SOME BOUNDARY PROBLEMS OF THE SPIN SYSTEMS AND THE SYSTEMS OF FERRO-MAGNETIC CHAIN Ⅰ. NONLINEAR BOUNDARY PROBLEMS

§1. Introduction The nonlinear system of partial differential equations and the corresponding system with diffusion term are interesting and meaningful both in physical and mathematical point of view, where z= (u, v, w) and f(x, t, z) are two three-dimensional vector functions and"× "denotes the cross product operator of two three-dimensional vectors. The system (1) contains the so-called Landau- Lifschitz equation of isotropic Heigenberg ferro-magnetio chain as the     

PROPERTIES IN THE LARGE OF QUADRATIC SYSTEMS IN THE PLANE

The author proves that any quadratic system in the plane can be changed into the quadratic system (E_1) or (E_2) by means of linear transformations, and then gives a necessary and sufficient condition for the systems (E_1) and (E_2) to be bounded for t≥0 and to have precisely one monotone unboundol orbit for t≥0 respectively.     

THE INTERACTIONS OF SHOCK WAVES OF NONSTRICTLY HYPERBOLIC SYSTEMS

The interaction of shock waves is investigated for the following nonstrictly hyperbolic system:The interaction of shock waves is complicated, with new types of shock waves, and new singula rities in the dependence of interaction on the relative positions of the three states separated by shock waves. Several ideas are introduced to helo organize and clarify the new phenomena.     

ON THE ALMOST PERIODIC SOLUTIONS OF HIGHER DIMENSIONAL NONAUTONOMOUS SYSTEMS

By means of the properties of almost periodic system and Lyapunov function, we give some criteria which guarantee the existence and uniqueness and stability of almost periodic solutions of higher dimensional nonautonomous system. The result is more convenient and effective than the related result in[1].     

ON THE SPECTRUM OF SINGULAR HAMILTONIAN DIFFERENTIAL SYSTEMS

In this paper, we study a bounded-below singular Hamiltonian system. Sufficient and necessary conditions are obtained for the existence and the number of eigenvalues on the left-axis. The main results of this paper are the extension and improvement of Weyl spectral theorem for singular second order differential equations.     

THE CONFOCAL INVOLUTIVE SYSTEM AND THE INTEGRABILITY OF THE NONLINEARIZED LAX SYSTEMS FOR AKNS HIERARCHY

The completely integrable Hamiltonian systems generated by the general confocal involutive system are proposed. It is proved that the nonlinearized eigenvalue problem for AKNS hierarchy is such an integrable system and showed that the time evolution equations for n≤3 obtained by nonlinearizing the time parts of Lax systems for AKNS hierarchy are Liouville integrable under the constraint of the spatial part.     

QUALITATIVE THEORY OF THE KUKLES SYSTEMS:(Ⅰ) NUMBER OF CRITICAL POINTS

The special cubic system (a natural and simPlest generaIization of thequadratic system of C1ass (l) to cubic system)f.'x = yl y = --x 6y a,x' a2xy a,y' a#x' a,x'y a.xy' a,y' (1)was first studied by the Russian Mathematician I.S.KukIes [ll on the Ilecessaryand sufficient conditions fot O(0,0) to be a ccnter of (1). After [ll, the sameproblem and also the order of O as a fine fOcus were also studied iIl [2--l0] and[13-19], so (l) is now called as a "Kukles system". In l988-…     

THE CONNECTIVITY OF Z-TRANSFORMATION GRAPHS OF PERFECT MATCHINGS OF HEXAGONAL SYSTEMS

Let H be a hexagonal system.The Z-transformation graph Z(H) is a graph where the vertices are perfect matchings of H and where two perfect matchings are joined by an edge provided their symmetric difference consists of six edges of a hexagon of H. We prove that the connectivity of Z(H) is equal to the minimum degree of vertices of Z(H).     

THE OPTIMAL FEEDBACK CONTROL OF LINEAR SYSTEMS WITH NON-CHEAP CONTROL

Consider the optimal feedback control problem of a linear system. The corresponding dynamic equation and performance index are     

A GENERAL PROPERTY OF THE QUADRATIC DIFFERENTIAL SYSTEMS

In this paper, it is proved that the quadratic differential systems with a weak saddle of order 2 or 3 have no closed or singular closed orbit. Then by the results of [3], it follows that the greatest order of the homoclinic loop bifurcation of a quadratic differential system is between 2 and 3 It means a homoclinic loop can be split into at most three limit cycles.     

