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中文摘要中文摘要由于通信网络以及分布式控制的快速发展，多智能体系统的一致性研究成为系统与控制领域的研究热点，受到了国内外学者的广泛关注。
多智能体系统的一致性是指系统中每个智能体通过通信网络传递信息，使其在位置或者速度等状态量上趋于渐近相同，呈现出行为状态的一致，被广泛的应用于多无人机编队、多卫星角度校正、多传感器网络同步等。
由于现实世界中存在噪声、空气阻力等非线性因素，这些因素常常会给多智能体系统的一致性造成一定的影响，因此考虑带有非线性因素的多智能体系统的一致性具有重要意义。
在一致性控制策略的设计中，采样控制策略能降低控制器的更新频率，但控制器在每个控制时间段内依然要连续工作，而间歇控制策略可以减少控制器的工作时间，因此将采样控制和间歇控制策略相结合有利于统一考虑控制器的更新频率和工作时间。
本文研究基于间歇采样控制策略的非线性多智能体系统一致性问题，具体内容如下：首先，针对带有非线性因素的一阶多智能体系统，分别采用了周期间歇采样控制策略和非周期间歇采样控制策略，通过矩阵理论以及不等式的证明等得到了系统实现一致的充分性条件，从理论上分析证明了所设计的控制策略的可行性。
最后利用数值仿真验证了理论结果的有效性，并通过仿真结果进一步剖析得知，通信宽度和采样宽度对系统状态达到一致起着至关重要的作用。
其次，在以运动学为背景的物理世界中，研究带有非线性因素的二阶多智能体系统更符合实际情况。
并运用间歇采样控制策略，通过严格的理论证明，得到了二阶非线性多智能体系统达到一致的充分条件。
最后利用数值仿真验证了一致性理论的有效性，使得多智能体一致性算法具有更强的实用价值。
最后，为了进一步验证所研究的一致性算法的实用性，基于Anylogic软件仿真平台，搭建了多无人机系统一致性的虚拟原型环境，模拟多智能体之间信息交流，最后通过一致性耦合运算实现了无人机系统的一致性运动，从而验证了一致性理论的可行性。
关键词：多智能体系统；非线性；间歇采样控制；一致性；Anylogic仿真IABSTRACTABSTRACTDue to the rapid development of communication network and distributed control,the research on the consensus of multi-agent systems has become a hot topic in the field of systems and control,which has been widely concerned by scholars at home and abroad.A multi-agent system is a set of systems that work in a network environment and have multiple autonomous individuals.Consensus means that each intelligence in a system transmits information through a communication network to make it asymptotically identical in terms of position or velocity,showing a consensus behavior,and is widely used in multi-UA V formation,multi-satellite angle correction,multi-sensor network synchronization and so on.Because there are nonlinear factors such as noise and air resistance in the real world,these factors often affect the consensus of multi-agent systems, so it is important to consider the consensus of multi-agent systems with nonlinear factors.In the consensus analysis of nonlinear multi-agent systems,the sampled-data control strategy can reduce the update frequency of the controller, but the controller still has to work continuously in each control time period,and the intermittent control strategy can reduce the working time of the controller, Therefore,the combination of sampled-data control and intermittent control strategy is beneficial to consider the update frequency and working time of the controller consensus.This paper studies the consensus problem of nonlinear multi-agent system based on intermittent sampled-data control strategy,the details of the paper are as follows:Firstly,for the first-order multi-agent system with nonlinear factors,the control strategy of periodic intermittent sampled-data and aperiodic intermittent sampled-data are adopted respectively.By means of matrix theory and the proof of inequality,we get the conditions for the system to achieve consensus adequacy.the feasibility of the designed control strategy is proved by theoretical analysis.Finally,numerical simulation is used to verify the validity of theIII非线性多智能体系统的间歇采样控制一致性研究theoretical results,and further analysis of the simulation results shows that the communication width and sampled-data width play a vital role in the system state to achieve consensus.Secondly,in the physical world with kinematics as the background,it is more realistic to study the second-order multi-agent system with nonlinear factors.By using the intermittent sampled-data control strategy,it is proved by strict theory that the consensus condition of the second-order nonlinear multi-agent system.Finally,the validity of the consensus theory is verified by numerical simulation,which makes the multi-agent consensus algorithm more practical.Finally,in order to further verify the practicability of the studied consensus algorithm,based on the above theoretical results,based on the Anylogic software simulation platform,a virtual prototype environment of multi-UA V system consensus is built,and the information exchange between multi-agent is simulated.Finally,the consensus motion of UAV system is realized by consensus coupling operation,which verifies the feasibility of consensus theory. Key words:Multi-agent systems;Nonlinear;Intermittent sampled-data control; Consensus;Anylogic simulationIV目录目录第一章绪论 (1)1.1课题背景及研究意义 (1)1.2多智能体系统一致性简介 (2)1.3一致性问题研究现状及分析 (3)1.4基于间歇采样控制的一致性研究概况 (6)1.4.1基于采样控制的一致性 (6)1.4.2基于间歇控制的一致性 (6)1.5本文研究内容及结构安排 (7)第二章预备知识 (9)2.1基本符号 (9)2.2代数图论 (10)2.3一致性相关理论 (13)2.4本章小结 (14)第三章一阶非线性多智能体系统间歇采样控制的一致性 (15)3.1引言 (15)3.2系统模型的建立及预备知识 (15)3.3周期间歇采样控制策略 (18)3.4非周期间歇采样控制策略 (20)3.5数值仿真 (23)3.6本章小结 (28)第四章二阶非线性多智能体系统间歇采样控制的一致性 (29)4.1引言 (29)4.2系统模型的建立及预备知识 (29)4.3理论分析与证明 (32)4.4数值仿真 (34)4.5本章小结 (39)第五章基于Anylogic的多智能体系统一致性仿真 (41)5.1引言 (41)5.2无人机系统仿真平台创建 (42)V非线性多智能体系统的间歇采样控制一致性研究5.3无人机系统仿真前端设计 (45)5.4无人机系统仿真实验结果 (49)5.5本章小结 (52)第六章总结与展望 (53)6.1全文总结 (53)6.2工作展望 (53)参考文献 (55)致谢 (61)攻读学位期间发表的学术论文目录 (63)VI第一章绪论1第一章绪论1.1课题背景及研究意义洞察自然界，随处可见许多奇妙有趣的现象。
基于复卷积双域级联网络的欠采样磁共振图像重建算法邱华禄;蔺素珍;王彦博;刘峰;李大威【期刊名称】《计算机应用》【年(卷),期】2024(44)2【摘要】目前,大多数加速磁共振成像(MRI)的重建算法通过重建欠采样幅值图像,利用实值卷积进行特征提取,没有考虑MRI数据本身是复数,从而限制了对MRI复值数据的特征提取能力。
为了提高对单个切片MRI复值数据特征提取能力,从而重建出细节更为清晰的单切片磁共振(MR)图像,提出复卷积双域级联网络(ComConDuDoCNet)。
将原始欠采样MRI数据作为输入,使用残差特征聚合(RFA)块交替提取MRI数据的双域特征,最终重建出具有清晰纹理细节的MR图像。
每个RFA块使用复卷积作为特征提取器。
不同域间通过傅里叶变换或逆变换进行级联,并加入数据一致性层实现数据保真。
在公开的膝关节数据集上进行实验,与双任务双域网络(DDNet)在采样率为20%的三种不同采样掩码下的对比结果表明,在二维高斯采样掩码下,所提算法的标准均方根误差(NRMSE)下降了13.6%,峰值信噪比(PSNR)提升了4.3%,结构相似性指数(SSIM)提升了0.8%;在泊松采样掩码下,所提算法的NRMSE下降了11.0%,PSNR提升了3.5%,SSIM提升了0.1%;在径向采样掩码下,所提算法的NRMSE下降了12.3%,PSNR提升了3.8%,SSIM提升了0.2%。
实验结果表明,ComConDuDoCNet结合复卷积与双域学习,能够重建出细节更加清晰、视觉效果更加逼真的MR图像。
【总页数】8页(P580-587)【作者】邱华禄;蔺素珍;王彦博;刘峰;李大威【作者单位】中北大学计算机科学与技术学院;昆士兰大学信息技术与电子工程学院;中北大学电气与控制工程学院【正文语种】中文【中图分类】TP391.41【相关文献】1.基于卷积神经网络的欠采样脑部核磁共振图像重建方法2.基于并行双域级联卷积网络的磁共振图像重建3.基于非下采样双树复小波域的双变量模型去噪算法4.基于残差图卷积神经网络的高倍欠采样核磁共振图像重建算法5.基于双域并行编解码网络的磁共振图像重建因版权原因，仅展示原文概要，查看原文内容请购买。
具有三个位势的广义耦合无色散可积方程的多孤子解（英文）2O11年5月第42巷第3期内蒙古大学(自然科学版)JournalofInnerMongoliaUniversity(NaturalScienceEdition)May2011V o1.42No.3ArticleID:1000—1638(2O11)03—0253-05 Multi—solitonSolutionsoftheGeneralized DispersionlessIntegrableEquationwithThreeCoupledPotentialsZhaqilao(C0llegeofMathematicsScience,InnerMongoliaNormalUniversity,Hohhot010022,Chin a)Abstract:AnewN—foldDarbouxtransformationofthegeneralizedcoupleddis—Dersionlessintegrableequationisderivedwiththeaidofthegaugetransforma—tionbetweencorresponding2×2matrixspectralproblemswiththreepoten—tla1s.AsanapplicationoftheDarbouxtransformation,N—solitonsolutionsof thegeneralizedcoupleddispersionlessintegrableequationareexplicitlygiven. Keywords:Darbouxtransformation;solitonsolution;coupleddispersionlessinte—grableequationCLCnumber:O175.2Documentcode:AIntr0ducti0n Thestudvofdispersionlesshierarchiesisoneofthemostprominentsubjectinthefieldofnon —linearscience,partlybecauseofdispersionlessequationsemergenceindiverseareasofmathe maticalandthe.retica1Dhysicssuchasquantumfieldtheory,conformalfieldtheory,stringtheory,sol itontheorv,etc.Inpast,thecoupleddispersionlessequationanditsgeneralizationhasalsoattracte da reatdealofinterestbecauseofitsniceintegrabilitystructureandsolitonsolutions. InRefs.[1—2],KonnoandKakuhataconsideredthegeneralformofthecoupleddispersionless integrableequation(CDIE)q+(rs)=0,,.