The Multi-sensitivity and Topological Sequence Entropy of Dynamical System with Group Action

In this paper,we study the relationship between the multi-sensitivity and the topological maximal sequence entropy of dynamical systems for general group action.Furthermore,we also discuss the consistency of multi-sensitivity of a dynamical system(G■X)and its hyperspace dynamical system G■K(X).Moreover,we research the relationship between the multi-sensitivity of two dynamical systems and the multi-sensitivity of their product space dynamical system.Finally,we prove that if the topological sequence entropy of G■X vanishes,then so does that of its induced system G■M(X);if the topological sequence entropy of G■X is positive,then that of its induced system G■M(X)is infinity.

Topology and topological sequence entropy

Let X be a compact metric space and T:X-→X be continuous.Let h*(T)be the supremum of topological sequence entropies of T over all the subsequences of Z+and S(X)be the set of the values h*(T)for all the continuous maps T on X.It is known that{0}■S(X)■{0,log 2,log 3,...}∪{∞}.Only three possibilities for S(X)have been observed so far,namely S(X)={0},S(X)={0,log 2,∞}and S(X)={0,log 2,log 3,...}∪{∞}.In this paper we completely solve the problem of finding all possibilities for S(X)by showing that in fact for every set{0}?A?{0,log 2,log 3,...}∪{∞}there exists a one-dimensional continuum XAwith S(XA)=A.In the construction of XAwe use Cook continua.This is apparently the first application of these very rigid continua in dynamics.We further show that the same result is true if one considers only homeomorphisms rather than continuous maps.The problem for group actions is also addressed.For some class of group actions(by homeomorphisms)we provide an analogous result,but in full generality this problem remains open.

RESEARCH ON ADAPTIVE COMPRESSION CODING FOR NETWORK CODING IN WIRELESS SENSOR NETWORK

Based on the sequence entropy of Shannon information theory, we work on the network coding technology in Wireless Sensor Network (WSN). In this paper, we take into account the similarity of the transmission sequences at the network coding node in the multi-sources and multi-receivers network in order to compress the data redundancy. Theoretical analysis and computer simulation results show that this proposed scheme not only further improves the efficiency of network transmission and enhances the throughput of the network, but also reduces the energy consumption of sensor nodes and extends the network life cycle.

PERIODIC WINDOWS OF NONLINEAR GEAR SYSTEM BASED ON SYMBOLIC DYNAMICS

The periodic window is researched by means of the symbolic dynamics and formal language. Firstly, the proper sampling period is taken and the orbital points of periodic motion are obtained through Poincar6 mapping. Secondly, according to the method of symbolic dynamics of one-dimensional discrete mapping, the symbolic sequence describing the periodic orbit is obtained. Finally, based on the symbolic sequence, the corresponding model of minimal finite automation is constructed and the entropy is obtained by calculating the maximal eigenvalue of Stefan matrix. The results show that the orbits in periodic windows can be strictly marked by using the method of symbolic dynamics, thus a foundation for control of switching between target orbits is provided.

Examples of entropy generating sequence

We introduce the notion of entropy generating sequence for infinite words and define its dimension when it exists. We construct an entropy generating sequence for each symbolic example constructed by Cassaigne such that the dimension of the sequence is the same as its topological entropy dimension. Hence the complexity can be measured via the dimension of an entropy generating sequence. Moreover, we construct a weakly mixing example with subexponential growth rate.

Measurable n-sensitivity and maximal pattern entropy

We introduce the notion of measurable n-sensitivity for measure preserving systems,and study the relation between measurable n-sensitivity and the maximal pattern entropy.We prove that,if(X,ℬ,μ,T)is ergodic,then(X,ℬ,μ,T)is measurable n-sensitive but not measurable(n+1)-sensitive if and only if h_(μ)^(*)(T)=log n,where h_(μ)^(*)(T)is the maximal pattern entropy of T.

Feedback structure based entropy approach for multiple-model estimation

The variable-structure multiple-model（VSMM）approach,one of the multiple-model（MM）methods,is a popular and effective approach in handling problems with mode uncertainties.The model sequence set adaptation（MSA）is the key to design a better VSMM.However,MSA methods in the literature have big room to improve both theoretically and practically.To this end,we propose a feedback structure based entropy approach that could fnd the model sequence sets with the smallest size under certain conditions.The fltered data are fed back in real time and can be used by the minimum entropy（ME）based VSMM algorithms,i.e.,MEVSMM.Firstly,the full Markov chains are used to achieve optimal solutions.Secondly,the myopic method together with particle flter（PF）and the challenge match algorithm are also used to achieve sub-optimal solutions,a trade-off between practicability and optimality.The numerical results show that the proposed algorithm provides not only refned model sets but also a good robustness margin and very high accuracy.

Chaos and null systems

A dynamical system is called a null system, if the topological sequence entropy along any strictly increasing sequence of non-negative integers is 0. Let 0≦p≦q≦1. A dynamical system is Dqp chaotic, if there is an uncountable subset in which any two different points have trajectory approaching time set with lower density p and upper density q. In this paper, we show that there is a null system which is also D3/41/4 chaotic.

Density-equicontinuity and Density-sensitivity

In this paper we introduce the notions of(Banach) density-equicontinuity and densitysensitivity. On the equicontinuity side, it is shown that a topological dynamical system is densityequicontinuous if and only if it is Banach density-equicontinuous. On the sensitivity side, we introduce the notion of density-sensitive tuple to characterize the multi-variant version of density-sensitivity. We further look into the relation of sequence entropy tuple and density-sensitive tuple both in measuretheoretical and topological setting, and it turns out that every sequence entropy tuple for some ergodic measure on an invertible dynamical system is density-sensitive for this measure;and every topological sequence entropy tuple in a dynamical system having an ergodic measure with full support is densitysensitive for this measure.