ASYMPTOTIC STABILITY OF A CLASS OF NONLINEAR GENERALIZED SYSTEMS

§ 1　IntroductionAs compared with the ODEs system,the generalized system is of many distinctiveproperties[1 ] ,forexam ple,the level structural problem of system state solutions,the uncon-sistent initial value problem,the uniqueness of initial value problem and so on.The asymp-totic stability problem of zero solution of generalized system involves the tractability andunimpulsivity.These properties are of great importance in engineering applications.Theconclusion on the ODEs system can not …     

ON THE REACHABLE SEMIGROUP OF BILINEAR CONTROL SYSTEMS ON LIE GROUP

This paper studies the reachability and the structure of reachable semigroup of bilinear control systems on Lie group. In the second section some equivalency lemmas are given, which not only simplify the proofs of the main results, but discover some properties of systems also. In the third section some conditions are advanced that the reachable semigroup of system is weakly symmetric by means of the study of one prameter subgroups. This study is discussed by manifold theory and matrix theory, respectively. In the last seetion, some topological properties of the reachable semigroup are advanced.     

THE NUMBER OF LIMIT CYCLESOF THE PLANE AUTONOMOUS SYSTEMS

In this paper the author discusses the vector fields satisfying the conditions (C1) and (C2) and proves some theorems for the number of limit cycles of the plane autonomous systems. Some applications of the theorems and lemmas in §1 are given in §2.     

DENSITY OF MARKOV SYSTEMS AND ZEROS OF CHEBYSHEV POLYNOMIALS

We raise and partly answer the question: whether there exists a Markov system with respectto which the zeros of the Chebyshev polynomials are dense, but the maximum length of a zerofree interval of the nth Chebyshev polynomial does not tends to zero. We also draw the conclu-tion that a Markov system, under an additional assumption, is dense if and only if the maxi-mum length of a zero free interval of the nth associated Chebyshev polynomial tends to zero.     

THE LOCAL AND GLOBAL EXISTENCE OF THE SOLUTIONS OF HYPERBOLIC-PARABOLIC SYSTEM MODELING BIOLOGICAL PHENOMENA

The authors prove the local existence and uniqueness of weak solution of a hyperbolic-parabolic system and establish the global existence of the weak solution for this system for the spatial dimension n = 1.     

STRUCTURAL STABILITY OF ASYMPTOTICALLY ELEMENTARY SYSTEMS

In this paper, an elementary system proves constructively to be structurally stable provided the system is asymptotically stable on some region. This conclusion favors system modelling because Lyapunov and gradient systems are all elementary.     

THE QUENCHING OF SOLUTION OF SEMILINEAR PARABOLIC SYSTEMS

This paper deals with the quenching of the solutions for semilinear parabolic systems Conditions under which quenching occurs in finite time and the condition for the existence of global solution are given.     

THE UNIQUENESS OF LIMIT CYCLE AND THE STRUCTURE OF CRITICAL POINT AT INFINITY FOR A CLASS OF CUBIC SYSTEM

A class of cubic system, which is an accompany system of a quadratic differential one, is studied. It is proved that the system has at most one limit cycle, and the critical point at infinity is a higher order one. The structure and algebraic character of the critical point at infinity are obtained.     

ON THE WEAK DISCONTINUITY SURFACES AND TAPES OF EQUATION SYSTEMS IN FLUID MECHANICS

The weak discontinuity surfaces for a system of quasi-linear differential equations of higher order are developed.The classification of equation systems in fluid mechanics is based on the propagative weak discontinuity surfaces.Types of equations for different flow models are discussed.The conclusion is as follows:(a) For incompressible nonviscous flow,incompressible viscous flow and compressible viscous flow,the types of equations are all parabolic in the unsteady case and elliptic in the steady case.(b) For compressible nonviscous flow,the type of equations is hyperbolic in the unsteady case or steady supersonic case,and the type is elliptic in the steady subsonic case.     

ON THE CENTER OF GENERALIZED LIENARD SYSTEM

This paper discusses the local and global center of the generalized Lienard system, gives two theorems which extend and improve the relative results of [1]-[3]. Moreover the lemmas in this paper are also useful to the study for the oscillation, stability and the existence of limit cycles of the generalized Lienard system.