QUALITATIVE THEORY OF THE QUADRATIC DIFFERENTIAL SYSTEMS (II)-ERGODICITY OF LIMIT CYCLES

Ａｓｉｓｗｅｌ－ｋｎｏｗｎｗｈｅｎａｒｅａｌｑｕａｄｒａｔｉｃｄｉｆｅｒｅｎｔｉａｌｓｙｓｔｅｍ：ｘ＝－ｙ＋δｘ＋ｌｘ２＋ｍｘｙ＋ｎｙ２＝Ｐ（ｘ，ｙ），ｙ＝ｘ（１＋ａｘ＋ｂｙ）＝Ｑ（ｘ，ｙ）（１）ｈａｓｆｏｕｒｆｉｎｉｔｅｃｒｉｔｉｃａｌｐｏｉｎｔｓ...     

QUALITIVE THEORY OF THE QUADRATIC DIFFERENTIAL SYSTEMS (I)

Westudyinthispaperqualitativepropertiesofthequadraticdifferentialsystemsfundertheconditions:Inthissectionwewillshowthatunder(4),theanti-saddleFI(xl,yi)of(2)(withxl>0)lyingon1 ax--y=0canbeaweakfocusfortwotimesas6varIes.*ProjectsupportedbytheNationalNatural…     

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.     

APPLICATIONS OF THE THREE CRITICAL POINTS THEOREM IN QUASILINEAR ELLIPTIC EQUATIONS

In this paper, using the three critical points theorem which is presented by Chang Gongqing (张恭庆) in reference [1], we prove the following homogeneous Dirichlet problem:     

NUMBER OF LIMIT CYCLES OF THE BOUNDED QUADRATIC SYSTEM WITH TWO FINITE SINGULAR POINTS

In this paper, the author proves the bounded quadratic system with two singular points at finite which corresponds to figures 12(a), 12(b), and 13(b) in[1] has at most one limit cycle, and shows under what conditions the limit cycle exists.     

POLYNOMIAL SYSTEMS POSSESSING FOUR FINITE ELEMENTARY CRITICAL POINTS AND AT MOST THREE CRITICAL POINTS AT INFINITY

ＰＯＬＹＮＯＭＩＡＬＳＹＳＴＥＭＳＰＯＳＳＥＳＳＩＮＧＦＯＵＲＦＩＮＩＴＥＥＬＥＭＥＮＴＡＲＹＣＲＩＴＩＣＡＬＰＯＩＮＴＳＡＮＤＡＴＭＯＳＴＴＨＲＥＥＣＲＩＴＩＣＡＬＰＯＩＮＴＳＡＴＩＮＦＩＮＩＴＹ叶彦谦南京大学，邮编：２１０００８ＰＯＬＹＮＯＭＩＡ...     

NUMBER OF BOUNDED SOLUTIONS IN CRITICAL CASE

In this paper, using the method of linearization, we determine the number of bounded solutions of ordinary differential equations in critical case.     

THE STRUCTURE OF THE HYPERBOLIC SET OF POINTS

ＴＨＥＳＴＲＵＣＴＵＲＥＯＦＴＨＥＨＹＰＥＲＢＯＬＩＣＳＥＴＯＦＰＯＩＮＴＳ￥ＬｉｎＺｈｅｎｓｈｅｎｇ（ＦｕｚｈｏｕＵｎｉｖｅｒｓｉｔｙ，）Ａｂｓｔｒａｃｔ：Ｉｎｔｈｉｓｐａｐｅｒｗｅｇｉｖｅｔｈｅｎｅｗｄｅｆｉｎｉｔｉｏｎｏｆｔｈｅｈｙｐｅｒｂｏｌ...     

ON THE QUALITATIVE BEHAVIOUR OF SOLUTIONS OF THE LIENARD EQUATION

ＯＮＴＨＥＱＵＡＬＩＴＡＴＩＶＥＢＥＨＡＶＩＯＵＲＯＦＳＯＬＵＴＩＯＮＳＯＦＴＨＥＬＩＥＮＡＲＤＥＱＵＡＴＩＯＮＣｈｅｎＸｉｕｄｏｎｇ（陈秀东）（ＤａｌｉａｎＵｎｉｖｅｒｓｉｔｙｏｆＴｅｃｈ．，大连理工大学，邮编：１１６０２４）ＬｉＪｉａｘｕ（Ｈｅｉ...     

