胞映射基本理论与算法实现

:2000-06-15:(:29777030):,,1954,:,刘信安王宗笠蒋启华肖维平张环(,400044):,,,HsuCS(),:,:;;;:O6654,O65732:A:1001-4160(2000)05-427-435ThePrincipleandAlgorithmofCellMappingTheoryLIUXin-anWANGZong-liJIANGQ-ihuaXIAOWe-ipingZHANGHuan(CollegeofEnvironmentalandChemicalEngineering,PhysicalCollege,ChongqingUniversity,Chongqing400044)Abstract:Cellmappingmethodisoneofthenumericalcomputingtechnologyforsolvingtheglobalbehaviorofcomplexnonlinearsystem,andthecomplexevolution,strangerattractorsandchaosbehaviorsofnonlineardynamicchemistrysystemwillbeeffectiveaccuratelydescribedinrealtimewiththecellmappingalgorithmsothatthesystemmotioninternalitycanbeprofundityunveiled.Inthepapertheprincipleconceptsforsimplecellmappingandgeneralizedcellmapping,algorithmflowchart,programmingmod-ulesandskillshavebeengivenindetail,andtheresearchingresultsshowsthatcellmappingtheoryhasquantitativeanalysisanduniversalismfeaturesfornonlinearsystemssothatwouldbeappliedtothevariousfieldsofenvironment,chemistryandbiology.Keywords:cellmappingtheory,nonlinearchemistry,algorithmrealization,chaos,,,,,,,,,,[1-3]80HsuCS,(Poincar)(MarkovChains){Xi},(),,,(),,,1752000928ComputersandAppliedChemistryVol17,No5September,20001胞映射原理与基本概念[4](1):x=F(x,t,),xRN,tR,RK(1)xN,t,K,Fx,t,N,XNN,,,,Nx,,hh,,,,N,1xi,i=1,2,Nziz1,z2,,zNz,zzi,i=1,2,N,F(z),1Fig1SimpleCellMapping2SCM()Fig2SCMperiodic(shaded)andtransientcells.11(SinkCell),:,,,12(SimpleCellMapping,SCM),:z(n+1)=C(z(n),),(2),C,NN13(GeneralizedCellMapping,GCM)z,42817,nz(n),,()Nc,,1NcNcpnp(n),p(n)ipi(n)niP,NcNc,Ppijji(3),P(n+1)=Pp(n)(3),,14(EquilibriumCellsandPeriodicMotions)C,z*z*=C(z*),z*CmC0mKz*(j),j{K},,K,z*(j)Kz*(m+1)=Cm(z*(1)),m{K-1},(4)z*(1)=CK(z*(1)),(4)P-K(P-K),P-K,P-12,1z1z2,z3,z4P-32z11,z2z3z433Fig3Domainsforattractionofsingle-domicilecellsandboundaryregionofmultiple-domicilecells4Fig4Flowchartforsimplecellmappingalgorithm15(TransientCellsandDomainsofAttraction)z*(j)P-KP-K,rCr(z)=z*(j),zKrzKP-K,P-KKr,Krrmax,rmaxrz3:Gr(z)P(z)St(z),02,,4295:3Gr(z)P(z)St(z)kkj,j,Dj,16(Simgle-DomicileandMultiple-DomicileTransientCells)jBh,Bhj,:;,317(Protectionthicknessofadomainofattractionforanattractor)Ag,Ag,Agd(A,A)-1,d(A,A)AAAAA,,,0,2胞映射算法21NNc,1,2,,Nc,SC(6),,:z(n+1)=C(z(n)),z(n),z(n+1)S(6)(a),,Nc;(b),1;(c)z:(c-1)z,z;(c-2)zr,zr,22z=1,z=Nc,,,:C(1),C(2),C(3),,C(Nc)(7),(7){Nc}Nc,{Nc},3Gr(z),P(z)St(z)z,3,C(z)[4],Gr(z),zS,0,zGr(z)=0,,,-1,-1-1,,,z,Gr(z),z,:zC(z)C2(z)Cm(z).(8)zmz1,2,,Nc43017z,(8)Ci(z):(I)Ci(z)Gr(Ci(z))=0,Ci(z)Ci+1(z),Gr(Ci(z))=-1,Ci(z),(II)Ci(z)=z,Ci(z),,Ci(z),,z,:St(Cj(z))=St(z)+i-j,j=0,1,,i.