Differential Polynomial Systems Research Articles

Integrability and dynamics of a simplified class B laser system.

A simplified class B laser system is a family of differential polynomial systems of degree two depending on the parameters a and b. Its rich dynamics has already been observed in 1980s, see Arecchi et al. [Opt. Commun. 51, 308-314 (1984)] and Politi et al. [Phys. Rev. A 33, 4055 (1986)], and still nowadays, it attracts the interest of the researchers. In this paper, we characterize its dynamics near infinity for all values of the parameters. When a=0, the partial integrability was already proved by Oppo and Politi [Z. Phys. B Con. Mat. 59, 111-115 (1985)]. Here, we prove that for a=0, it is completely integrable with two independent first integrals given by Liouvillian functions, and we present a complete study of its dynamics. When a≠0, we study its dynamics in the Poincaré ball B3, i.e., the interior of this ball is identified with R3 and its boundary the two-dimensional sphere S2 is identified with the infinity of R3.

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.

Ritt-Wu Characteristic Set Method for Laurent Partial Differential Polynomial Systems

In this paper, a Ritt-Wu characteristic set method for Laurent partial differential polynomial systems is presented. The concept of Laurent regular differential chain is defined and its basic properties are proved. The authors give a partial method to decide whether a Laurent differential chain A is Laurent regular. The decision for whether A is Laurent regular is reduced to the decision of whether a univariate differential chain A1 is Laurent regular. For a univariate differential chain A1, the authors first give a criterion for whether A1 is Laurent regular in terms of its generic zeros and then give partial results on deciding whether A1 is Laurent regular.

A triangular decomposition algorithm for differential polynomial systems with elementary computation complexity

In this paper, a new triangular decomposition algorithm is proposed for ordinary differential polynomial systems, which has triple exponential computational complexity. The key idea is to eliminate one algebraic variable from a set of polynomials in one step using the theory of multivariate resultant. This seems to be the first differential triangular decomposition algorithm with elementary computation complexity.

Decomposition of Reflexive Differential-Difference Polynomial Systems

This paper presents a zero decomposition algorithm for reflexive difference and differential polynomial systems. The problem of negative exponent of difference operators in DD-polynomials is solved by means of generalized term order. Our algorithm would be used to decompose the zero set of a finitely generated reflexive DD-polynomial set into the union of zero sets of coherent chains.

Jacobi’s bound for independent systems of algebraic partial differential equations

In this paper we prove Jacobi’s bound for systems of n partial differential polynomials in n differential variables, which are independent over a prime differential ideal . This generalizes on the one hand our result (Kondratieva et al. in On Jacobi’s bound for systems of differential polynomials, algebra. Moscow University Press, Moscow, pp 79–85, 1982) about Jacobi’s bound for ordinary differential polynomials independent over a prime differential ideal and, on the other hand, the result by Tomasovic (A generalized Jacobi conjecture for arbitrary systems of algebraic differential equations. PhD thesis, Columbia University, 1976), who proposed a version of Jacobi’s bound for partial differential polynomials and proved it in the linear case. In Kondratieva et al. (Differential and difference dimension polynomials, Kluwer, Dordrecht, 1999) we gave another proof of this result by Tomasovic. The exposition in the present paper follows our proof in Kondratieva et al. (Differential and difference dimension polynomials, Kluwer, Dordrecht, pp 273–280, 1999).

An optimization of Seidenberg's elimination algorithm in differential algebra

Abstract Seidenberg's elimination algorithm computes a projection of the algebraic differential manifold associated with a system of differential polynomials (here with coefficients in an ordinary differential field of characteristic zero). Found in 1956, the original version of the procedure cannot be efficiently carried out on a computer. In this paper, we describe a new version of the algorithm with a simple and efficient termination proof.