一2r一0,s一25g一0(1) andobtainedthesingleanddoublesolitonsolutionsofCDIE(1)bytheinversescatteringtransf orm(IST).CDIE(1)hastwospecialcases:r=andr—.Theformercasewasfoundtobeequivalent tothesine—Gordone0uation[.andthelattertothePohlmeyer—gund--Reggeequation 们.Forr—s,CDIE(1)reducestothemodelwithtwopotentials.Inthepastyears,manyeffortshavebeen dedicatedtothereducedequationandrichfamiliesofexactanalyticalsolutionshavebeenobta inedeffectivemethods,suchasIST,Painlev6analysis∞,B/icklundtransformation(BT)"andDarbouxtransformation(DT)C783.However,toourknowledge,verylittleresearchhas beendoneonCDIE(1)viatheN—foldDTmethod. TheaimofthispaPeristoobtainN—solitonsolutionsofCDIE(1)withthehelpofspectral*Receiveddate:2011-05-06Foundationitem:SupportedbytheNaturalScienceFoundationofInnerMongolia(GrantNo 2009MS0108);the HighEducationScienceResearchofInnerMongoIiaAutonomousRegion(GrantNo?NJ10045)Biography:Zhaqila.(1971一),ma1e(Mongolian),anativeofOrdosofInnerMongolia,Viceprofesser,Ph?D?E_ mail:zhaqilao@.254内蒙古大学(自然科学版)2011矩problemsandtheDarbouxmatrixmethod(9-131.Tothisend,westartfromthecorresponding spec—tralproblemsofCDIE(1)andachieveanN—foldDTinsection1.ResortingtoDTinsection1.N—solitonsolutionsofCDIE(1)areobtainedinsection2.Mostimportantly,thedetailedstructure softhesinglesolitonanddoublesolitonsolutionsofCDIE(1)aregivenbothanalyticallyandgrap hical—ly.DarbouxtransformationInthissection,weshal1constructaThespectralproblemsofCDIE(1)whereU:=——iXnewN—foldDTforCDIE(1).aregivenas一U,≯===qfr5—q,V一(:--.r)十i(～0)(2)(3)Throughaa"lrectcalculation,thezerocurvatureequationUf—V+U,厂一【,『一0giverisetoCDIE(1).TheDarbouxtransformationisactuallyagau—ofspectralproblems(2).Itisrequiredthatalso一,一(+TU)T～,whereU一一iq尸sI—qlgetransformation(4)satisfiesthesamespectralproblemst—,V一(+)T-(5),V—O—+～o)Nowwediscussaconcretetransformation.LetmatrixTin(6)beintheformofT一丁c一A暑)whereN一1N～1N—lN--1A—一+∑A,B一∑B,c一∑CA一,D—一+∑D,0^=0=0k=OA,B,C^,Dkarefunctionsofzandt.A^,B,CandD^aregivenbyalinearalgebraicsystem N—l一一,∑(cI+3jD)—as117景0(6)(7)(8)怒蒋,,≤≤2N/1(9)1()+J1(J)'where一(l,2),一(1,)aretWObasicsolutionsofspectralproblems(2),and(≠J,:≠,asi≠)aresomeparameterssuitablychosensuchthatthedeterminantofcoefficientsfor (8)iSnonzero.Hence,ifwetakeDl,一--AN-一1.(10)therestA^,B^,CkandD^(1≤J≤N一1)areuniquelydeterminedby(8).Eq.(7)showsthatdetT()isa(一2N)th—orderpolynomialin,anddetT()=A()D()一B(,)C().Ontheotherhand,from(8)wehaveA()一一a,B(),CO,)一一D().,,,B+A,L∑h.代w第3期Zhaqila0Multi--solitonSolutionsoftheGeneralizedCoupled (255)Therefore,itholdsthatdet-/(,)一0whichimpliesthat,(1≤J≤2N)are2Nrootsofdet丁(),thatisdetT()===yⅡ(—)(11)whereyisindependentof.Inawaysimilartotheproofin[13],wecanverifythefollowingprop.sitions.Proposition1MatrixdeterminedbythesecondexpressionofEq.(5)hasthesameformasU.wherethetransformationformulafromtheoldpotentialsq,,-,intonewonesaregivenby ……一一邶一r+i一一瓮whereAN--1iscoefficientsof(8),i.e.AN一111…l__N÷1282),7….:LN+;;;:'.;j12N2N--N1….=L--2NN-.-2Ni.+(13)and△^…,△BareproducedfromGbyreplacingits(2N一1)th,2Nthcolumnwith(一,…, 一^.--2NN),respectively.△c1isproducedfrom△lbyreplacingits(2N一1)thcolumnwith(一,…,一2N).Accordingto(10),allJ(】≤≤2N)mustsatisfyconditions一一(一1,3,5,7,…).Proposition2MatrIxdefinedbythefourthexpressionofEq.(5)hasthesameformasV,m whchtheo1dpotentia1sg,r,saremappedintonewpotentials,r,,accordingtothesameDT (12).Propositions1and2showthatthetransformation(12)changesspectralproblems(2)intoan—otherspectra1problems(5),withU,Vand—U,一Vhavingthesameform.Thereforebothofthespec—tra1DroblemsleadtothesameCDIE(1),SO(12)istheDarbouxtransformationofCDIE(1)?F romPropositions1and2wehavethefollowingtheorem?Theorem1Let(q,r,s)beasolutionofCDIE(1).Thenfunction(,尸,)determinedbyDT (12)iSanewsolutionofCDIE(1).2SolitonsolutionsNowwecho.sethetrivialsolutionq.=fiz(|8isanarbitrarynon—zeroconstant)andro—s.一0,thecorrespondingcompatibIebasicsolutionofspectralproblems(2)canbewrittenas (.z,t,)一exp(一￡)00exp($i)(14)where￡一if1)!(一赢?Using(9),when,≠0andj≠0,wehave一exp(2岛),(:1,2)(15)Substituting(15)into(13),(12)denotesaunifiedandexplicitformulationofallk～solit.n.一lution(1≤是≤N),fr.mwhichitiseasytogetN—solitonsolutionsofCDIE(】)?Forsimplicity,wesha1】discl】sstwospectralcasesofN一1andN一2.256内蒙古大学(自然科学版)2011在Case1(N一1).Let,,,(一1,2)and2=::--A1.From(12),wehaveanexplicitsolutionof CDIE(1)asfollowing—fix-一一一一c16)'一一'where一lexp(2￡-i一蠢_1'2).If2一一1andl,2arepureimaginaryparameters,l,2arerealparameters,thesolutionde—terminedby(16)isthesinglesolitonsolutionq一—Itanhi,r一一sechi,s一一1sechi,一砉(￡一2Za12X)(17)Case2(N一2).Let,J(一1,2,3,4)and2一--;t1,4一--A3.From(12),wehaveanewso—lutionofCDIE(1)qz一卢z+i,r一i,s一i等c8whprA1一2j3 AA.△c1一一_『一一一一l一一3i一4l2iA3I23 AB.=一——.I_一where(一1,2,3,4)aredeterminedin(15).If,uz一--/zl,/14一一3and,(一1,2,3,4)arepureimaginaryparameters,J(一1,2,3,4)are realparameters,thesolutiondeterminedby(18)isthedoublesolitonsolution(seeFig.1).-4020(a】(b)(c)Fig.1Doublesolitonsolution(18)withr—s一0,一A1一2—5i,l一--p2—1,卢=0.O1,ps:一{一1,一A3=^=3iIteratingtheabovemethod,wecanobtainaseriesofmulti—solitonsolutionsofCDIE(1).3ConclusionByextendingelementsoftheDTmatrixintonegative--orderpolynomialsin,wedeveloped theDTmatrixmethod"todirectlyconstructanexplicitdeterminantformu1aforN—foldDTofCDIE(1).ThisN—foldDTformulacanbeinterpretedasanonlinearsuperpositionofasingleDT. Moreover,notonlytheconstructionofN—foldDTisverynaturalandmuchsimplerthanthatob一11111111乱1111融111l第3期ZhaqilaoMuIti—solitonSolutionsoftheGeneralizedCoupled (257)tainedintheusualwayEl-z],butalsotheN—foldDTisverysuitableforgeneratingaseriesofexplic—itsolutionsbysymboliccomputationonacomputer.References:[1][2][3][4][5]1-6][7][83[9][1o][11][12][13]KonnoK,KakuhataH.InteractionAmongGrowing,DecayingandStationarySolitonsforC oupledIntegrableDis—persionlessEquations[J].J.Phys.Soc.Jpn.,1995,64(8):2707-2709.KonnoK,KakuhataH.NovelSolitonicEvolutionsinaCoupledIntegrable,DispersionlessS ystem[J].J.Phys.Soc.Jpn.,1996,65(3):713—721.HirotaR,TsujimotoS.Noteon"NewCoupledIntegrableDispersionlessEquations"[J],J.Ph ys.Soc.Jpn.,I994,63(9):3533-3533.KotlyarovVP.OnEquationsGaugcEquivalenttOtheSine-GordonandPohlmeyer—Iund —ReggeEquations[J].J.Phys.Soc.Jpn.,1994,63:3535-3537.KonnoK,OonoH.NewCoupledIntegrableDispersionlessEquations[J].J.Phys.Soc.Jpn.,1 994,63:377—378.AlagesanT,ChungY,NakkeeranK.B~cklundtransformationandsolitonsolutionsfortheco upleddispersionlessequations[J].Chaos,SolitonsandFractals,2004,21:63—67.CbenAH,LiXM.Solitonsolutionsofthecoupleddispersionlessequation[J].Phys.Lett.A,2 007,370:281—286.HassanHJ.Darbouxtransformationofthegeneralizedcoupleddispersionlessintegrablesys tem[JⅢ.Phys.A:Math.Gen.,2009,42:l一11.NeugebaureG,MeineIR.GeneralN—so!itonsolutionoftheAKNSclassonarbitrarybackground[J].Phys.Lett.A,1984,100:467-470.MatveevVB,SalleMA.DarbouxTransformationsandSolitons[M].Berlin:Springer,1991. GuCH,HuHS,ZbouZx.DarbouxTransformationinSolitonTheoryandItsGeometricAppli cations[M].Shanghai:ShanghaiScienceandTechnologyPublishingHouse,2005.FanE(;.ComputerAlgebraandIntegrableSystems[M].Beijing:SciencePress,2004. Zhaqilao,ChenY,IiZB.Darbouxtransformationandmulti—solitonsolutionsforsomesolitonequations[J].Chaos,SolitonsandFractals.2009,41:661-670.(责任编委孙炯)具有三个位势的广义耦合无色散可积方程的多孤子解扎其劳(内蒙古师范大学数学科学学院,呼和浩特010022)摘要:利用具有三个位势的2×2矩阵谱问题的规范变换,给一个广义耦合无色散方程构造了一种新的N重达布变换.作为达布变换的应用,获得了该广义耦合无色散方程的N一孤子解.关键词:达布变换;孤子解;广义耦合无色散可积方程中图分类号:O175.2文献标志码:A收稿日期:2010—05-06基金项目:内蒙古自然科学基金项目(2009MS0108);内蒙古自治区高等学校科学研究项目(NJ10045)作者简介:扎其劳(1971一),男(蒙古族),内蒙古鄂尔多斯市人,副教授,博士.。
