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1、October4,201111:56WSPC/S0218-127403003InternationalJournalofBifurcationandChaos,Vol.21,No.9(2011)2695–2712cWorldScientificPublishingCompanyDOI:10.1142/S0218127411030039HOMOCLINICANDHETEROCLINICORBITSANDBIFURCATIONSOFANEWLORENZ-TYPESYSTEMXIANYILI∗andHAIJUNWANGCollegeofMa
2、thematicsandComputationalScience,ShenzhenUniversity,Shenzhen,Guangdong518060,P.R.China∗xyli@szu.edu.cnReceivedAugust26,2010;RevisedMay27,2011Inthispaper,anewLorenz-typesystemwithchaoticattractorisformulated.Thestructureofthechaoticattractorinthisnewsystemisfoundtobecomp
3、letelydifferentfromthatintheLorenzsystemortheChensystemortheL¨usystem,etc.,whichmotivatesustofurtherstudyindetailitscomplicateddynamicalbehaviors,suchasthenumberofitsequilibrium,thestabil-ityofthehyperbolicandnonhyperbolicequilibrium,thedegeneratepitchforkbifurcation,the
4、Hopfbifurcationandthelocalmanifoldcharacter,etc.,whenitsparametersvaryintheirspace.Theexistenceornonexistenceofhomoclinicandheteroclinicorbitsofthissystemisalsorigor-ouslyproved.Numericalsimulationevidencesarealsopresentedtoexaminethecorrespondingtheoreticalanalyticalre
5、sults.Keywords:Lorenz-typesystem;centermanifoldtheorem;degeneratepitchforkbifurcation;Hopfbifurcation;homoclinicandheteroclinicorbit.1.Introductioncircumstances,thechaosisundesirableandshouldbeweakenedoreliminated,whichisnowalsocalledSincethefirstremarkablechaoticmodel,i
6、.e.thechaoscontrol.Therefore,inordertomakechaosLorenzsystem[Lorenz,1963;Sparow,1982],wasbyCITYUNIVERSITYOFHONGKONGon03/11/14.Forpersonaluseonly.moreuseful,peopleneedtothoroughlyunderstandpresented,alotofeffortshavebeendevotedtoInt.J.BifurcationChaos2011.21:2695-2712.Down
7、loadedfromwww.worldscientific.comseekingsomenewsystemsthatarerelatedtothephysicalessenceofchaos.Lorenz-typechaoticsystemsandinvestigatingtheirDuringrecentyears,thereportsontheinves-complicateddynamicalbehaviors[Chen&Ueta,tigationsintochaoshaveconcentratedonnotonly1999;L
8、¨u&Chen,2002;Celikovsk´ˇy&Chen,2002a;proposingnewandinterestingchaoticsystems(theYangetal.,2006;Yang&Chen,2008