Polynomial inverse integrating factors for quadratic differential systems

We characterize all the quadratic polynomial differential systems having a polynomial inverse integrating factor and provide explicit normal forms for such systems and for their associated first integrals. We also prove that these families of quadratic systems have no limit cycles.     

Planar polynomial vector fields having a polynomial first integral can be obtained from linear systems

We consider in this work planar polynomial differential systems having a polynomial first integral. We prove that these systems can be obtained from a linear system through a polynomial transformation of variables.     

多项式微分系统的周期解

给出了多项式微分系统的反射函数第一分量不依赖于第二变量的的充要条件,并应用所得结论研究了周期多项式系统周期解的几何性态.     

Integrability and solvability of polynomial Liénard differential systems

We provide the necessary and sufficient conditions of Liouvillian integrability for Liénard differential systems describing nonlinear oscillators with a polynomial damping and a polynomial restoring force. We prove that Liénard differential systems are not Darboux integrable excluding subfamilies with certain restrictions on the degrees of the polynomials arising in the systems. We demonstrate that if the degree of a polynomial responsible for the restoring force is greater than the degree of a polynomial producing the damping, then a generic Liénard differential system is not Liouvillian integrable with the exception of linear Liénard systems. However, for any fixed degrees of the polynomials describing the damping and the restoring force we present subfamilies possessing Liouvillian first integrals. As a by-product of our results, we find a number of novel Liouvillian integrable subfamilies. In addition, we study the existence of nonautonomous Darboux first integrals and nonautonomous Jacobi last multipliers with a time-dependent exponential factor.     

First fall degree and Weil descent

Polynomial systems arising from a Weil descent have many applications in cryptography, including the HFE cryptosystem and the elliptic curve discrete logarithm problem over small characteristic fields. Understanding the exact complexity of solving these systems is essential for the applications. A first step in that direction is to study the first fall degree of the systems. In this paper, we establish a rigorous general bound on the first fall degree of polynomial systems arising from a Weil descent. We also provide experimental data to study the tightness of our bound in general and its plausible consequences on the complexity of polynomial systems arising from a Weil descent.     

On rational integrability of Euler equations on Lie algebra so(4, ℂ)

We consider the Euler equations on the Lie algebra so(4, ℂ) with a diagonal quadratic Hamiltonian. It is known that this system always admits three functionally independent polynomial first integrals. We prove that if the system has a rational first integral functionally independent of the known three ones so called fourth integral, then it has a polynomial first integral that is also functionally independent of them. This is a consequence of more general fact that for these systems the existence of Darboux polynomial with no vanishing cofactor implies the existence of polynomial fourth integral.     

Polynomial first integrals for weight-homogeneous planar polynomial differential systems of weight degree 3

Our main result is the classification of all weight-homogeneous planar polynomial differential systems of weight degree 3 having a polynomial first integral.     

Shape preserving polynomial curves

We introduce particular systems of functions and study the properties of the associated Bézier-type curve for families of data points in the real affine space. The systems of functions are defined with the help of some linear and positive operators, which have specific properties: total positivity, nullity diminishing property and which are similar to the Bernstein polynomial operator. When the operators are polynomial, the curves are polynomial and their degrees are independent of the number of data points. Examples built with classical polynomial operators give algebraic curves written with the Jacobi polynomials, and trigonometric curves if the first and the last data points are identical.     

The Kowalevskaya exponents and rational integrability of polynomial differential systems

This paper primarily grows from the paper of Llibre and Zhang [J. Llibre, X. Zhang, Polynomial first integrals for quasi-homogeneous polynomial differential systems, Nonlinearity 15 (2002) 1269-1280] with the following essential generalizations: (i) we prove that the link established in the mentioned paper between the Kowalevskaya exponents and the degree of the polynomial first integrals holds not only for (1,…,1)-2 type systems but also for any (s₁,…,sn)-d type systems. (ii) by using different methods, we obtain necessary and sufficient conditions for planar (s₁,s₂)-d systems to have rational first integrals, whereas in the mentioned paper, only (s₁,s₂)-2 type systems and only polynomial integrability are considered.As an application of the methods and the results, we present an illustrative and well studied example to show its non-existence of polynomial first integrals.     