建议收藏 | 肿瘤免疫机制的研究思路背景知识肿瘤是一个复杂的生态系统，其中癌细胞和宿主细胞之间的相互作用影响疾病进展和治疗反应。
除了癌细胞，免疫细胞可以说是实体瘤中最复杂的参与者，其活性范围可以从抗致瘤性到致瘤性。
在肿瘤进展期间，癌细胞会采用多种手段来逃避免疫攻击，例如下调抗原呈现机制或诱导抑制性免疫检查点分子表达。
同时，癌细胞会劫持免疫细胞（如中性粒细胞，巨噬细胞和调节性T细胞（Treg细胞）），来协调免疫抑制性的TME。
这反过来促进免疫逃逸，促进脉管系统和细胞外间质的重塑，并支持癌症进展和治疗抵抗。
因此，异常免疫反应被认为是癌症的标志，并为癌症治疗提供了靶点，最好的例子是免疫检查点阻断（Immune-Checkpoint Blockade，ICB）疗法的成功。
近些年绘制肿瘤细胞结构图和评估单个细胞分子特征的技术快速发展。
这些技术，尤其是单细胞分析、空间转录组分析，为TME的异质性提供了前所未有的见解。
同时，这些技术也揭示了特定癌症类型的内部特征与癌症免疫景观间的异质性。
小编在下文中重点关注遗传变异、表观遗传修饰、细胞信号通路和代谢改变，这些都是塑造癌症免疫景观的癌细胞固有特征[1]。
遗传异常与肿瘤免疫癌症基因组通常会发生多种体细胞改变，例如点突变、DNA片段的插入或缺失、基因组扩增和重排。
这些改变中的一些作为“Driver”，在肿瘤的发生和进展中起作用，而另一些作为“Passenger”，对于癌细胞生长没有影响。
近年来，科研人员将高通量测序技术与高分辨率免疫图谱相结合来分析人类肿瘤，以确定遗传畸变与免疫表型之间的关联，免疫肿瘤学领域从中受益匪浅。
这些研究也发现了新的肿瘤基因与免疫表型相关性。
这一部分的重点往往是影响肿瘤免疫结构和免疫治疗反应的特定癌基因、抑癌基因和DNA损伤机制。
这些遗传畸变会影响免疫格局和肿瘤对免疫治疗的反应。
此外，有临床证据已经证实其免疫调节作用（图1）。
图1癌症基因分型与免疫表型的关系MYC是人类肿瘤中最常见的激活癌蛋白之一，也是目前研究最多的一类癌基因。
[Word]半导体行业的英文单词和术语半导体行业的英文单词和术语安全地线 sfe ground wire安全特性 security feture安装线 hook-up wire按半周进行的多周期控制 multicycle controlled by hlf-cycle 按键电话机push-button telephone set 按需分配多地址 demnd ssignment multiple ccess(DM) 按要求的电信业务 demnd telecommuniction service 按组编码encode by groupB八木天线 Ygi ntenn白噪声 white Gussin noise白噪声发生器 white noise genertor半波偶极子 hlfwve dipole半导体存储器 semiconductor memory半导体集成电路 semiconductor integrted circuit 半双工操作 semi-duplex opertion半字节 Nib包络负反馈 pek envelop negtive feed-bck 包络延时失真 envelop dely distortion薄膜 thin film薄膜混合集成电路 thin film hybrid integrted circuit 保护比(射频) protection rtio (RF)保护时段 gurd period保密通信 secure communiction报头 heder报文分组 pcket报文优先等级 messge priority报讯 lrm备用工作方式 spre mode背景躁声 bckground noise倍频 frequency multipliction倍频程 ctve倍频程滤波器 octve filter被呼地址修改通知 clled ddress modified notifiction 被呼用户优先priority for clled subscriber 本地PLMN locl PLMN本地交换机 locl exchnge本地移动用户身份 locl mobile sttion identity ( LMSI) 本地震荡器 locl oscilltor比功率(功率密度) specific power比特 bit比特并行 bit prllel比特号码 bit number (BN)比特流 bit strem比特率 bit rte比特误码率 bit error rte比特序列独立性 bit sequence independence 必要带宽 necessry bndwidth闭环电压增益 closed loop voltge gin 闭环控制 closed loop control 闭路电压 closed circuit voltge 边瓣抑制 side lobe suppression 边带sidebnd边带非线性串扰 sidebnd non-liner crosstlk 边带线性串扰 sidebnd liner crosstlk 边带抑制度 sidebnd suppression边角辐射 boundry rdition编号制度 numbering pln编解码器 codec编码 encode编码律 encoding lw编码器 encoder编码器输出 encoder output编码器总工作时间 encoder overll operte time 编码效率 coding efficiency编码信号 coded signl编码约束长度 encoding constrint length 编码增益 coding gin编译程序 compiler鞭状天线 whip ntenn变频器 converter变频损耗 converter conversion loss 变容二极管 vrible cpcitnce diode 变形交替传号反转 modified lternte mrk inversion便携电台 portble sttion便携设备 portble equipment便携式载体设备 portble vehicle equipment 标称调整率(标称塞入率) nominl justifiction rte (nominl stuffing rte)标称值 nominl vlue标称呼通概率 nominl clling probbility 标准码实验信号 stndrd codetest signl (SCTS) 标准模拟天线 stndrd rtificil ntenn 标准频率 stndrd frequency标准时间信号发射 stndrd-time-signl emission 标准实验调制 stndrd test modultion标准输出功率 stndrd power output标准输入信号 stndrd input signl标准输入信号电平 stndrd input-signl level 标准输入信号频率 stndrd input-signl frequency 标准信躁比 stndrd signl to noise表面安装 surfce mounting表示层 presenttion lyer并串变换器 prllel-seril converter (serilizer) 并馈垂直天线 shunt-fed verticl ntenn 并行传输 prllel trnsmission并行终端 prllel terminl拨号错误概率 diling mistke probbility 拨号后延迟 post-diling dely 拨号交换机 dil exchnge拨号线路 dil-up line拨号音 diling tone拨号终端 dil-up terminl波动强度(在给定方向上的) cymomotive force (c. m. f) 波段覆盖 wve coverge波峰焊 wve soldering波特 bud泊送过程 Poisson process补充业务 supplementry service (of GSM) 补充业务登记 supplementry service registrtion 补充业务询问 supplementry service interrogtion 补充业务互连 supplementry service interworking 捕捉区(一个地面接收台) cpture re (of terrestril receiving sttion)捕捉带 pull-in rnge捕捉带宽 pull-in bnwidth捕捉时间 pull-in time不连续发送 discontinuous trnsmission (DTX) 不连续干扰 discontinuous interference 不连续接收 discontinuous reception (DRX) 不确定度uncertinty步谈机 portble mobile sttionC采样定理 smpling theorem采样频率 smpling frequency采样周期 smpling period参考边带功率 reference side bnd power 参考差错率 reference errorrtio参考当量 reference equivlent参考点 reference point参考结构 reference configurtion参考可用场强 reference usble fiend-strength 参考灵敏度 reference sensibility参考频率 reference frequency参考时钟 reference clock参考输出功率 reference output power残余边带调制 vestigil sidebnd modultion 残余边带发射 vestigil-sidebnd emission 操作维护中心 opertion mintennce center (OMC) 操作系统opertion system (OS)侧音消耗 sidetone loss层2转发 lyer 2 rely (L2R)插入组装 through hole pchnology插入损耗 insertion loss查号台 informtion desk差错控制编码 error control coding差错漏检率 residul error rte差分脉冲编码调制(差分脉码调制) differentil pulse code modultion (DPCM)差分四相相移键控 differentil qudrture phse keying (DQPSK)差分相移键控 differentil phse keying (DPSK) 差模电压，平衡电压differentil mode voltge, symmetricl voltge差拍干扰 bet jmming差频失真 difference frequency distortion 长期抖动指示器 long-term flicker indictor 长期频率稳定度 long-term frequency stbility 场强灵敏度field intensity sensibility 场效应晶体管 field effect trnsistor (FET) 超长波通信 myrimetric wve communiction 超地平对流层传播 trnshorizon tropospheric 超地平无线接力系统 trnshorizon rdio-rely system 超高帧hyperfrme超帧 superfrme超大规模集成电路 very-lrge scle integrted circuit (VLSI)超再生接收机 super-regenertor receiver 车载电台 vehicle sttion撤消 withdrwl成对不等性码(交替码、交变码)pired-disprity code (lterntive code, lternting code)承载业务 berer service城市交通管制系统 urbn trffic control system 程序设计技术 progrmming technique程序设计环境 progrmming environment程序优化 progrm optimiztion程序指令 progrm commnd充电 chrge充电率 chrge rte充电效率 chrge efficiency充电终止电压 end-of chrge voltge 抽样 smpling抽样率 smple rte初级分布线路 primry distribution link 初始化 initiliztion处理增益 processing gin传播时延 propgtion dely传播系数 propgtion coefficient 传导干扰 conducted interference 传导杂散发射 conducted spurious emission 传递函数 trnsfer function 传递时间 trnsfer time传声器 microphone传输保密 trnsmission security传输层协议 trnsport lyer protocol 传输集群 trnsmission trunking 传输结束字符 end of trnsmission chrcter 传输媒体 trnsmission medium 传输损耗 trnsmission loss传输损耗 (无线线路的) trnsmission loss (of rdio link)传输通道 trnsmission pth传输信道 trnsmission chnnel传真 fcsimile, FX船舶地球站 ship erth sttion船舶电台 ship sttion船舶移动业务 ship movement service 船上通信电台 on-bord communiction sttion ,ship communiction sttion船用收音机 ship rdio串并变换机 seril to prllel (deserilizer) 串并行变换 seril-prllel conversion 串话 crosstlk垂直方向性图 verticl directivity pttern 唇式传声器 lip microphone 磁屏蔽 mgnetic shielding次级分布线路 secondry distribution link 猝发差错 burst error猝发点火控制 burst firing control 存储程序控制交换机 stored progrm controlled switching systemD大规模集成电路 lrge scle integrted circuit (LSI) 大信号信躁比 signl-to-noise rtio of strong signl 带成功结果的常规操作 norml opertion with successful outcome带宽 bndwidth带内导频单边带 pilot tone-in-bnd single sidebnd 带内谐波 in-bnd hrmonic带内信令 in-bnd signlling带内躁声 in-bnd noise带通滤波器 bnd-pss filter带外发射 out-of-bnd emission带外功率 out-of-bnd power带外衰减 ttenution outside chnnel带外信令 out-bnd signlling带状线 stripline单边带发射 single sidebnd (SSB) emission 单边带发射机 single side-bnd (SSB) trnsmitter 单边带调制 single side bnd modultion 单边带解调 single side bnd demodultion 单边带信号发生器 single side bnd signl generltor 单端同步 single-ended synchroniztion 单工、双半工simplex, hlfduplex单工操作 simplex opertion单工无线电话机 simplex rdio telephone单呼 