Integrability and linearizability of the Lotka-Volterra systems

Integrability and linearizability of the Lotka-Volterra systems are studied. We prove sufficient conditions for integrable but not linearizable systems for any rational resonance ratio. We give new sufficient conditions for linearizable Lotka-Volterra systems. Sufficient conditions for integrable Lotka-Volterra systems with 3:−q resonance are given. In the particular cases of 3:−5 and 3:−4 resonances, necessary and sufficient conditions for integrable systems are given.     

Linearizability conditions of a time-reversible quartic-like system

In this paper we study the linearizability problem of polynomial-like complex differential systems. We give a reduction of linearizability problem of such non-polynomial systems to the problem of polynomial systems. Applying this reduction, we find some linearizability conditions for a time-reversible quartic-like complex system and derive from them conditions of isochronous center for the corresponding real system.     

Generalized isochronous centers for complex systems

In this paper, the definition of generalized isochronous center is given in order to study unitedly real isochronous center and linearizability of polynomial differential systems. An algorithm to compute generalized period constants is obtained, which is a good method to find the necessary conditions of generalized isochronous center for any rational resonance ratio. Its two linear recursive formulas are symbolic and easy to realize with computer algebraic system. The function of time-angle difference is introduced to prove the sufficient conditions. As the application, a class of real cubic Kolmogorov system is investigated and the generalized isochronous center conditions of the origin are obtained.     

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.     

Computing centre conditions for certain cubic systems

We present necessary and sufficient conditions for a critical point of certain two-dimensional cubic differential systems to be a centre. Extensive use of the computer algebra system REDUCE is involved. The search for necessary and sufficient conditions for a centre has long been of considerable interest in the theory of nonlinear differential equations. It has proved to be a difficult problem, and full conditions are known for very few classes of systems. Such conditions are also required in the investigation of Hilbert's sixteenth problem concerning the number of limit cycles of polynomial systems.     

Polynomial Poisson Algebras with Regular Structure of Symplectic Leaves

We study polynomial Poisson algebras with some regularity conditions. Linear (Lie–Berezin–Kirillov) structures on dual spaces of semisimple Lie algebras, quadratic Sklyanin elliptic algebras, and the polynomial algebras recently described by Bondal, Dubrovin, and Ugaglia belong to this class. We establish some simple determinant relations between the brackets and Casimir functions of these algebras. In particular, these relations imply that the sum of degrees of the Casimir functions coincides with the dimension of the algebra in the Sklyanin elliptic algebras. We present some interesting examples of these algebras and show that some of them arise naturally in the Hamiltonian integrable systems. A new class of two-body integrable systems admitting an elliptic dependence on both coordinates and momenta is among these examples.     

Linearizability of the polynomial differential systems with a resonant singular point

Integrability and linearizability of polynomial differential systems are studied. The computation of generalized period constants is a way to find necessary conditions for linearizable systems for any rational resonance ratio. A method to compute generalized period constants is given. The algorithm is recursive and easy to realize with computer algebraic system. As the application, we discuss linearizable conditions for several Lotka-Volterra systems, and where this is the first time that the linearizability is considered for 3:−4 and 3:−5 resonances.     

<Emphasis Type="Italic">A</Emphasis>-Orbital Linearization of Affine Systems

The problem of transformation of an affine system into a linear controllable system is considered. For affine systems with a single control, the notion of A-orbital linearizability is introduced, which generalizes the notion (well known for affine systems) of orbital linearizability to the case where the control-dependent changes of independent variable are used. A necessary and sufficient condition for the A-orbital linearizability is proved, and an algorithm for determining linearizable transformations is proposed based on the construction of the derived series of the codistribution associated with the original system.     

On the period function of planar systems with unknown normalizers

A necessary and sufficient condition for the period function's monotonicity on a period annulus is given. The approach is based on the theory of normalizers, but is applicable without actually knowing a normalizer. Some applications to polynomial and Hamiltonian systems are presented.

     

Non–limit-circle and limit-point criteria for symplectic and linear Hamiltonian systems

Several necessary and/or sufficient conditions for the existence of a non–square-integrable solution of symplectic dynamic systems with general linear dependence on the spectral parameter on time scales are established and a sufficient condition for the limit-point case is derived. Almost all presented results are new even in the continuous and discrete cases, that is, for the linear Hamiltonian differential systems and for the discrete symplectic systems, respectively.     

