Development Trend of Exhibition Industry of Wuhan in The E-commerce Environment

With the development of Web Economics, network exhibition came into being. Network and reality exhibition complement each other and develop together. New opportunities and challenges are promoting exhibition industry of Wuhan, which based on innovation, make full use of modern technology, electronic commerce and service, efforts to expand its geographical space and space. Exhibition industry of Wuhan should speed up the construction of virtual exhibition business platform, which is a unique design, complete functions, convenient operation platform and it is feasible

Solutions of equivariance for iterative differential equations

AbstractBy equivariance under the action of a group of invertible linear transformations on a Euclidean space, we describe symmetries of mappings. Based on known results on existence of solutions for iterative differential equations, in this paper, we discuss the special class of solutions which possess equivariance on R. Existence, uniqueness, and smooth dependence are given by using fixed-point theorems and by virtue of properties of finitely generated Lie groups

Smooth invariant foliations without a bunching condition and Belitskii's C1C^{1} linearization for random dynamical systems

Smooth linearization is one of the central themes in the study of dynamical systems. The classical Belitskii's $C^1$ linearization theorem has been widely used in the investigation of dynamical behaviors such as bifurcations, mixing, and chaotic behaviors due to its minimal requirement of partial second order non-resonances and low regularity of systems. In this article, we revisit Belitskii's $C^1$ linearization theorem by taking an approach based on smooth invariant foliations and study this problem for a larger class of dynamical systems ({\it random dynamical systems}). We assumed that the linearized system satisfies the condition of Multiplicative Ergodic Theorem and the associated Lyapunov exponents satisfy Belitskii's partial second order non-resonant conditions. We first establish the existence of $C^{1,\beta}$ stable and unstable foliations without assuming the bunching condition for Lyapunov exponents, then prove a $C^{1,\beta}$ linearization theorem of Belitskii type for random dynamical systems. As a result, we show that the classical Belitskii's $C^1$ linearization theorem for a $C^{2}$ diffeomorphism $F$ indeed holds without assuming all eigenspaces of the linear system $DF(0)$ are invariant under the nonlinear system $F$, a requirement previously imposed by Belitskii in his proof

Averaging approach to cyclicity of Hopf bifurcation in planar linear-quadratic polynomial discontinuous differential systems

Agraïments: The first author is supported by NSFC grant #11471228. The third author is supported by NSFC grants #11231001, #11221101.It is well known that the cyclicity of a Hopf bifurcation in continuous quadratic polynomial differential systems in \R^2 is 3. In contrast here we consider discontinuous differential systems in \R^2 defined in two half--planes separated by a straight line. In one half plane we have a general linear center at the origin of \R^2, and in the other a general quadratic polynomial differential system having a focus or a center at the origin of \R^2. Using averaging theory, we prove that the cyclicity of a Hopf bifurcation for such discontinuous differential systems is at least 5. Our computations show that only one of the averaged functions of fifth order can produce 5 limit cycles and there are no more limit cycles up to sixth order averaged function