single cll单频双工 single frequency duplex单频信令 single frequency signlling单相对称控制 symmetricl control (single phse) 单相非对称控制symmetricl control (single phse) 单向 one-wy单向的 unidirectionl单向控制 unidirectionl control单信道地面和机载无线电分系统 SINCGRS单信道无绳电话机 single chnnel cordless telephone 单信号方法 single-signl method单音 tone单音脉冲 tone pulse单音脉冲持续时间 tone pulse durtion单音脉冲的单音频率 tone frequency of tone pulse 单音脉冲上升时间tone pulse rise time单音脉冲下降时间 tone pulse decy time单音制 individul tone system单元电缆段(中继段) elementry cble section (repeter section)单元再生段 elementry regenertor section (regenertor section)单元增音段，单元中继段 elementry repeter section当被呼移动用户不回答时的呼叫转移 cll forwrding on no reply (CFNRy) 当被呼移动用户忙时的呼叫转 clling forwrding on mobile subscriber busy (CFB)当漫游到原籍PLMN国家以外时禁止所有入呼 brring of incoming clls when roming outside the homePLMN country (BIC-Rom)当前服务的基站 current serving BS当无线信道拥挤时的呼叫转移clling forwrd on mobile subscriber not rechble (CENRc) 刀型天线 blde ntenn导频 pilot frequency导频跌落pilot fll down倒L型天线 inverted-L ntenn等步的 isochronous等幅电报 continuous wve telegrph等权网(互同步网) democrtic network (mutully synchronized network) 等效比特率 equivlent bit rte等效地球半径 equivlent erth rdius等效二进制数 equivlent binry content等效全向辐射功率 equivlent isotropiclly rdited power (e. i. r. p.) 等效卫星线路躁声温度 equivlent stellite link noise temperture 低轨道卫星系统 LEO stellite mobile communiction system 低气压实验 low tmospheric pressure test低时延码激励线性预测编码 low dely CELP (LD-CELP)低通滤波器 low pss filter低温实验 low temperture test低躁声放大器 low noise mplifier地-空路径传播 erth-spce pth propgtion地-空通信设备 ground/ir communiction equipment地波 ground wve地面连线用户 lnd line subscriber地面无线电通信 terrestril rdio communiction地面站(电台) terrestril sttion第N次谐波比 nth hrmonic rtio第二代无绳电话系统 cordless telephone system second genertion (CT-2) 第三代移动通信系统 third genertion mobile systems点波束天线 spot bem ntenn点对地区通信 point-re communiction点对点通信 point-point communiction点至点的GSM PLMN连接 point to point GSM PLMN电报 telegrphy电报电码 telegrph code电波衰落 rdio wve fding电池功率 power of bttery电池能量 energy cpcity of bttery电池容量 bttery cpcity电池组 bttery电磁波 electromgnetic wve电磁波反射 reflection of electromgnetic wve 电磁波饶射 diffrction of electromgnetic wve 电磁波散射 scttering of electromgnetic wve 电磁波色射dispersion of electromgnetic wve 电磁波吸收 bsorption of electromgnetic wve 电磁波折射 refrction of electromgnetic wve 电磁场 electromgnetic field电磁发射 electromgnetic field电磁辐射 electromgnetic emission电磁干扰 electromgnetic interference (EMI) 电磁感应 electromgnetic induction电磁环境 electromgnetic environment 电磁兼容性 electromgnetic comptibility (EMC) 电磁兼容性电平 electromgnetic comptibility level 电磁兼容性余量 electromgnetic comptibility mrgin 电磁脉冲 electromgnetic pulse (EMP) 电磁脉冲干扰 electromgnetic pulse jmming 电磁敏感度electromgnetic susceptibility 电磁能 electromgnetic energy 电磁耦合 electromgnetic coupling电磁屏蔽 electromgnetic shielding电磁屏蔽装置 electromgnetic screen电磁骚扰 electromgnetic disturbnce 电磁噪声 electromgnetic noise 电磁污染 electromgnetic pollution电动势 electromotive force (e. m. f.) 电话机 telephone set电话局容量 cpcity of telephone exchnge 电话型电路 telephone-type circuit电话型信道 telephone-type chnnel电离层 ionosphere电离层波 ionosphere wve电离层传播 ionosphere propgtion电离层反射 ionosphere reflection电离层反射传播 ionosphere reflection propgtion 电离层散射传播ionosphere sctter propgtion 电离层折射 ionosphere refrction 电离层吸收 ionosphere bsorption电离层骚扰 ionosphere disturbnce电流探头 current probe电路交换 circuit switching电屏蔽 electric shielding电视电话 video-telephone, viewphone, visul telephone电台磁方位 mgnetic bering of sttion 电台方位 bering of sttion电台航向 heding of sttion电文编号 messge numbering电文队列 messge queue电文格式 messge formt电文交换 messge switching电文交换网络 messge switching network 电文结束代码 end-of-messge code电文路由选择 messge routing电小天线 electroniclly smll ntenn 电信管理网络 telecommuniction mngement network (TMN)电信会议 teleconferencing电压变化 voltge chnge电压变化持续时间 durtion of voltge chnge 电压变化的发生率 rte of occurrence of voltge chnges 电压变化时间间隔 voltge chnge intervl 电压波动 voltge fluctution电压波动波形 voltge fluctution wveform 电压波动量 mgnitude of voltge fluctution 电压不平衡 voltge imblnce, voltge unblnce 电压浪涌 voltge surge电压骤降 voltge dip电源 power supply电源电压调整率 line regultion电源抗扰性 mins immunity电源持续工作能力 continuous opertion bility of the power supply 电源去耦系数 mins decoupling fctor电源骚扰 mins disturbnce电子干扰 electronic jmming电子工业协会 Electronic Industries ssocition (EI) 电子系统工程electronic system engineering 电子自动调谐 electronic utomtic tuning 电子组装 electronic pckging电阻温度计 resistnce thermometer跌落试验 fll down test顶部加载垂直天线 top-loded verticl ntenn 定长编码 block code定期频率预报 periodicl frequency forecst 定时 clocking定时超前 timing dvnce定时电路 timing circuit定时恢复(定时抽取) timing recovery (timing extrtion)定时截尾试验 fixed time test定时信号 timing signl定数截尾试验 fixed filure number test 定向天线 directionl ntenn 定型试验 type test动态频率分配 dynmic frequency lloction 动态信道分配 dynmic chnnel lloction 动态重组 dynmic regrouping动态自动增益控制特性 dynmic GC chrcteristic 抖动 jitter独立边带 independent sidebnd独立故障 independent fult端到端业务 teleservice短波传播 short wve propgtion短波通信 short wve communiction短路保护 short-circuit protection 短期抖动指示器 short-term flicker indictor 短期频率稳定度 short-term frequency stbility 短时间中断(供电电压) short interruption (of supply voltge)段终端 section termintion对称二元码 symmetricl binry code对地静止卫星 geosttionry stellite对地静止卫星轨道 geosttionry stellite orbit 对地同步卫星geosynchronous stellite 对讲电话机 intercommunicting telephone set 对空台 eronuticl sttion对流层 troposphere对流层波道 troposphere duct对流层传播 troposphere propgtion对流层散射传播 troposphere sctter propgtion 多次调制 multiple modultion多点接入 multipoint ccess多电平正交调幅 multi-level qudrture mplitude modultion (QM)多分转站网 multidrop network多服务器队列 multiserver queue多工 multiplexing多工器 nultiplexer多功能系统 MRS多级处理 multilevel processing多级互连网络 multistge interconnecting network 多级卫星线路 multi-stellite link多径 multipth多径传播 multipth propgtion多径传播函数 nultipth propgtion function 多径分集 multipth diversity 多径时延 multipth dely多径衰落 multipth fding多径效应 multipth effect多路复接 multiplexing多路接入 multiple ccess多路信道 multiplexor chnnel多脉冲线性预测编码 multi-pulse LPC (MPLC) 多频信令 multifrequency signlling多普勒频移 Doppler shift多跳路径 multihop pth多信道选取 multichnnel ccess (MC)多信道自动拨号移动通信系统multiple-chnnel mobile communiction system with utomtic diling多优先级 multiple priority levels多帧 multifrme多址呼叫 multiddress cll多址联接 multiple ccess多重时帧 multiple timefrme多用户信道 multi-user chnnelE额定带宽 rted bndwidth额定射频输出功率 rted rdio frequency output power 额定使用范围 rted operting rnge额定音频输出功率 rted udio-frequency output power 额定值 rted vlue 爱尔兰 erlng恶意呼叫识别 mlicious cll identifiction (MCI) 耳机(受话器) erphone 耳机额定阻抗 rted impednce of erphone 二十进制码 binry-coded deciml (BCD) code 二十进制转换 binry-to-deciml conversion 二十六进制转换 binry-to-hexdeciml conversion 二进制码 binry code二进制频移键控 binry frequency shift keying (BFSK) 二进制数 binry figure二频制位 binry digit(bit)二频制 two-frequency system二维奇偶验码 horizontl nd verticl prity check code 二线制 two-wire system二相差分相移键控 binry different phse shift keying (BDPSK)二相相移键控 binry phse shift keying (BPSK)F发报机 telegrph trnsmitter发射 emisssion发射(或信号)带宽 bndwidth of n emission (or signl) 发射机 trnsmitter 发射机边带频谱 trnsmitter sidebnd spectrum 发射机额定输出功率 rted output power of trnsmitter 发射机合路器 trnsmitter combiner 发射机冷却系统 cooling system of trnsmitter 发射机启动时间trnsmitter ttck time发射机效率 trnsmitter frequency发射机杂散躁声 spurious trnsmitter noise 发射机之间的互调 iner-trnsmitter intermodultion 发射机对答允许频(相)偏trnsmitter mximum permissible frequency(phse) devition发射类别 clss