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NonlinearDyn(2014)78:2517–2531DOI10.1007/s11071-014-1607-7ORIGINALPAPERLocalbifurcationanalysisandultimateboundofanovel4Dhyper-chaoticsystemJiezhiWang·QingZhang·ZengqiangChen·HangLiReceived:19April2014/Accepted:15July2014/Publishedonline:6August2014©SpringerScience+BusinessMediaDordrecht2014AbstractThispaperpresentsanewfour-dimensionalsmoothquadraticautonomoushyper-chaoticsystemwhichcangeneratenoveltwodouble-wingperiodic,quasi-periodicandhyper-chaoticattractors.TheLya-punovexponentspectrum,bifurcationdiagramandphaseportraitareprovided.Itisshownthatthissys-temhasawidehyper-chaoticparameter.ThepitchforkbifurcationandHopfbifurcationarediscussedusingthecentermanifoldtheory.Theellipsoidalultimateboundofthetypicalhyper-chaoticattractorisobserved.Numericalsimulationsaregiventodemonstratetheevolutionofthetwobifurcationsandshowtheultimateboundaryregion.KeywordsHyper-chaos·Pitchforkbifurcation·Hopfbifurcation·UltimateboundJ.Wang(B)·Q.Zhang·Z.ChenCollegeofScience,CivilAviationUniversityofChina,Tianjin300300,People’sRepublicofChinae-mail:wjzh7845@gmail.comZ.ChenDepartmentofAutomation,NankaiUniversity,Tianjin300071,People’sRepublicofChinae-mail:chenzq@nankai.edu.cnH.LiEconomicsandManagementCollege,CivilAviationUniversityofChina,Tianjin300300,People’sRepublicofChina1IntroductionThefirstfour-dimensional(4D)autonomoushyper-chaoticattractorwasreportedbyRösslerin[1].Hyper-chaoticattractorhasmorethanonepositiveLya-punovexponent(LE),whichmeansthedynamicsofthehyper-chaoticattractorexpendsinmorethanonedirection.So,inengineering,hyper-chaoticattrac-torhasagreatpotentialofapplication[2,3].Con-structingnewhyper-chaoticsystems,especiallythe4Dsmoothquadraticautonomoushyper-chaoticsystems,hasbecomeahottopic.Butithasnotaunifiedandeffectivemethodtogeneratehyper-chaoticsystems.Bynow,theusualtechniquetogetanew4Dsmoothquadraticautonomoushyper-chaoticsystem[4–10]istoaddonemorestatevariabletothethreedimensionalchaoticsystem,justastheLorenzsystem[11],Chensystem[12],Lüsystem[13],Qisystem[14]andsoon.Therealsosomehyper-chaoticsystemisconstructeddirectly[15–18],butthisisdifficult.Thehighdimensionalchaoticsystemsusuallyhascomplexnonlineardynamics.Thespecificstudyofthelocalstaticbifurcationanddynamicbifurcationcanhelptoin-depthandmulti-angleanalyzeandclarifytheevolutionofthecomplicateddynamicbehaviorofanonlinearsystem.Pitchforkbifurcationofequilib-riumpositionsisanimportantlocalstaticbifurcationandHopfbifurcationisanimportantlocaldynamicbifurcation.Thebifurcationpointplaysveryimportantroleinthestudyingofthestabilityoftheequilibria[5,7,19–21].1232518J.Wangetal.Ontheotherhand,itismeaningfultofindtheulti-mateboundaryofachaoticattractor.Itisachallengeworktoestimatetheultimateboundaryregionofachaoticattractor.Oneusualwayisthroughtheopti-mizationtechniqueandLyapunovfunctiontoachievetheultimateboundaryregionsofsomeexistingchaoticsystems[18,22–26].Recently,alsoutilizingtheLya-punovfunction,stabilitytheoryandtheoptimizationtechnique,aunifiedapproachwasconstructedbyWangetal.[27]toestimatetheultimateboundaryregionsofaclassofhighdimensionalquadraticautonomousdynamicalsystem(HDQADS).Thisapproachcansolvetheultimateboundproblemofsomechaoticsys-temswithmorecomplexdynamicequation.Buttoapplythisunifiedapproach,observingtheexamplesin[27],itoftenrequiresthecoefficientofthesinglei-thstatevariableinthei-thequationofthesmoothquadraticautonomoussystemhavethesamesign.Inthispaper,anew4Dsmoothquadraticautonomoushyper-chaoticsystemisconstructeddirectly.Itcangen-erateattractorswithunusualtype,whichauthorscalledtwodouble-wingattractorhere.Thisshapeisdiffer-entfromtheshapeofthegeneraltwo-wingattractorandfour-wingattractor.Thisnew4DsystempossesseslocalstaticpitchforkbifurcationandlocaldynamicHopfbifurcation.Thesetwotypesofbifurcationpointsarebothfoundthroughrigoroustheoreticalanalysisandcomplexmathematicalcalculation.Theexhibitofthebifurcationsisclearlyshowsthetrendofthedevel-opmentofsystemdynamicbehavior.Morespecially,inthenewsystem,thecoefficientofthesinglei-thstatevariableinthei-th(i=1,2,3,4)stateequa-tionisnegative.Justliketheexamplesystemsin[27],thisnewsystempossessesthespecialconditiontousetheunifiedstandardapproach[27].Accordingto[27],theultimateboundofanewhyper-chaoticattractorisfound,whichisusefultorealizethesynchronizationbetweentwochaoticsystems.Itneedstobepointedoutthatthemethodin[26]isnotverysuitabletobeappliedtoestimatetheultimateboundofthenewsys-tem.Therestofthispaperisorganizedasfollows.Section2introducesthenewhyper-chaoticsystemandparticularlyshowssometwodouble-wingattractorswithsimulations.Section3researchesthepitchforkbifurcationandHopfbifurcationofthenewhyper-chaoticsystemusingthecentermanifoldtheoryandthebifurcationtheory.Section4estimatestheellip-soidalultimateboundofthenewtypicalhyper-chaoticattractorandprovidethenumericalsimulation.Section5drawstheconclusions.2Thenovelhyper-chaoticsystemandsomebasicpropertiesThenew4Dsmoothquadraticautonomoushyper-chaoticsystemhasthefollowingdynamicequation:⎧⎪⎪⎨⎪⎪⎩˙x=a(y−x)+w,˙y=bx−ey+xz,˙z=−cz−xy,˙w=−dw−bx−xz,(1)where(x,y,z,w)T∈R4,b
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