周期时变布尔网络的完全同步化
韩吉;张化光;田辉
【摘 要】This paper mainly investigates the complete synchronization of two Boolean networks (BNs) coupled uni-directionally in the drive-respon configuration, where the drive BN is periodically time-variant. We discuss it in two cas under the algebraic framework of logical systems. For each ca, a necessary and sufficient criterion for complete synchronization is prented. Accordingly, two general design approaches to a synchronizing respon BN are developed. Finally, numerical examples are given in order to illustrate the effectiveness of our results.%本文主要研究了驱动–响应结构下的布尔网络的完全同步化，其中驱动系统是一个周期性时变的布尔网络。对于上述问题，本文基于逻辑系统的代数形式下分两种情况讨论。对于每种情况，都将给出一个完全同步的充要条件。相应地，提出了两个响应布尔网络的同步化方案。最后，通过一些数值例子来说明本文结果的有效性。
【期刊名称】《控制理论与应用》
【年(卷),期】2016(033)007
【总页数】上海口译7页(P863-869)
【关键词】布尔网络;周期性时变布尔网络;完全同步
【作 者】韩吉;张化光;田辉
【作者单位】东北大学信息科学与工程学院，辽宁沈阳110819;东北大学信息科学与工程学院，辽宁沈阳110819;东北大学信息科学与工程学院，辽宁沈阳110819; 河南理工大学数学与信息科学学院，河南焦作454000detective
【正文语种】中 文
【中图分类】TP273
Boolean network was first introduced by Kauffman in 1969.Ithasbeen w idely ud inmodeling complex systems such as gene regulatory networks，biological evolutionmodelsand neuralnetworks.Ateach discrete timepoint，every node ina BN can t
akeoneof two binary values，1 or0，where 1means that the node isactive and 0 inactive.Every nodeupdatesstate on thebasisof a logical relationship with the form of a Boolean function.Although themodelsof BNsare very simple，they can describe basic dynam ic behavior ofmany realsystemsand provide uful information.Therefore，lotsof issuesaboutBNshave been investigated such asstability，controllability，optimal controland so on[1-6].
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As we know，the synchronization problem of systemshasbeen attractingmany rearchers from various fields，e[7-9].The complete synchronization of BN-swhich can be equivalently transformed into the linear discrete-time systemsby utilizing them i-tensorproduct（STP）[10]also caud a lot of attention and many results were provided.For example，a necessary and sufficientcriterion forcompletesynchronization of BNs wasprovided in[11]，ageneralapproach for the design of a respon BN was given in[12]and the synchronization problem of BNswith some special cas such asdifferentupdate，time-delay and output-coupledwere investigated in[13-16]，respectively，and so on.artistic是什么意思
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Most of rearchers just considered the situation that the BNs were time-invariant Boolean networks（TIBNs），while the synchronization problem of timevariantBoolean networks（TVBNs）was rarely studied. TVBNs havemany esntial properties different from TIBNs.Asa specialcaof TVBNs，periodically timevariantBoolean networks（PTVBNs）can beud to describemany systems such as the sw itched BNswhich have periodical sw itching signals[17]，the Boolean con-trol networks（BCNs）which have dynam ical control lers[18]，the perturbed BNswhich have periodical function perturbations[19]，and so on.Therefore，the study of synchronization of PTVBNs is significant.A lthough the problem of PTVBNs have been studied in few papers[20-21]，the problem of complete synchronization of PTVBNshas notbeen solved well.This is themotivation of our paper.
澳大利亚留学条件雅思In this paper，we investigate a special ca of the complete synchronization problem for two BNs that the drive BN isperiodically time-variant.At first，we study the cycles of PTVBNs and provide some special propertieswhich never exist in the cycles of TIBNs.Then，according to the properties，the PTVBNs are divided into two categories.
butA fter that，the synchronization problemsaresolved forboth two categoriesby utilizing STPand the results are turned out to be quite different from each other.At last，two examples for both categories are provided to support our points.Themajor contributions of this paper：1）provide two necessary and sufficient criteria for complete synchronization of PTVBNs；2）make two design approaches to a synchronizing respon BN，which can complete synchronize with the drive BN in finite steps.
Thispaper isorganized as follows.Problem formulation and some prelim inariesaregiven in Section 2.In Section 3，the periodically time-variant drive Boolean network is discusd in two casand some criteria for completesynchronization and somegeneralapproaches for the design of a synchronizing respon BN are provided.Two numericalexamples are given in Section 4 and a conclusion isdrawn in Section 5.
正规翻译公司First，we give some notations.
饼干英文
1）N reprents thetofnon-negative integers，N+reprents the tof positive integersand N[λ1，λ2]reprents thetof integers fromλ1toλ2whichmeans that N[λ1，λ2]
=｛λ1，λ1+1，………，λ2｝.
2）In∈Rn×nreprents the identity matrix，reprents the i-th column of In.
3）∆nreprents the t of，where i=1，2，………，n.
4）ε（x）reprents a bijection from∆2nto N[1，2n]such that
5）Col（A）reprents the tof all columns of A，where A∈Rm×n.