of emission发射频段 trnsmit frequency bnd发射余量 emission mrgin发送 sending发送响度评定值 send loudness rting (SLR) 繁忙排队/自动回叫 busy queuing/ cllbck 反馈控制系统 feedbck control system 反射功率 reflection power反射卫星 reflection stellite反向话音通道 reverse voice chnnel (RVC) 反向控制信道 reverse control chnnel (RECC) 泛欧数字无绳电话系统 digitl Europen cordless telephone 方舱 shelter方向性系数 directivity of n ntenn防爆电话机 explosion-proof telephone set 防潮 moisture protection 防腐蚀 corrosion protection防霉 mould proof仿真头 rtificil hed仿真耳 rtificil er仿真嘴 rtificil mouth仿真天线 dummy ntenn放大器 mplifier放大器线性动态范围 liner dynmic rnge of mplifier 放电 dischrge放电电压 dischrge voltge放电深度 depth of dischrge放电率 dischrge rte放电特性曲线 dischrge chrcter curve 非等步的 nisochronous非归零码 nonreturn to zero code (NRZ) 非均匀编码 nonuniform encoding 非均匀量化 nonuniform quntizing非连续干扰discontinuous disturbnce “非”门 NOT gte非强占优先规则 non-preemptive priority queuing discipline非受控滑动 uncontrolled slip非线性电路 nonliner circuit非线性失真 nonlier distortion非线性数字调制 nonliner digitl modultion 非占空呼叫建立 off-ir-cll-set-up (OCSU) 非专用控制信道 non-dedicted control chnnel 非阻塞互连网络non-blocking interconnection network分贝 decibel (dB)分辨力 resolution分布参数网络 distributed prmeter network 分布式功能 distributed function分布式数据库 distributed dtbse分别于是微波通信系统 distributed microwve communiction system分布式移动通信系统 distributed mobile communiction system分布路线 distribution link分段加载天线 sectionl loded ntenn 分机 extension分集 diversity分集改善系数 diversity improvement fctor 分集间隔 diversity seprtion 分集增益 diversity gin分集接收 diversity reception分接器 demultiplexer分频 frequency division分散定位 distributed chnn。
第４４卷 第１期系统工程与电子技术Ｖｏｌ．４４ Ｎｏ．１２０２２年１月ＳｙｓｔｅｍｓＥｎｇｉｎｅｅｒｉｎｇａｎｄＥｌｅｃｔｒｏｎｉｃｓＪａｎｕａｒｙ ２０２２文章编号：１００１ ５０６Ｘ（２０２２）０１ ０２７９ ０６ 网址：ｗｗｗ．ｓｙｓ ｅｌｅ．ｃｏｍ收稿日期：２０２１０１２９；修回日期：２０２１０５１６；网络优先出版日期：２０２１０７１２。
网络优先出版地址：ｈｔｔｐｓ：∥ｋｎｓ．ｃｎｋｉ．ｎｅｔ／ｋｃｍｓ／ｄｅｔａｉｌ／１１．２４２２．ＴＮ．２０２１０７１２．１６２０．０１８．ｈｔｍｌ基金项目：国家自然科学基金（６１８６７００５，６１８０６２０９，６１７７３３８７）资助课题 通讯作者．引用格式：罗哲，权婉珍，张朴睿，等．单边Ｌｉｐｓｃｈｉｔｚ非线性多智能体系统一致性追踪控制［Ｊ］．系统工程与电子技术，２０２２，４４（１）：２７９ ２８４．犚犲犳犲狉犲狀犮犲犳狅狉犿犪狋：ＬＵＯＺ，ＱＵＡＮＷＺ，ＺＨＡＮＧＰＲ，ｅｔａｌ．Ｃｏｎｓｅｎｓｕｓｔｒａｃｋｉｎｇｃｏｎｔｒｏｌｆｏｒｏｎｅ ｓｉｄｅＬｉｐｓｃｈｉｔｚｎｏｎｌｉｎｅａｒｍｕｌｔｉ ａｇｅｎｔｓｙｓｔｅｍｓ［Ｊ］．ＳｙｓｔｅｍｓＥｎｇｉｎｅｅｒｉｎｇａｎｄＥｌｅｃｔｒｏｎｉｃｓ，２０２２，４４（１）：２７９ ２８４．单边犔犻狆狊犮犺犻狋狕非线性多智能体系统一致性追踪控制罗 哲１，权婉珍１， ，张朴睿２，杨小冈１（１．火箭军工程大学导弹工程学院，陕西西安７１００２５；２．国防科技大学航天科学与工程学院，湖南长沙４１００７３） 摘 要：针对单边Ｌｉｐｓｃｈｉｔｚ非线性多智能体系统，提出了一种分布式一致性控制方法。
首先，构建了领导跟随者动力学结构，用于实现单边Ｌｉｐｓｃｈｉｔｚ多智能体系统的追踪控制。
然后，设计了单边Ｌｉｐｓｃｈｉｔｚ非线性多智能体系统的一致性控制协议，可根据智能体之间局部交互信息构建分布式反馈控制，并将系统的一致性追踪问题转化为系统的稳定性问题。
Nonlinear Dyn(2015)80:845–854DOI10.1007/s11071-015-1910-yORIGINAL PAPERMulti-switching combination synchronization of chaotic systemsU.E.Vincent·A.O.Saseyi·P.V.E.McClintockReceived:11August2014/Accepted:6January2015/Published online:20January2015©Springer Science+Business Media Dordrecht2015Abstract A novel synchronization scheme is pro-posed for a class of chaotic systems,extending the concept of multi-switching synchronization to com-bination synchronization such that the state variables of two or more driving systems synchronize with dif-ferent state variables of the response system,simulta-neously.The new scheme,multi-switching combina-tion synchronization(MSCS),represents a significant extension of earlier multi-switching schemes in which two chaotic systems,in a driver-response configuration, are multi-switched to synchronize up to a scaling fac-tor.In MSCS,the chaotic driving systems multi-switch a response chaotic system in combination synchroniza-tion.For certain choices of the scaling factors,MSCS reduces to multi-switching synchronization,implying that the latter is a special case of MSCS.A theoretical approach to control design,based on backstepping,is presented and validated using numerical simulations. Keywords Multi-switching·Combination Synchro-nization·Chaos·BacksteppingU.E.Vincent·A.O.SaseyiDepartment of Physical Sciences,Redeemer’s University, Redemption City,NigeriaU.E.Vincent(B)·P.V.E.McClintockDepartment of Physics,Lancaster University,Lancaster LA14YB,UKe-mail:u.vincent@ 1IntroductionThe possibility of realizing synchronization in coupled or forced chaotic systems came as a major breakthrough in nonlinear science.Until1990,it had appeared impos-sible,due to the well-known divergence of trajecto-ries caused by the sensitivity of chaotic systems to ini-tial conditions,but the pioneering work of Pecora and Carroll[1]showed that this negative expectation was wrong.Since then,the synchronization of chaotic sys-tems has attracted much attention.In addition to its own intrinsic interest,and its rich variety of intriguing fea-tures,chaos synchronization has acquired a wide range of important interdisciplinary applications,including time series analysis,secure communication systems, modeling cardiac rhythm and brain activity,and earth-quake dynamics[2–6].These have provided the moti-vation and driving force for the huge effort currently being devoted to ways of achieving chaos synchroniza-tion.In general,for coupled or interacting chaotic sys-tems with state space variables x1(t)and x2(t),a complete/identical synchronization manifold x1(t)= x2(t)exists if the condition,lim t→∞ x1(t)−x2(t) = 0,∀t≥0,is satisfied[1].However,as proposed by Mainieri and Rehacek[7],two chaotic systems can synchronize up to a scaling factor,φsuch that lim t→∞ x1(t)−φx2(t) =0,∀t≥0.In this case,they achieve projective synchronization(PS). Projective synchronization has been investigated with increasing interest in recent years due to the possibil-846U.E.Vincent et al.ity of achieving faster communication with its scaling feature,and a wide variety of PS schemes have been proposed(See Refs.[8–19]and references therein);all of them are concerned with a single driving—single-response configuration.Very recently,however,a novel form of projective synchronization was proposed by Luo et al.[20],in which three classic chaotic systems were made to syn-chronize simultaneously via systematically designed nonlinear controls;two of which were driving a single-response system,in a kind of double-driving/single-response arrangement.The implication of combination synchronization proposed in[20]for communication is such that a signal can be split into two,each loaded and transmitted between two drive systems or at dif-ferent intervals.Further developments in this direction are reported in Refs.[21–27];in particular,compound synchronization[28,29],double compound synchro-nization[30],combination-combination synchroniza-tion[23,24,26],finite-time combination–combination synchronization[24–27],finite-time stochastic combi-nation synchronization[22],hybrid and reduced-order hybrid combination synchronization[31,32]have been proposed and investigated.It is noteworthy that in all these previous works,the goals were to achieve syn-chronization between state variables of the driver sys-tems and that of identical response system.This con-strained the coupled systems to evolve in predeter-mined and predictable directions simultaneously.In order to improve the security of information trans-mission via synchronization,it may be required that in master–slave synchronization,different states of the slave system are synchronized with desired state of the master system in a multi-switching manner.This form of synchronization was proposed by Ucar et al.