(9),,S,,(III)Ci(z)=z-1,Ci(z),,,Ci(z)Ci(z)(j+1),,Ci(z)=Cj(z),ji(i-j),(i-j),(10a),(10b):St(Ck(z))=j-k,k=0,1,,j-1,(10a)St(Ck(z))=0,k=j,j+1,,i-1,(10b),S,,45,SSt(z),5Fig5FlowchartfortwosubroutinesofSCM6Fig6Determinatingofpersistentgrounp3广义胞映射的算法及计算模块31,,GCM,GCM,,SCM4315:326:1GCM,:z1Nc,CCI(z)z,CTC(z,i)zi,CCP(z,i)zi,ziCTC(z,i)CCP(z,i)i1CCI(z)CCI(z),CTC(z,i),CCP(z,i)31Nc,,,3,,,,,z,,:,,z,z(1),,,,[5-6],[7],Delphi40,:CellFun(hh,yxz,rrr);x2:=rrr[0];y2:=rrr[1];FxyToIxy(x2,y2,i,j);IxyToSn(i,j,k);IF((k=0)AND(kbigcellnum))THENk=kELSEk=bigcellnum;kk=Discern(k,CTC[z]);IFkk0thenBEGINCCI[z]=CCI[z]+1;Setlength(CTC[z],CCI[z]);Setlength(CCP[z],CCI[z]);CTC[z,CCI[z]-1]=k;CCP[z,CCI[z]-1]=1;ENDELSECCP[z,kk]=CCP[z,kk]+1;END;hh,yxz,rrr:(a)CellFun(),,;(b)Fxy-ToIxy(),;(c)IxyToSn(),FxyToIxy;(d)Discern(),,,,,k,z,k[1,Nc],,zk,z,k=bigcellnum,,kk0,kCTC[z],zCCI[z]1,setlength()CTC[z]CCP[z],kCTC[z],1;kk0,kz,zkCCP[z,kk]1,3CCI(z),CTC(z,i)CCP(z,i)2,zzCRI(z)z,RTC(z,j)zj,RTC(z,j)z1Nc,j1CRI(z),RTC(z,j)1CTC(z,i)CCI(z)RTC(z,j)CRI(z)RTC(z,j),1NcCRI(z)01zi=1,2,,CCI(z),CTC(z,i),zi,CTC(z,i)=z,zz,zzCRI(z)RTC(z),CRI(z))(11),,RTC(z,j),,j1CRI(z)CRI(z)CRI(z)+1RTC(z,CRI(z))Z(11)432172CRI(z)RTC(z,j),3SCM1,,;,(),,,,,34,,,CCI(z),CCI(z)1()CTC(z,i)=z(),,,,2:z,:,z,z1,z2,z3,3,z33,,,,12,3:1,2,12,,,,,,,,45GCM,,,33:,:31,4,5,,Ar5,16,1Ar,6Ar,615,15Ar,6,5,54,435,316,3Ar,1143,Ar,6,Ar1,4,5,6,3,2,26,,,,,4335:,4,1,2,,kkGr(z)gg,,B(0,g);0N(0,g),K(g)-2,Gr(z)Nc,K(g)N(0,g)k5k()N(0,g)K(g)B(0,g);g1k,,5,,,,;,5,6645,,6,:,45,6k4一个非线性运动方程的胞映射实现具体步骤Rossler,,:x=-(y+z)y=x+ayz=b+z(x-c)(12),:dxdtxt=x(n+1)-x(n)t,:-[y(n)+z(n)]:x(n+1)=x(n)+{-[y(n)+z(n)]t}(12),t001,,,Posi,xnum,ynumXY,xmin,ymin,xmax,ymaxXY,:ProcedurePosiToIXY(Posi:integer;varx,y:integer);//(1)beginif(Posidivxnum)=ynumthenx:=xnumelsex:=Posimodxnum;y:=Posidivxnum;end;,,;ProcedureIxyToFxy(x1,y1:integer;varx2,y2:real);//(2)beginx2:=(x1/xnum*(xmax-xmin)+xmin);y2:=(y1/ynum*(ymax-ymin)+ymin);43417end;(3),ProcedureFxyToIxy(x2,y2:real;varx1,y1:integer);//(3)beginx1:=round((x2-xmin)/(xmax-xmin)*xnum-05);y1:=round((y2-ymin)/(ymax-ymin)*ynum-05);end;(4);ProcedureIxyToPosi(x,y:integer;varPosi:integer);//(4)beginif(x=0)and(y=0)thenPosi:=y*xnum+xelsePosi:=bigcellnum;end;4,,,,,,(zi-05)hxi(zi+05)hzixi,hxizi,,(2)(3)xminzi5结论与展望,,,,,,:,,,[7-8]1HsuCS.Cel-lto-CellMapping:AMethodofGlobalAnalysisforNon-linearSystem.NewYork:Springer-Verlag,19872HsuCS,GuttaluRS.Anunravellingalgorithmforglobalanalysisofdynamicalsystems:Anapplicationofcel-lto-cellmappings.ASME,JApplMech,1980,47:940-9483HsuCS,Gu