Geometry of cascade feedback linearizable control systems

Cascade feedback linearization provides geometric insights on explicit integrability of nonlinear control systems with symmetry. A central piece of the theory requires that the partial contact curve reduction of the contact sub-connection be static feedback linearizable. This work establishes new necessary conditions on the equations of Lie type - in the abelian case - that arise in a contact sub-connection with the desired static feedback linearizability property via families of codimension one partial contact curves. Furthermore, an explicit class of contact sub-connections admitting static feedback linearizable contact curve reductions is presented, hinting at a possible classification of all such contact sub-connections. Key tools in proving, and stating, the main results of this paper are truncated versions of the total derivative and Euler operators. Additionally, the Battilotti-Califano system with three inputs is used as a clarifying example of both cascade feedback linearization and the new necessary conditions.     

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.     

Optimization of higher order differential inclusions with initial value problem

This paper concerns the sufficient conditions of optimality for initial value problem with higher order differential inclusions (HODIs) and free endpoint constraints. Formulation of the transversality conditions plays a substantial role in the next investigations without which hardly any necessary or sufficient conditions would be obtained. In terms of Euler–Lagrange and Hamiltonian forms the sufficient conditions of optimality both for convex and “non-convex” HODIs are based on the apparatus of locally adjoint mappings. Moreover, by applying the main result to a Bolza problem described by a polynomial differential operator with constant coefficients in terms of the adjoint differential operator the sufficient condition of optimality is obtained.     

斜对角无穷维Hamilton算子的点谱和特征函数系辛结构的非退化性

本文研究斜对角无穷维Hamilton算子$H=\begin{pmatrix}0&B\\C&0\end{pmatrix}$的点谱和特征函数系辛结构的非退化性, 给出斜对角无穷维Hamilton算子$H$的特征函数系具有非退化辛结构的充分必要条件. 基于此, 进一步刻画了斜对角无穷维Hamilton算子$H$的点谱分别包含于实轴、虚轴以及其它区域的充分必要条件. 最后, 以板弯曲问题和弦振动问题中导出的斜对角无穷维Hamilton算子为例, 验证了所得结论的正确性.     

Darboux integrability and algebraic limit cycles for a class of polynomial differential systems

This paper deals with the existence of Darboux first integrals for the planar polynomial differential systems x=x-y+P n+1(x,y)+xF2n(x,y),y=x+y+Q n+1(x,y)+yF2n(x,y),where P i(x,y),Q i(x,y)and F i(x,y)are homogeneous polynomials of degree i.Within this class,we identify some new Darboux integrable systems having either a focus or a center at the origin.For such Darboux integrable systems having degrees 5and 9 we give the explicit expressions of their algebraic limit cycles.For the systems having degrees 3,5,7 and 9and restricted to a certain subclass we present necessary and sufficient conditions for being Darboux integrable.     

On the factorization of polynomial matrices over the domain of principal ideals

We consider the problem of decomposition of polynomial matrices over the domain of principal ideals into a product of factors of lower degrees with given characteristic polynomials. We establish necessary and, under certain restrictions, sufficient conditions for the existence of the required factorization.     

矩阵多元多项式的带余除法及其应用

给出矩阵多元多项式的带余除法,从而用微分代数的观点,得到把一类微分方程(组)化为无穷维Hamilton系统的充要条件及其具体无穷维Hamilton系统形式。再把此方法和吴方法相结合获得构造一类微分方程(组)的通解的新方法。几个例子表明这些方法都很有效的。     

VARIATIONS ON A THEME BY EULER

1.IntroductionAHallliltolliansystemofdifferentialequationsonRZnisgivedbyP~~H,(P,q),q=HP(P,q),(1)wherep=(pl,'.,P.),q=(ql,',q.)eR"arethegeneralizedcoordinatesandmolllentarespectivelyandH(P,q)istheellergyofthesystem.Thesystem(1)canberewrittenasthecompactf…     

R~n和RP~n上的多项式系统

本文讨论了ｎ维欧氏空间Ｒｎ（ｎ＞２）上的多项式向量场集合的系数拓扑不变量，可分为具有不同全局拓扑性质的两类不交的子集合；证明了Ｒｎ上的多项式向量场可连续地延拓成ｎ维射影空间ＲＰｎ上的连续多项式向量场的充要条件，反应了其次数与系数相关的拓扑性质；还证明了平面上的多项式向量场的赤道是闭轨线和不变集的充要条件．     

Banach空间中线性离散时间系统的一致多项式膨胀性

给出了Banach 空间中线性离散时间系统一致多项式膨胀性的概念,并讨论了其离散特征。借助Lyapunov函数给出了线性离散时间系统满足一致多项式膨胀的充要条件。所得结论将一致指数稳定性、指数膨胀性及多项式稳定性中的若干经典结论推广到了一致多项式膨胀性的情形。