一类含参数激励和强迫激励的时滞反馈系统的分岔分析

25220076()J.ofXuzhouNormalUniv.(NaturalScienceEdition)Vol.25,No.2Jun.,2007:2007201218:E-(E03004):(1980-),,,,.1,2,1(1.,276005;2.,277160):,12-.,.:;;:O175:A:100726573(2007)0220028203,,.,.[1],,,.[2].10.[3,4].[5,6]vanderPol2Duffing,.[7],,.,,.:x-(-x2)Ûx+(2+2cos2t)(x-x3)=Fcost+(gpx(t-1)+gdÛx(t-2)),(1)0n1,,,,F,,,1,2,gp,gd.,,.12-.112-[8](1)x=x0(T0,T1)+x1(T0,T1)+,(2)T0,T1,T0=t,T1=t.:ddt=55T0+55T1+=D0+D1+,(3)d2dt2=D20+2D0D1+.(4)12-,=1+,=2,.(2)(4)(1),,©1994-2007ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.=0,(5)D20x1+x1=-2D0D1x0+(-x20)D0x0+x30-2x0-2x0cos2T0+Fcos2T0+gpx0(T0-1)+gdD0x0(T0-2).(6)(5)x0=A(T1)eiT0+A(T1)e-iT0,(7)AA.(7)(6)D20x1+x1=(-2iD1A+i(-AA)A+3A2A-2A-A)eiT0+(gp(cos1-isin1)A+gd(sin2+icos2)A)eiT0+(-iA3+A3-A)ei3T0+F2ei2T0+cc,(8)cc.,-2iD1A+i(-AA)A+3A2A-2A-A+gp(cos1-isin1)A+gd(sin2+icos2)A=0,(9)A=aei,aT1,(9),dadT1=12(-gpsin1+gdcos2)a-12a3+12asin2,addT1=-12gpcos1-12gdsin2a-32a3+12acos2.(10),dadT1=ddT1=0,12(-gpsin1+gdcos2)-12a3+12asin2=0,-12gpcos1-12gdsin2a-32a3+12acos2=0,(11)(11)e,e,e=-gpsin1+gdcos2,e=-12gpcos1-12gdsin2,(12),12ea-12a32+ea-32a32=14a2.(13)2(12)1,2,,1=,2=+,e=-gpsin+gdcos(+),e=-12gpcos-12gdsin(+),(14)(14),e,e(e-)2+(2e-2)2=g2p+g2d+2gpgdsin,(15)€e=2e,(15)(e,€e),(,2),,gp+gd,min{0,|gp-gd|}.(13)a6+-12e-3e24+942a4+142e+2e-1424+942a2=0,(16)r=a2,h1=-12e-3e24+942,h2=142e+2e-1424+942,922,:©1994-2007ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.(16)r2+h1r+h2=0.1(e,€e)Fig.1Localbifurcationsetsin(e,€e)parametricplane[9],h1=0,-12e-3e=0,h(r,h2)=r2+h2,h2.-12e-3e=0(e,€e),(e-)2+(2e-2)2=g2p+g2d+2gpgdsin(e,€e)(,2),1,A,B,C(e-)2+(2e-2)2=g2p+g2d+2gpgdsin.(e,€e),.,.12-;,.12-,.:[1].[J].:,2006,24(3):20.[2],.[J].,2006,36(1):17.[3],.[J].,2006,55(2):617.[4],,.VanderPolHopf[J].,1999,20(4):297.[5],,.VanderPol2DuffingHopf[J].,2000,32(1):112.[6],.Duffing[J].,2004,17(3):365.[7]XuJ,ChungKW.EffectsoftimedelayedpositionfeedbackonavanderPol2Duffingoscillator[J].PhysicaD,2003,180(2):17.[8],.[J].,1991,23(4):464.[9]GolubitskyM,SchaefferDG.Singularitiesandgroupsinbifurcationtheory[M].NewYork:Spring2Verlag,1985.BifurcationofNonlinearSystemsInvolvingTimeDelaysUnderCombinedParametricandForcingExcitationZHANGDong2mei1,LIFeng2wei2,XUHan1(1.DepartmentofMathematics,LinyiNormalUniversity,Linyi,Shandong,276005,China;2.DepartmentofMathematicsandInformationScience,ZaozhuangUniversity,Zaozhuang,Shandong,277160,China)Abstract:Inthispaper,theproblemsofbifurcationofnonlinearsystemsinvolvingtimedelaysundercombinedparametricandforcingexcitationfor12subharmonicresonance2primaryparametricresonancearestudied.Byusingthemethodofmultiplescales,thebifurcationequationwithtimede2laysisobtained,andthebifurcationequationisdiscussedbymeansofsingularitytheory.Sotheeffectoftimedelaysonthesteadyresponsecanbefound.Keywords:methodofmultiplescales;bifurcation;dynamicsofnonlinearsystemsinvolvingtimedelays03()25©1994-2007ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.