REGION QUALITATIVE ANALYSIS OF PREDATOR-PREY SYSTEMS ON TIME SCALES

We first investigate some basic properties of dynamic equations on time scales,and propose contained curves to describe the jump direction of the discrete points.Then we perform qualitative analysis regarding the planar predator-prey systems on time scales,thereby obtain two theorems of this system.At last,we emulate application examples to discuss the parameters of the system.     

ON THE SPACES OF THE MAXIMAL POINTS

For a continuous domain D, some characterization that the convex powerdomain CD is a domain hull of Max(CD) is given in terms of compact subsets of D. And in this case, it is proved that the set of the maximal points Max(CD) of CD with the relative Scott topology is homeomorphic to the set of all Scott compact subsets of Max(.D) with the topology induced by the Hausdorff metric derived from a metric on Max(D) when Max(D) is metrizable.     

THE THEORY OF STATISTICAL ELASTICITY

§1. Introduction In simple words, what the discipline of rook mechanics studies is the inner regularity in inevitable or certain mechanical phenomena; while statistical or stochastic rock mechanics, the statistical regularity in occassional or stochastic mechanical phenomena. Rock mechanics is a discipline derived from geology, but the situation of which might be better than geology. After all, being a young     

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     

THE MORSE CRITICAL GROUPS OF THE MINMAX THEOREM

The author gives a new min-max theorem and studies the local homology propertiesof critical points that is got by min-max theorem.In particular,this result includes manymin-max theorem of link type,which can be applicable to nonlinear analysis.     

THE STRUCTURE OF Ⅱ SPACES(Ⅰ)

The spectral theory of selfadjoint operators and unitary operators in Hilbert spacehas been successfully generalized to Π_k space.However,there are only a few results for thespectral theory of selfadjoint operators and unitary operators in Π space.One of theimportant reasons is that the structure of Π space is more complex than that of Π_k space.This paper and the forthcoming paper “The structure of Π space(Ⅱ)”will mainly be dealtwith the structure of Ⅱ spaces,which will be used to further study the operators in Ⅱspaces.     

ON THE CROSSING NUMBER OF THE COMPLETE TRIPARTITE GRAPH K_(1,8,n)

The well known Zarankiewicz' conjecture is said that the crossing number of the complete bipartite graph Km,n (m≤n) is Z(m,n). where Z(m,n) = [m/2] [(m-1)/2] [n/2] [(n-1)/2](for and real number x, [x] denotes the maximal integer no more than x). Presently, Zarankiewicz' conjecture is proved true only for the case m≤G. In this article, the authors prove that if Zarankiewicz' conjecture holds for m≤9, then the crossing number of the complete tripartite graph K1,8,n is Z(9, n) 12[n/2].     

THE PREDICTION THEORY OF STATIONARY RANDOM FIELDS (Ⅱ): NONSYMMETRICAL HALF-PLANE PREDICTION

In this paper, we consider the nonsymmetrical half-plane prediction. A prediction theoretic proofof the fundamental formula about one step prediction error is given. A Wold-type decomposition andits spectral representation theorem are proved. Spectral extraction problems of the half-plane innova-tions and its singular components are investigated.     

ON THE NUMBER OF EULERIAN PLANAR MAPS

This paper presents the number of combinatorially distinct rooted Eulerian planar maps with the number of non-root-vertices and the number of non-root-faces as two parameters. The parametric expressions for determining the number in the loopless Eulerian case are also obtained.     

THE TOTAL CHROMATIC NUMBER OF PSEUDO-OUTERPLANAR GRAPHS

A Planar graph g is called a ipseudo outerplanar graph if there is a subset v.∈V(G),[V.]=i,such that G-V. is an outerplanar graph in particular when G-V.is a forest ,g is called a i-pseudo-tree .in this paper.the following results are proved;(1)the conjecture on the total coloring is true for all 1-pseudo-outerplanar graphs;(2)X1(G) 1 fo any 1-pseudo outerplanar graph g with △(G)≥3,where x4(G)is the total chromatic number of a graph g.     

THEORY OF SINGULAR POINTS OF ORDINARY DIFFERENTIAL EQUATIONS IN COMPLEX DOMAIN

In this paper, the topological of integral surfaces near certain of Lyapunov type singularpoints and certain type of nodes of ordinary differential equations in complex domain are studied.We introduce Briot-Bouquet transformation, in order to study the topological structure of integralsurfaces near higher order singular points. At last we give an estimate of the makimum numberof isolated limit integral surfaces passing through certain type of higher order singular points.