Transcendental solutions of systems of complex differential equations北大核心CSCD

Using maximum modulus principle, we investigate the problem of the existence of the transcendental meromorphic solutions of system of complex algebraic differential equations, and obtain a result which the system only the algebraic solution under certain conditions. Examples show that our results are sharp. ©, 2015, Chinese Academy of Sciences. All right reserved.     

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.     

Jacobi’s bound for systems of algebraic differential equations

This review paper is devoted to the Jacobi bound for systems of partial differential polynomials. We prove the conjecture for the system of n partial differential equations in n differential variables which are independent over a prime differential ideal \mathfrakp\mathfrak{p}. On the one hand, this generalizes our result about the Jacobi bound for ordinary differential polynomials independent over a prime differential ideal \mathfrakp\mathfrak{p} and, on the other hand, the result by Tomasovic, who proved the Jacobi bound for linear partial differential polynomials.     

The asymptotic stability of multistep multiderivative methods for systems of delay differential equations

IntroductionFor many years, many papers investigated the linear stabilit}' of delay differential equation(DDE) solvers and a significant number of important results have already been found for bothRunge-Kutta methods and linear multistep methods (see, for example, [l--8]). In this paper,we firstly consider stability of numerical methods with derivative for DDEs. It is shown thatA-stability of multistep multiderivative methods for ordinary differential equations (ODEs) isequit,alent to p-s…     

On deformations of linear systems of differential equations and the Painlevé property

We consider systems of linear differential equations discussing some classical and modern results in the Riemann problem, isomonodromic deformations, and other related topics. Against this background, we illustrate the relations between such phenomena as the integrability, the isomonodromy, and the Painlevé property. The recent advances in the theory of isomonodromic deformations presented show perfect agreement with that approach.     

A decision method for the integrability of differential–algebraic Pfaffian systems

We prove an effective integrability criterion for differential–algebraic Pfaffian systems leading to a decision method of consistency with a triple exponential complexity bound. As a byproduct, we obtain an upper bound for the order of differentiations in the differential Nullstellensatz for these systems.     

Generalizations of Darbo’s fixed point theorem and solvability of integral and differential systems

In this paper, we establish some new generalizations of Darbo’s fixed point theorem for multivalued mappings. Moreover, we prove the existence of solutions for a class of integral equations by Darbo’s fixed point theorem and the existence of solutions for a class of differential inclusions using a generalization of Darbo’s fixed point theorem.     

Maximum principle for differential games of forward–backward stochastic systems with applications

This paper is concerned with a maximum principle for both zero-sum and nonzero-sum games. The most distinguishing feature, compared with the existing literature, is that the game systems are described by forward–backward stochastic differential equations. This kind of games is motivated by linear-quadratic differential game problems with generalized expectation. We give a necessary condition and a sufficient condition in the form of maximum principle for the foregoing games. Finally, an example of a nonzero-sum game is worked out to illustrate that the theories may find interesting applications in practice. In terms of the maximum principle, the explicit form of an equilibrium point is obtained.     

Bounding the number of limit cycles of discontinuous differential systems by using Picard–Fuchs equations

In this paper, by using Picard–Fuchs equations and Chebyshev criterion, we study the upper bounds of the number of limit cycles given by the first order Melnikov function for discontinuous differential systems, which can bifurcate from the periodic orbits of quadratic reversible centers of genus one (r19): $\dot{x}=y?12x^2+16y^2$, $\dot{y}=?x?16xy$, and (r20): $\dot{x}=y+4x^2$, $\dot{y}=?x+16xy$, and the periodic orbits of the quadratic isochronous centers $(S_1):\dot{x}=?y+x^2?y^2$, $\dot{y}=x+2xy$, and $(S_2):\dot{x}=?y+x^2$, $\dot{y}=x+xy$. The systems (r19) and (r20) are perturbed inside the class of polynomial differential systems of degree n and the system $(S_1)$ and $(S_2)$ are perturbed inside the class of quadratic polynomial differential systems. The discontinuity is the line $y=0$. It is proved that the upper bounds of the number of limit cycles for systems (r19) and (r20) are respectively $4n?3(n\ge 4)$ and $4n+3(n\ge 3)$ counting the multiplicity, and the maximum numbers of limit cycles bifurcating from the period annuluses of the isochronous centers $(S_1)$ and $(S_2)$ are exactly 5 and 6 (counting the multiplicity) on each period annulus respectively.     