Algebraic structure of the manifold of solutions of the $N$-body problem

The Theorem of Ritt on the decomposition of the perfect differential ideal generated by a single irreducible differential polynomial is, here, generalized to system of polynomials satisfying certain conditions. We use these results to prove that all solutions of the N-body problem, excepting the solutions for which one or more of the r,; (the distance between the masses Mi, Mj) is zero, belong to one irreducible manifold. Introduction. In [3, p. 22] Ritt proved that the manifold of zero's of a system of differential polynomials is a finite union of irreducible manifolds. If the system consists of a single irreducible differential polynomial F, then the manifold of F breaks up into an irreducible general manifold and a certain number, possibly zero, of singular manifolds. In this paper the latter result is generalized to systems of polynomials: F= F1, * * * , F., n > 1 satisfying certain conditions. These conditions are almost always satisfied in problems of particle mechanics. When we apply these results to the N-body problem, we prove that all solutions of the N-body problem, excepting those for which one or more of the rij (the distance between the point masses Mj, Mj) is zero, belong to one irreducible manifold (the general manifold). The solutions of the N-body problem for which one or more of the rij is zero constitute the singular manifolds. In particular, the equilateral triangle, isosceles triangle, straight line solutions of the three-body problem belong to the general manifold. However, this paper still does not answer the question whether any singular solution belongs to the general manifold. For the applications to mechanics only Theorem 1 and Corollary 1 is needed and the number of variables can be assumed to be the same as the number of equations. In order to obtain a proper generalization of Ritt's Theorem on the decomposition of a single differential polynomial in an arbitrary number of variables we have proven our results for the case where the number of variables is > the number of equations, and extended these results to systems of greater generality than that required for problems in particle mechanics. Received by the editors July 23, 1969. AMS Subject Classifications. Primary 1280, 8520.

Jacobi’s bound for the order of systems of first order differential equations

Let A 1 , … , A n {A_1}, \ldots ,{A_n} be a system of differential polynomials in the differential indeterminates y ( 1 ) , … , y ( n ) {y^{(1)}}, \ldots ,{y^{(n)}} , and let M \mathcal {M} be an irreducible component of the differential variety M ( A 1 , … , A n ) \mathcal {M}({A_1}, \ldots ,{A_n}) . If dim ⁡ M = 0 \dim \mathcal {M} = 0 , there arises the question of securing an upper bound for the order of M \mathcal {M} in terms of the orders r i j {r_{ij}} of the polynomials A i {A_i} in y ( j ) {y^{(j)}} . It has been conjectured that the Jacobi number \[ J = J ( r i j ) = max { ∑ i = 1 n r i j i : j 1 , … , j n is a permutation of 1, … , n } J = J({r_{ij}}) = \max \{\sum \limits _{i = 1}^n {{r_{i{j_i}}}} :{j_1}, \ldots ,{j_n}{\text { is a permutation of 1,}} \ldots ,n \} \] provides such a bound. In this paper J J is obtained as a bound for systems consisting of first order polynomials. Differential kernels are employed in securing the bound, with the theory of kernels obtained in a manner analogous to that of difference kernels as given by R. M. Cohn.

On the basis theorem for differential systems

As originally proved by Hilbert, this theorem applied to polynomials whose coefficients were either elements of a field, or rational integers. In keeping with the modern tendency toward abstraction, however, one now finds the theorem proved for polynomials whose coefficients are elements of a commutative ring with unit element in which every set has a finite basis. When one turns to differential polynomials and differential ideals one finds that the exact analogue of the Hilbert theorem is lacking('). It is not true that every system of differential polynomials z contains a finite subset F1, * *, F8 such that

On a type of algebraic differential manifold

The manifolds(') to be investigated, which are manifolds of systems of differential polynomials in a single unknown, possess a degree of analogy to bounded sets of numbers. They are manifolds which may be said to contain infinity as a solution; more definitely, zero is not a limit of reciprocals of solutions. For manifolds of this type, which will be called limited, operations of addition, multiplication and differentiation will be studied. Given two manifolds(2) 91k and 92, their arithmetic sum is secured by completing into a manifold the totality of functions each of which is, in some area, the sum of a solution in 9Nh And a solution in 9)2. Multiplication is defined similarly. It turns out that if 9Th and 9)2 are general solutions of equations of the first order, and are limited, their sum and product are limited. On the other hand, as is shown by examples based on the theory of the elliptic functions, when 9Th and 9N2 involve more than one arbitrary constant their limited character may not be communicated to their sum and product; what is equivalent to this, as far as multiplication is concerned, is the rather unexpected result that the product of two manifolds may contain zero even if neither manifold does. The derivative of a limited manifold proves to be limited in all cases.

On the basis theorem for infinite systems of differential polynomials

On the basis theorem for infinite systems of differential polynomials