Polynomial stability of positive switched homogeneous systems with time delay

Based on the logarithm contraction average dwell-time method, this paper investigates the polynomial stability of positive switched homogeneous time-delay systems whose vector fields are of different degrees with respect to a dilation map. Using the analytical skills developed in positive systems, an explicit polynomial stability criterion is established for the first time for the involved system under the logarithm contraction average dwell-time switching. Moreover, the main result is applied to the polynomial stability of Persidskii-type switched systems.     

Centers of quasi-homogeneous polynomial differential equations of degree three

We characterize the centers of the quasi-homogeneous planar polynomial differential systems of degree three. Such systems do not admit isochronous centers. At most one limit cycle can bifurcate from the periodic orbits of a center of a cubic homogeneous polynomial system using the averaging theory of first order.     

Reduction of integrable planar polynomial differential systems

The integrability problem consists in finding the class of functions, a first integral of a given planar polynomial differential system must belong to. We recall the characterization of systems which admit a Darboux, elementary, Liouvillian or Weierstrass first integral. The reduction problem of an integrable planar system consists in finding the class of functions, a map that reduces the original system (transforms into a simple system or equation) must belong to. We identify the class of functions of this map for polynomial, rational, Darboux, elementary, Liouvillian and Weierstrass integrable systems.     

一个在无穷远点分支出八个极限环的多项式微分系统

本文研究一类高次系统无穷远点的中心条件与极限环分支问题．作者首先推出一个计算系统无穷远点奇点量的线性递推公式，并利用计算机代数系统计算出该系统在无穷远点处的前11个奇点量，从而导出无穷远点成为中心和最高阶细焦点的条件，在此基础上作者首次给出了多项式系统在无穷远点分支出8个极限环的实例。     

Extensions of nonnatural Hamiltonians

Theoretical and Mathematical Physics - The concept of extended Hamiltonian systems allows a geometric interpretation of several integrable and superintegrable systems with polynomial first...     

On the number of limit cycles of a generalized Abel equation

New results are proved on the maximum number of isolated T-periodic solutions (limit cycles) of a first order polynomial differential equation with periodic coefficients. The exponents of the polynomial may be negative. The results are compared with the available literature and applied to a class of polynomial systems on the cylinder.     

Quadratic systems with a rational first integral of degree three: a complete classification in the coefficient space ℝ12

A quadratic polynomial differential systemcan be identified with a single point of ?¹² through its coefficients. The phase portrait of the quadratic systems having a rational first integral of degree 3 have been studied using normal forms. Here using the algebraic invariant theory, we characterize all the non-degenerate quadratic polynomial differential systems in ?¹² having a rational first integral of degree 3. We show that there are only 31 different topological phase portraits in the Poincaré disc associated to this family of quadratic systems up to a reversal of the sense of their orbits, and we provide representatives of every class modulo an affine change of variables and a rescaling of the time variable. Moreover, each one of these 31 representatives is determined by a set of algebraic invariant conditions and we provide for it a first integral.     

A SIMPLE EXAMPLE OF A NON-FINITELY BASED SYSTEM OF POLYNOMIAL IDENTITIES

ABSTRACT

A system of polynomial identities is called finitely based if it is equivalent to some finite system of polynomial identities. Every system of polynomial identities in associative algebras over a field of characteristic 0 is finitely based: this is a celebrated result of Kemer. The first non-finitely based systems of polynomial identities in associative algebras over a field of a prime characteristic have been constructed recently by Belov, Grishin and Shchigolev. These systems of identities are relatively complicated. In the present paper we construct a simpler example of such a system and give a simple self-contained proof of the fact that the system is non-finitely based.     

The Ruelle Transfer Operator in the Context of Orthogonal Polynomials

We redefine the Ruelle transfer operator, a classical tool from dynamical systems theory, in terms of orthogonal polynomial sequences. This transfer operator will be given via the preimages of the Chebyshev polynomials of the first kind and we will show that function spaces determined by the Chebyshev polynomials of the first kind are left invariant while function spaces determined by various other orthogonal polynomial sequences are not.     

Linear Estimation of the Number of Zeros of Abelian Integrals for Some Cubic Isochronous Centers

This paper consists of two parts. In the first part we study the relationship between conic centers (all orbits near a singular point of center type are conics) and isochronous centers of polynomial systems. In the second part we study the number of limit cycles that bifurcate from the periodic orbits of cubic reversible isochronous centers having all their orbits formed by conics, when we perturb such systems inside the class of all polynomial systems of degree n.