[33] based on active control formalism.Despite its clear relevance to information security,only a few studies of this kind of synchronization have been reported[34–40].To the best of our knowledge,all the work on multi-switching synchronization of this kind reported in the literature have related to single-driver/single-response systems.In this paper,we propose a multi-switching com-bination synchronization scheme,wherein two driver chaotic systems are multi-switched in diverse ways with a single-response chaotic system.The possibil-ity of realizing such a form of synchronization would present varieties of synchronization directions between the driver systems’and response system’s variables,thereby ensuring better security when employed in communication applications.The rest of the paper is organized as follows:In Sect.2,the definitions and basic formulation of multi-switching combination synchronization(MSCS)are presented.In Sect.3,an example of the MSCS of three classic chaotic systems is formulated.The cor-responding numerical simulation results are presented in Sect.4.The paper is summarized and concluded in Sect.5.2Definition and formulation of MSCSConsider the following master–slave n-dimensional chaotic systems,where the master systems are given by˙x1m=f1x(x1m,...),˙x2m=f2x(x1m,...),...,˙x nm=f nx(x1m,...)(1) and˙y1m=f1y(y1m,...),˙y2m=f2y(y1m,...),...,˙y nm=f ny(y1m,...)(2) and the controlled slave system is given by˙z1s=g1z(z1s,...)+U1,˙z2s=g2z(z1s,...)+U2,...,˙z ns=g nz(z1s,...)+U n;(3) where x jm,y km,z is(i,j,k=1,2,...n)∈R n arestate space vectors of the systems,f j x,f ky,g iz: R n→R n are three continuous vector functions com-posed of linear and nonlinear components,and U i(i= 1,2,...,n):R n→R n is a nonlinear control func-tion.The indices m and s stand for master and slave systems,respectively.Definition1[20]If there exists three constant matri-ces A,B,C∈R n and C=0,such thatlimt→∞||Cz is−Ax jm−By km||=0,where||.||is the matrix norm and A,B,C are scaling matrices,then,systems(1),(2)and(3)are said to be in combination synchronization.Comment1The error states in relation to the def-inition1are strictly chosen to satisfy the definition, e i jk(i=j=k),where i,j and k are the indices of the error.Multi-switching Combination synchronization847Definition2If the error states in relation to Definition 1are redefined such that i=j=k or i=k=j,or j=k=i or i=j=k,i=j=k or j=i=k and limt→∞||Cz is−Ax jm−By km||=0,then systems(1),(2)and(3)are said to be in multi-switching combination synchronization.Comment2We refer to the conditions,i=j=k or i=k=j,j=k=i or i=j=k,as generic con-ditions that must be met.In addition,there are several non-generic cases where the above possible generic cases are combined in a mixed mode.To formulate the active backstepping procedure,we define a typical multi-switching synchronization error for a3-dimensional system,ase123=γ1z1s−α2x2m−β3y3m,e231=γ2z2s−α3x3m−β1y1m,(4) e312=γ3z3s−α1x1m−β2y2m,and obtain the following error dynamical system˙e123=γ1g1z+γ1U1−α2f2x−β3f3y,˙e231=γ2g2z+γ2U2−α3f3x−β1f1y,(5)˙e312=γ3g3z+γ3U3−α1f1x−β2f2y,whereαj,γi,βk(i,j,k=1,2,3)are scaling fac-tors.In principle and by some algebraic manipulations, the error dynamical system(5)can be expressed in terms of the synchronization error,e123,e231and e312, because g iz,f j x,and f ky consist of linear and nonlin-ear parts.Thus,the synchronization problem reduces to that of asymptotic stabilization of Eq.(5)with appro-priate control inputs.Here,we use the active back-stepping technique,because it provides a systematic design approach for both control and synchronization and guarantees global stability of the closed loop sys-tem.The main feature of this approach is that it allows forflexibility in the construction of control laws,so that the control strategy can be extended very eas-ily to higher-dimensional systems.We now provide a description of a simple design procedure for active backstepping-based multi-switching combination syn-chronization.Letν1=e123,so that we obtain theν1-subsystem ˙ν1=F1(ν1,f xyz,U1);(6) where f xyz is a nonlinear function derived from g iz(z is),f j x(x jm),and f ky(y km).Considering the error variable e231as a virtual control input via a classical Lyapunov function V1,if V1satisfies the conditionsV1(e231)>0if e231=0,V1(e231)=0if e231=0,(7) and˙V1(e231)<0if e231=0,˙V1(e231)=0if e231=0,(8)then theν1-subsystem of Eq.(6)is asymptotically sta-ble.When e231has been designed,we can obtain the following(ν1,ν2)-subsystem˙ν1=F1(ν1,f xyz,U1),˙ν2=F2(ν1,ν2,e123,g xyz,U2),(9)by settingν2=e231−α1(ν1),whereα1(ν1)is a virtual control and g xyz is a nonlinear function derived from g iz(z is),f j x(x jm)and f ky(y km).In other to stabilize the(ν1,ν2)-subsystem,a sec-ond positive Lyapunov function V2is chosen.The process continues by consideration of the variable e312=α2(ν1,ν2)as the virtual control input for the (ν1,ν2)-subsystem,and so on.Then,if˙V2is nega-tive definite,we can conclude that(ν1,ν2)-subsystem is asymptotically stable.Finally,the full(ν1,ν2,ν3)-system can be constructed and stabilized in the same manner.3MSCS of three chaotic systemsIn order to generalize the concept,we examine multi-switching combination synchronization of three strictly different chaotic systems,namely,Rössler,Newton-Leipnik,and Lorenz systems.The Rössler and Newton-Leipnik systems provide the driving and are repre-sented by the state variables x,y as˙x1=−x2−x3,˙x2=x1+a1x2,(10)˙x3=b1+x3(x2−c1),and˙y1=−a2y1+y2+10y2y3,˙y2=−y1−0.4y2+5y1y3,(11)˙y3=b2y3−5y1y2;848U.E.Vincent et al.while the response of the Lorenz system represented by the state variable z is given by˙z1=a3(z2−z1)+U1,˙z2=b3z1−z2−z1z3+U2,(12)˙z3=z1z2−c3z3+U3,where U1,U2and U3are controllers to be designed.There are several possible generic switching combi-nations that could exist for the drive-response system (10),(11)and(12),some of which are given below.For i=j=k,we have:e112,e221,e331and e113,e223,e332.For i=k=j,we have:e121,e212,e313and e131,e232,e323.For j=k=i,we have:e122,e211,e311and e133,e233,e322.For i=j=k,we have:e123,e213,e312and e132,e231,e321.For i=j=k,we have e122,e233,e311and e133,e211,e322.For j=i=k,we have e121,e212,e313.In this paper,we present results for some particular switching combinations,randomly selected from the combinations given above.They are:e112=γ1z1−α1x1−β2y2,e213=γ2z2−α1x1−β3y3,Switch1(13) e311=γ3z3−α1x1−β1y1;e123=γ1z1−α2x2−β3y3,e213=γ2x2−α1x1−β3y3,Switch2(14) e323=γ3z3−α2x2−β3y3;e112=γ1z1−α1x1−β2y2,e223=γ2z2−α2x2−β3y3,Switch3(15) e321=γ3z3−α2x2−β1y1;e113=γ1z1−α1x1−β3y3,e221=γ2z2−α2x2−β1y1,Switch4(16) e312=γ3z3−α1x1−β2y2.The notations for the scaling factorsαi,γj,βk(i,j,k= 1,2,3)are set for convenient and may assume differ-ent or same values in applications.For simplicity and reference,we refer to the error dynamics Eq.(13)as Switch1,Eq.(14)as Switch2,Eq.(15)as Switch3, and Eq.(16)as Switch4.3.1Switch1For Switch1Eq.(13),the time derivative of the errors is given by˙e112=γ1˙z1−α1˙x1−β2˙y2,˙e213=γ2˙z2−α1˙x1−β3˙y3,(17)˙e311=γ3˙z3−α1˙x1−β1˙y1.Substituting for˙z1,˙x1,˙y2,˙z2,˙y3,˙z3and˙y1from Eqs.