This paper is concerned with 2-D discrete systems of the form xm-1, n=f(xm,n, xm,n+1), where f:R2→R is a function, m, n e N0={0, 1, 2, . . .}. Some sufficient conditions for this system to be stable and a verification of this system to be chaotic in the sense of Devaney, respectively, are derived.

This paper is concerned with chaos induced by strictly turbulent maps in noncompact sets of complete metric spaces. Two criteria of chaos for such types of maps are established, and then a criterion of chaos, characterized by snap-back repellers in complete metric spaces, is obtained. All the maps presented in this paper are proved to be chaotic either in the sense of both Li-Yorke and Wiggins or in the sense of both Li-Yorke and Devaney. The results weaken the assumptions in some existing criteria of chaos. Several illustrative examples are provided with computer simulation.

研究Banachl空间上连续Frechet可微映射导出的离散动力系统之混沌。建立一个由正则非退化同宿轨道产生混沌的判定定理，并对n维实空间上的离散动力系统的混沌进行了讨论，建立了两个由非退化返回扩张不动点产生混沌的判定定理，其中一个为Marotto定理的修正定理。特别地，分别给出了一般Banach空间及n维实空间上的连续可微映射不动点为扩张的充分必要条件，彻底解决了多年以来人们对n维实空间上连续可微映射不动点的扩张性与其Jacobi矩阵特征值之间关系的困惑。

In this paper, the Glazman-Krein-Naimark theory for a class of discrete Hamiltonian systems is developed. A minimal and a maximal operators, GKN-sets, and a boundary space for the system are introduced. Algebraic characterizations of the domains of self-adjoint extensions of the minimal operator are given. A close relationship between the domains of self-adjoint extensions and the GKN-sets is established. It is shown that there exist one-to-one correspondences among the set of all the self-adjoint extensions, the set of all the d-dimensional Lagrangian subspaces of the boundary space, and the set of all the complete Lagrangian subspaces of the boundary space.

This paper is concerned with the symplectic structure of discrete nonlinear Hamiltonian systems. The results are related to an open problem that was first proposed by C. D. Ahlbrandt [J. Math. Anal. Appl. 180 (1993), 498-517] discussed elsewhere in the literature. But we give a different statement and different proof. Under a solvable condition, we show that the solution operator of a discrete nonlinear Halmiltonian system is symplectic. Then its phase flow is a discrete one-parameter family of symplectic transformations and preserves the phase volume.