Comment on “Symmetry breaking of systems of linear second-order ordinary differential equations with constant coefficients”

The present paper corrects the way of using Jordan canonical forms for studying the symmetry structures of systems of linear second-order ordinary differential equations with constant coefficients applied in [1]. The approach is demonstrated for a system consisting of two equations.     

Limiting Bourgain–Brezis estimates for systems of linear differential equations: Theme and variations

J. Bourgain and H. Brezis have obtained in 2002 some new and surprising estimates for systems of linear differential equations, dealing with the endpoint case L ¹ of singular integral estimates and the critical Sobolev space \({W^{1,n}(\mathbb{R}^n)}\) . This paper presents an overview of the results, further developments over the last ten years and challenging open problems.     

Global classification of the planar Lotka–Volterra differential systems according to their configurations of invariant straight lines

In this article, we study the Lotka–Volterra planar quadratic differential systems. We denote by LV systems all systems which can be brought to a Lotka–Volterra system by an affine transformation and time homotheties. All these systems possess invariant straight lines. We classify the family of LV systems according to their geometric properties encoded in the configurations of invariant straight lines which these systems possess. We obtain a total of 65 such configurations which are distinguished, roughly speaking, by the multiplicity of their invariant lines and by the multiplicities of the singularities of the systems located on these lines. We determine an algebraic subvariety of \mathbbR¹²{\mathbb{R}^{12}} which contains all these systems and we find the bifurcation diagram of the configurations of LV systems within this algebraic subvariety, in terms of polynomial invariants with respect to the group action of affine transformations and time homotheties. This geometric classification will serve as a basis for the full topological classification of LV systems.     

On the completeness of systems of eigenfunctions and associated functions of differential operators of orders 2 ? �� and 1 ? ��

In the space of vector-functions, we consider a boundary-value problem for differential operators of fractional orders (2 ? ??) and (1 ? ??) and prove the completeness of the system of eigenfunctions and associated functions of this problem in the space $L_1 \left( {\left[ {0,1} \right],\,\mathbb{C}^p } \right)$ .     

Construction of asymptotic representations of solutions from the classL ρ τ for systems of differential equations by the method of characteristic matrix functions

By the method of characteristic matrix functions, we construct asymptotic representations of solutions of a system ofq linearn th-order differential equations with a singularity of rankp/r, p, r , in a sector of the complex plane whose central angle does not exceed r/p.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 9, pp. 1148–1155, September, 1994.     

An analog to Kamenkov’s critical stability case for nonstationary systems of impulsive differential equations

A new approach to the investigation of the stability of nonlinear nonautonomous differential equations with impulse effects in critical cases is proposed. The approach is based on the direct method of Lyapunov with the use of piecewise differentiable functions. The sufficient conditions of the asymptotic stability of the critical position of equilibrium in one case are obtained. The case is analogous to Kamenkov’s critical case.     

Multi-resurgence and connection-to-Stokes formulæ for some linear meromorphic differential systems

In this article, we consider a linear meromorphic differential system with several levels $r_1<...<r_p$. For any k, we prove that the Borel transforms of its $r_k$-reduced formal solutions are resurgent and we give the general form of all their singularities. Next, under some convenient hypotheses on the geometric configuration of singular points, we display exact formulæ to express some Stokes multipliers of level $r_k$ of initial system in terms of connection constants in the Borel plane, generalizing thus formulæ already obtained by M. Loday-Richaud and the author for systems with a single level. As an illustration, we develop one numerical example.     

Riemann-Hilbert and Poincaré problems with discontinuous boundary conditions for some model systems of partial differential equations

We study the solvability of the Riemann-Hilbert and Poincaré problems for systems of Cauchy-Riemann and Bitsadze equations in Sobolev spaces. For a generalized system of Cauchy-Riemann equations, we pose a boundary value problem and prove its unique solvability in the Sobolev space W ₂¹ (D). By supplementing the Riemann-Hilbert boundary conditions with some new conditions, we obtain a statement of the Poincaré problem with discontinuous boundary conditions for a system of second-order Bitsadze equations; we also prove the unique solvability of this problem in Sobolev spaces.