(10),(11)and(12),the error dynamical system for Switch1can be written as:˙e112=γ1a3γ2e213−a3e112+f+γ1U1˙e213=γ2b3γ1e112−γ2γ1γ3e112e311−γ2γ1γ3[α1x1+β1y1]e112−e213−γ2γ1γ3[α1x1+β2y2]e311+g+γ2U2(18)˙e311=γ3γ1γ2[α1x1+β3y3]e112+γ3γ1γ2e112e213+γ3γ1γ2[α1x1+β2y2]e213−c3e311+h+γ3U3 where,f=γ1a32−a3α1x1+γ1a3β32y3−a3β2y2+α1x2 +α1x3+β2y1+0.4β2y2−5β2y1y3g=γ2b3γ1(α1x1+β2y2)+α1(x2+x3)−β3[b2y3−5y1y2]−[α1x1+β3y3]−γ2γ1γ3[(α1x1+β2y2)(α1x1+β1y1)] h=γ3γ1γ2(α1x1+β2y2)(α1x1+β3y3)+α1(x2+x3)−c3(α1x1+β1y1)+β1(+a2y1−y2−10y2y3)] Theorem1If the control functions U1,U2and U3are chosen such thatU1=−1γ1f,U2=−1γ2γ2b3γ1−γ2γ1γ3(α1x1+β1y1)ν1+g,U3=−1γ3γ3γ1γ2(α1x1+β3y3)ν1+γ3γ1γ2(α1x1+β2y2)ν2+γ3γ1γ2ν1ν2+h,(19)whereν1=e112,ν2=e213,then the drive systems (10)and(11)will achieve multi-switching combination synchronization with the response system(12).Proof1We use the active backstepping technique to prove the above theorem.Multi-switching Combination synchronization 849Let ν1=e 112;its derivative is given by˙ν1=˙e 112=γ1a 32e 213−a 3e 112(20)where e 213≡α1(ν1)can be regarded as a virtual con-troller.For the design of α1(ν1)to stabilize the ν1-subsystem defined by Equ.(20),we consider the fol-lowing Lyapunov function V 1=12ν21.(21)Its time derivative is given by ˙V1=ν1˙ν1=ν1γ1a 32α1(ν1)−a 3ν1.(22)The virtual control α1(ν1)is an estimated control input,and it can take any convenient values that yields the desired control function.In practice,and for the pur-pose of applications,α1(ν1)should be chosen such that the overall controller complexity is reduced as much aspossible.Suppose α1(ν1)≡0,then ˙V 1=−a 3ν21≤0is negative definite,and according to Lyapunov stabil-ity theorem,the ν1-subsystem is asymptotically stable.If we denote the error between e 213and α1(ν1)by ν2,i.e.,ν2=e 213−α1(ν1),then we have the (ν1,ν2)-subsystem˙ν1=γ1a 3γ2ν2−a 3ν1,˙ν2=γ2b 3γ1−γ2γ1γ3[α1x 1+β1y 1] ν1−ν2,−γ2γ1γ3ν1e 311−γ2γ1γ3[α1x 1+β2y 2]e 311+g +γ2U 2.(23)To stabilize the (ν1,ν2)-subsystem,e 311≡α2(ν1,ν2)can be regarded as a virtual controller.Assuming the Lyapunov functionV 2=V 1+12ν22,and its time derivative given by˙V 2=−a 3ν21−ν22+ν2 γ2b 3γ1−γ2γ1γ3[α1x 1+β1y 1] ν1−γ213(ν1+α1x 1+β2y 2)α2(ν1,ν2)+g +γ2U 2 .(24)Similarly,if we let α2(ν1,ν2)=0and the controlfunction U 2is chosen as in Theorem 1,then ˙V2=−a 3ν21−ν22<0,implying that the (ν1,ν2)-subsys-tem is asymptotically stable.Finally,suppose ν3≡e 311−α2(ν1,ν2).We then obtain˙ν3=γ3γ1γ2[α1x 1+β3y 3]ν1+γ3γ1γ2ν1ν2+γ3γ1γ2[α1x 1+β2y 2]ν2−c 3ν3+h +γ3U 3.(25)This allows us to stabilize the full dimensional system(ν1,ν2,ν3),by taking the Lyapunov function as V 3=V 2+12ν23.(26)Using Theorem 1,its time derivative is given by˙V 3=−a 3ν21−ν22−c 3ν23<0.(27)Since ˙V 3=−a 3ν21−ν22−c 3ν23<0,we can con-clude,based on the Lyapunov stability theorem,that the equilibrium (0,0,0)of the full dimensional system (ν1,ν2,ν3)given by˙ν1=γ1a 3γ2ν2−a 3ν1˙ν2=−ν2−γ2γ1γ3ν1ν3−γ2γ1γ3[α1x 1+β2y 2]ν3(28)˙ν3=−c 3ν3is asymptotically stable and that global multi-switching combination synchronization has been achieved.This ends the proof. Comment 3The following corollaries can be easily obtained from Theorem 1,but their proofs are omitted here for brevity.If we let β1=β2=β3=0,γ1=γ2=γ3=1,then we have Corollary 1.Corollary 1If the controllers are chosen asU 1=−α1x 2−α1x 3,U 2=−(b 3−α1x 1)ν1−(b 3−1)α1x 1+α21x 21+α1(x 2+x 3),(29)U 3=−(ν1+ν2−c 3)α1x 1−ν1ν2−α21x 21−α1(x 2+x 3),then the drive system (10)will achieve multi-switchingprojective synchronization with the response system (12),with α1,α2and α3being the scaling factors.850U.E.Vincent et al. If we letα1=α2=α3=0,γ1=γ2=γ3=1,then we obtain Corollary2.Corollary2If the controllers are designed asU1=−a3(β3y3−β2y2)−β2y1−0.4β2y2+5β2y1y3,U2=−(b3−β1y1)ν1−b3β2y2+β3y3+β2y2β1y1+β3(b2y3−5y1y2),(30)U3=−β3y3ν1−β2y2ν2−ν1ν2−β2y2β3y3−c3β1y1+β1(a2y1−y2−10y2y3),then the drive system(11)and the response system(12)will reach multi-switching projective synchronization,withβ1,β2andβ3being the scaling factors.Supposeα1=α2=α3=0,β1=β2=β3=0,γ1=γ2=γ3=1,then one gets Corollary3.Corollary3If the controllers are chosen asU1=0U2=−b3ν1(31)U3=−ν1ν2then the equilibrium point(0,0,0)of the response sys-tem(12)will be asymptotically stable,and stabilizationof the system(12)is achieved.3.2Switch2,Switch3,and Switch4Following the same procedure presented in Sect.3.1,and considering Switch2given by Eq.(14),we cangive the following Theorem2.Theorem2If the control functions U1,U2and U3arechosen such thatU1=−1γ1fU2=−1γ2γ2b3γ1−γ2γ1γ3(α2x2+β3y3)ν1+gU3=−1γ3γ3γ1γ2(α1x1+β3y3)ν1+γ3γ1γ2×(α2x2+β3y3)ν2+γ3γ1γ2ν1ν2+hwhereν1=e123,ν2=e213,then the drive systems (10)and(11)will achieve multi-switching combination synchronization with the response system(12). Comment4The following corollaries are easily obtained from Theorem2,but their proofs are omit-ted for brevity.If we letβ1=β2=β3=0,γ1=γ2=γ3=1,then we have Corollary4.Corollary4If the controllers are chosen asU1=(a3+a1)α2x2−a3α1x1+α2x1,U2=α22x22−(b3−α2x2)ν1+α1(−x2−x3)−b3α2x2−α1x1,(32) U3=α2(x1+a1x2)−ν1ν2−(ν2+α1x1−c3)α2x2, then,the driven system(10)will achieve multi-switching projective synchronization with the response system (12),withα1,α2andα3being the scaling factors.If we letα1=α2=α3=0,γ1=γ2=γ3=1,then we obtain Corollary5.Corollary5If the controllers are designed asU1=−(a3−b2)β3y3−5β3y1y2,(33) U2=−(b3−β3y3)ν1−(b3−1−β3y3)β3y3+β3(b2y3−5y1y2),(34) U3=−[ν1ν2+(ν2+ν1−c3)β3y3+β23y23−β3(b2y3−5y1y2)],then the drive system(10)and the response system(12) will reach multi-switching projective synchronization, withβ1,β2andβ3being the scaling factor. Supposeα1=α2=α3=0,β1=β2=β3=0,γ1=γ2=γ3=1,then one gets Corollary6. Corollary6If the controllers are chosen asU1=0,U2=−b3ν1,(35) U3=−ν1ν2,then the equilibrium point(0,0,0)of response system (12)will be asymptotically stable.Similarly for Switch3,given by Eq.(15),we give the following theorem.Theorem3If the control functions U1,U2and U3are chosen such thatU1=−1γ1fU2=−12γ2b31−γ213(α2x2+β1y1)ν1+gU3=−1γ3γ3γ1γ2(α2x2+β3y3)ν1+γ3γ1γ2(α1x1+β2y2)ν2+γ3γ1γ2ν1ν2+hMulti-switching Combination synchronization851whereν1=e112,ν2=e223,then the drive systems (10)and(11)will achieve multi-switching combination synchronization with the response system(12).Comment5The following corollaries are easily obtained from Theorem3,but their proofs are omit-ted.If we letβ1=β2=β3=0,γ1=γ2=γ3=1, then we have corollary7.Corollary7If the controllers are chosen asU1=a3α1x1−(α1+a3α2)x2−α1x3,U2=(α1+x1α2x2)−(b3−α2x2)ν1+b3α1x1 +α2(x1+a1x2)−α2x2,(36) U3=−ν1ν2−α1x1ν2−(ν1+α1x1−c3)α2x2 +α2(x1+a1x2),then the drive system(10)will achieve multi-switching projective synchronization with the response system (12),withα1,α2andα3being the scaling factors.If we letα1=α2=α3=0,γ1=γ2=γ3=1,then we obtain corollary8.Corollary8:If the controllers are chosen asU1=a3β2y2−a3β3y3−β2y1−0.4β2y2+5β2y1y3, U2=−(b3−β1y1)ν1−(b3−β1y1)β2y2+β3y3 +β3(b2y3−5y1y2),(37) U3=−ν1ν2−β2y2ν2−(ν1+β2y2)β3y3+c3β1y1−β1(a2y1−y2−10y2y3),then the drive system(11)will achieve multi-switching projective synchronization with the response system (12),withβ1,β2andβ3being the scaling factor. Supposeα1=α2=α3=0,β1=β2=β3=0,γ1=γ2=γ3=1,then one gets corollary9. Corollary9If the controllers are chosen asU1=0,U2=−b3ν1,(38) U3=−ν1ν2,then the equilibrium point(0,0,0)of