is generally larger than the solving degree in experiments, and an accurate estimation cannot be obtained. In this article, we propose a new theoretical value for the complexity of the Rainbow-Band-Separation attack using a Gröbner basis algorithm, which provides a more precise estimation compared to that using the degree of regularity. This theoretical value is deduced by the two-variable power series∏i=1m(1−t1di1t2di2)(1−t1)n1(1−t2)n2. Since the two-variable power series coincides with the one-variable power series at t1=t2 deriving the degree of regularity, the theoretical value is less than or equal to the degree of regularity under a certain condition. Moreover, we show a relation between the Rainbow-Band-Separation attack using the hybrid approach and the HighRank attack. By considering this relation and our theoretical value, we obtain a new complexity estimation for the Rainbow-Band-Separation attack. Furthermore, applying our theoretical value to the complexity formula used in the NIST PQC 2nd round, we show that a slight modification of the proposed Rainbow parameter sets is required. Consequently, we provide a new theoretical value for generally estimating the solving degree of a bi-graded polynomial system, which can influence the parameter selection of Rainbow in the NIST PQC standardization project.

Exotic Bifurcations in Three Connected Populations with Allee Effect

We consider three connected populations with strong Allee effect, and give a complete classification of the steady state structure of the system with respect to the Allee threshold and the dispersal rate, describing the bifurcations at each critical point where the number of steady states change. One may expect that by increasing the dispersal rate between the patches, the system would become more well-mixed, hence simpler. However, we show that it is not always the case, and the number of steady states may (temporarily) go up by increasing the dispersal rate. Besides sequences of pitchfork and saddle-node bifurcations, we find triple-transcritical bifurcations and also a sun-ray shaped bifurcation where twelve steady states meet at a single point then disappear. The major tool of our investigations is a novel algorithm that decomposes the parameter space with respect to the number of steady states and finds the bifurcation values using cylindrical algebraic decomposition with respect to the discriminant variety of the polynomial system.

On Behavior of the Periodic Orbits of a Hamiltonian System of Bifurcation of Limit Cycles

In light of the previous recent studies by Jaume Llibre et al. that dealt with the finite cycles of generalized differential Kukles polynomial systems using the first- and second-order mean theorem such as (Nonlinear Anal., 74, 1261–1271, 2011) and (J. Dyn. Control Syst., vol. 21, 189–192, 2015), in this work, we provide upper bounds for the maximum number of limit cycles bifurcating from the periodic orbits of Hamiltonian system using the averaging theory of first order.

Limit cycles appearing from a generalized heteroclinic loop with a cusp and a nilpotent saddle

This paper studies the limit cycle bifurcation problem of a class of piecewise smooth differential polynomial systems of degree n by perturbing a piecewise cubic polynomial system having a generalized heteroclinic loop with a cusp and a nilpotent saddle. First, we provide all possible phase portraits of the unperturbed system on the plane with crossing periodic orbits and obtain a condition for the appearance of a generalized heteroclinic loop with a cusp and a nilpotent saddle by qualitative theoretical knowledge. Then, we investigate the algebraic structure of the first order Melnikov function and give its asymptotic expansion near the generalized heteroclinic loop with the help of analytical skills. Finally, we employ the expansion together with its coefficients to obtain the existence of at least 3n−1 limit cycles.

Center condition and bifurcation of limit cycles for quadratic switching systems with a nilpotent equilibrium point

In this work, a new perturbation approach is developed based on Bogdanov-Takens bifurcation theory, which enables the Poincaré-Lyapunov method for switching systems with linear type centers to be applied for studying the center conditions of planar switching polynomial systems associated with a nilpotent equilibrium point. The new method is then applied to consider a class of quadratic switching nilpotent systems, and a complete classification is given on the conditions of the nilpotent equilibrium point to be a center. Moreover, based on one of the center conditions, an example is constructed to show the existence of seven small-amplitude limit cycles around the nilpotent equilibrium point, which is a new lower bound on the number of limit cycles in such systems.

Bifurcations of Heteroclinic Contours in Two-Parameter Planar Systems: Overview and Explicit Examples

Smooth planar vector fields containing two hyperbolic saddles may possess contours formed by heteroclinic connections between these saddles. We present an overview of known results on bifurcations of these contours. Additionally, two new explicit polynomial systems containing such contours are derived, which are studied using the bifurcation software matcont and are shown to exhibit the theoretically predicted phenomena, including series of heteroclinic connections.

Learning multivariate functions with low-dimensional structures using polynomial bases

In this paper we propose a method for the approximation of high-dimensional functions over finite intervals with respect to complete orthonormal systems of polynomials. An important tool for this is the multivariate classical analysis of variance (ANOVA) decomposition. For functions with a low-dimensional structure, i.e., a low superposition dimension, we are able to achieve a reconstruction from scattered data and simultaneously understand relationships between different variables.

A generalized multivariable Newton method

It is well known that the Newton method may not converge when the initial guess does not belong to a specific quadratic convergence region. We propose a family of new variants of the Newton method with the potential advantage of having a larger convergence region as well as more desirable properties near a solution. We prove quadratic convergence of the new family, and provide specific bounds for the asymptotic error constant. We illustrate the advantages of the new methods by means of test problems, including two and six variable polynomial systems, as well as a challenging signal processing example. We present a numerical experimental methodology which uses a large number of randomized initial guesses for a number of methods from the new family, in turn providing advice as to which of the methods employed is preferable to use in a particular search domain.

Linearizability of planar polynomial Hamiltonian systems

Isochronicity and linearizability of two-dimensional polynomial Hamiltonian systems are revisited and new results are presented. We give a new computational procedure to obtain the necessary and sufficient conditions for the linearization of a polynomial system. Using computer algebra systems we provide necessary and sufficient conditions for linearizability of Hamiltonian systems with homogeneous non-linearities of degrees 5, 6 and 7. We also present some sufficient conditions for systems with nonhomogeneous nonlinearities of degrees two, three and five.