response system (12)will be asymptotically stable.4Numerical resultsWe now present the results of numerical simulations. These were done using a fourth-order Runge-Kutta method with variable time-step.As stated earlier,the main interest is to achieve multi-switching combina-tion synchronization of the Rössler system,Newton-Leipnik system,and Lorenz system.The systems’para-meters were chosen as a1=0.2,b1=0.2,c1=5.7, a2=0.4,b2=0.175,a3=10.0,b3=23.0and c3=8.0/3,in order to ensure the existence of chaotic attrac-tors,with the initial states for the master system and for the slave system taken arbitrarily to be(x1,x2,x3)= (0.0,1.0,−2.0),(y1,y2,y3)=(0.349,0.0,−0.16)and (z1,z2,z3)=(1.0,0.40,0.80),respectively.Although the system parameters may be chosen so that the sys-tems are non-chaotic,we emphasize that the chaotic synchronized state is a special case:It is of tremendous importance in thefield of secure communications,and the achievement of a synchronization state is indepen-dent of the choice of initial conditions or of the parame-ters of the system since the control inputs are dependent on the system parameters.Thus,any set of initial condi-tions leading to either a chaotic or a periodic orbit would give synchronization.For the control parameters,we assume that the master systems scaling factorsα1=α2 =α3=1,β1=β2=β3=−2.Note thatγi is the scal-ing factor for the slave system,and that its value should be set to unity to ensure that only the master systems are projected onto the slave[7].Thus,γ1=γ2=γ3=1, whileαj andβk may take on some convenient val-ues according to the required projections,ensuring bounded solutions.Boundedness of the chaotic attrac-tors should be preserved to ensure that the orbits stay within the basins of attraction.From our numerical experiments using different values ofαj andβk,we found that,in general,a regime ofαj andβk exists for which the solution is bounded and synchronization is reachable.Typically,this lies approximately in the control parameter range−8<αj,βk<8;otherwise, the solutions are unbounded,because the orbits tends to exit the basins of attraction.It is interesting to note that this regime encloses a variety of possible synchro-nization phenomena.For instance,whenαj,βk<0, we have projective multi-switching combination anti-synchronization,αj,βk=1yields complete multi-switching combination synchronization,and when0<αj,βk>1,projective multi-switching combination anti-synchronization is achieved.852U.E.Vincent et al.-20-10 0 10 2030 40 012345e i j ktimee 112e 213e 311Fig.1Synchronization error for switch 1with error states e 112,e 213,and e 311.Control was activated at time,t ≥1.The parame-ters of the systems were chosen as a 1=0.2,b 1=0.2,c 1=5.7,a 2=0.4,b 2=0.175,a 3=10.0,b 3=23.0and c 3=8.0/3For switch 1,the control inputs given in Theorem 1were programmed to turn on at time,t ≥1,simultane-ously.The result is shown in Fig.1,where we can see that multi-switching combination synchronization has clearly been achieved.Similar results were obtained using corollaries 1–3.In addition,Fig.2illustrates the temporal behavior of the synchronizing variables z 1,x 1+y 2,z 2,x 1+y 3,and z 3,x 1+y 1in the multi-switching compound synchronization state,with simul-taneous activation of the controls at t ≥25.In Fig.3,we illustrate the stabilization of the response system 12to the equilibrium point (0,0,0)on the application of Corollary 3.For the other cases,namely switches 2,3and 4,the control functions U i (i =1,2,3)were programmed to turn on at different times,namely:t ≥1.5,for switch 2;t ≥2,for switch 3and t ≥1,for switch 4,respectively.The numerical results corresponding to these cases are shown in Fig.4.In all these cases,synchronization have been achieved.5ConclusionsIn summary,we have introduced,analyzed,and val-idated a novel form of chaotic synchronization that can involve three or more dynamical systems,namely multi-switching combination synchronization (MSCS)of three chaotic systems,based on the backstep-ping nonlinear control approach.In this new syn--20-10 0 10 2030 40 0 10 20 30 40 50z 1,x 1+y 2time(a)z 1x 1+y 2-20-101020300 10 20 30 40 50z 2,x 1+y 3time(b)z 2x 1+y 3-20-10 0 10 2030 40 0 10 2030 40 50z 3,x 1+y 1time(c)z 3x 1+y 1Fig.2Temporal behavior of the synchronizing variables (a )z 1,(x 1+y 2),(b )z 2,(x 1+y 3),and (c )z 3,(x 1+y 1)in the multi-switching compound synchronization state,with simultaneous activation of the controls at t ≥25.The parameters of the systems were chosen as a 1=0.2,b 1=0.2,c 1=5.7,a 2=0.4,b 2=0.175,a 3=10.0,b 3=23.0and c 3=8.0/3chronization scheme,the state space variables of the three systems are multi-switched in different ways,Multi-switching Combination synchronization853-20-10 0 10 2030 40 010 20 30 4050z i (i =1,2,3)timez 1z 2z 3Fig.3Stabilization of state variables z 1,z 2,z 2of the Lorenz system,when the controllers in Corollary 3were activated at time,t ≥25-20-10 0 10 2030 40 01 2 3 4 5e i j ktimee 113e 221e 312e 223e 321e 112e 323e 213e 123Fig.4Multi-switching synchronization error for switches 2,3and 4.The controllers were activated at different times,switch 2at time t ≥1.5,Switch 3at t ≥2and Switch 4at t ≥1.The error states are e 112,e 223,e 321,e 123,e 213,e 323,e 113,e 221,and e 312such that their mutual synchronization takes place between different state variables.When synchroniza-tion is achieved in this manner in the communications context,it would be difficult or even impossible for an intruder to predetermine the vector space in which synchronization would occur,thereby enhancing infor-mation security.Numerical simulations using there the Lorenz ,Newton-Leipnik and Rössler systems have verified the theories presented.This synchronization scheme is applicable to all chaotic systems,including those of higher order that exhibits hyperchaotic behav-ior.Higher-dimensional systems are attractive in this context because they present more switching options for constructing the error space vector due to the larger number of variables that are then available for this pur-pose.Finally,the present results pave way for new directions in the study of various kinds of chaotic syn-chronization.For instance,considering the uncertain-ties in system parameters,the possibilities of function scaling factors,parameter uncertainty and the effect of noise on such synchronization schemes would be inter-esting directions for future work.Acknowledgments UEV is supported by the Royal Society of London through their Newton International Fellowship Alumni scheme.We acknowledge and thank all the reviewers for their constructive and critical comments that were very useful for improving the quality of this paper.References1.Pecora,L.M.,Carroll,T.L.:Synchronization in chaotic sys-tems.Phys.Rev.Lett.64,821–824(1990)2.Pikovsky,A.,Rosenblum,M.,Kurths,J.:Synchronization:A Universal Concept in Non-Linear Sciences.Cambridge University Press,United Kingdom (2001)3.Eisencraft,M.,Fanganiello,R.D.,Grzybowski,J.M.V .,Sori-ano,D.C.,Attux,R.,Batista,A.M.,Macau,E.E.N.,Mon-teiro,L.H.A.,Romano,J.M.T.,Suyama,R.,Yoneyama,T.:Chaos-based 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