Hopf Bifurcation Analysis for the Comprehensive National Strength Model with Time Delay Xiao-hong Wang 1, Yan-hui Zhai 2

Transcription

1 opf Bfurcaon Analy for h Comprhnv Naonal Srnh ol wh m Dlay Xao-hon an Yan-hu Zha School of Scnc ann Polychnc Unvry ann Abrac h papr manly mof an furhr vlop h comprhnv naonal rnh mol By mofyn h bac comprhnv naonal rnh mol can mor accuraly llura h ocy phnomna wh m lay Fr w rarch h ynamc of h mof wh m lay By mployn h normal form hory an cnr manfol mho w oban om abl rul on h u h concluon confrm ha a opf bfurcaon occur u o h xnc of ably wch whn h lay var Fnally om numrcal mulaon ar vn o llura h ffcvn of our rul Kywor opf bfurcaon Sably Comprhnv naonal rnh mol Cnr manfol Normal form I E ESABLISEN OF E ODEL Ornary ffrnal uaon u o uy h comprhnv naonal rnh of a counry ovrall uaon whch raually vlop n rcn yar h arcl brn n m lay n h comprhnv naonal rnh mol B h comprhnv naonal rnh mol mol ar al analy x x har powr rnh n whch a compo ncaor h lvl of h maral cvlaon rourc conomy mlary cnc an chnoloy c of a counry A x br h maral cvlaon mor proprou y of powr rnh n whch h prual cvlaon of a counry y an for ocal vl con-makn rror ucaon falur offcal corrupon popl al c A h m of powr ha obacl o h ocal vlopmn y man of powr upror w con-makn naonal ualy c ha ha a promon ffc on ocal vlopmn m conan So n h lraur [] comprhnv naonal rnh hown by h follown ffrnal uaon: x x x y y y m x x A known o all om chan of h of powr can b rflc on h har powr afr a cran pro of m h lraur [] ha ablh h follown lay mol of h comprhnv naonal rnh x x x y y y m x x pov an h ohr paramr ar h am a of hl mol of ha conr h m lay n h ral worl har powr rnh no only rly on har powr rnh of of powr n h pa bu alo rly on of powr In orr o mak h mol mor accura w mofy h ym of o h follown form: x x x ky y y y m x x 3 Pa 74

2 k h oranaon of h papr a follow: rarn a bfurcaon paramr w uy h ably of h ulbrum pon of h ym 3 an opf bfurcaon of h ulbrum pnn on hn ba on h nw normal form of h ffrnal-albrac ym nrouc by Chn al [3] an h normal form approach hory an cnr manfol hory nrouc by aar al [4] w rv h formula for rmnn h propr of opf bfurcaon of h ym n h hr con Numrcal mulaon am a ufyn h horcal analy wll b rpor n Scon 4 Fnally h papr n wh a cuon II SABILIY AND LOCAL OPF BIFURCAION ANALYSIS h ably an opf concluon for h ym of can b oban rcly from h lraur [56] From ym 3 w can ha hr x ulbrum E m m m Y E X Accorn o h praccal nfcanc of h mol hr w only cu h problm of h hopf bfurcaon an ably for h ol pov ulbrum pon E S u x X u y Y h ym 3 bcom u p u u ku u u u m p u u m hr p An h lnaraon of ym a E u p u ku u u u m p u h characrc uaon of ym a follow c f 3 c p m p f p m p k o uy h ably of h ulbrum pon E an bfurcaon ym w only n o cu h rbuon of h roo for characrc E 3 If h all roo of uaon 3 hav nav ral par h ulbrum pon E ay If h uaon ha a roo ha conan pov ral par h ulbrum pon E no abl In orr o uy h rbuon of h roo for E 3 Conrn n fr characrc uaon 3 c 4 hr p m p c Pa 75

3 Obvouly all roo of uaon of 3 ha nav ral par f af So h ym n ulbrum E locally aympocally abl for Now w nva h local ably aroun h pov ulbrum pon for h E whn ym 3 an h xnc of opf bfurcaon occurrn a h ulbrum pon Lmma For h ym 3 E ha a par of purly manary roo for whn af Proof If a oluon of h characrc uaon whn an only whn m c co n h paraon of h ral an manary par yl f f co c n 5 whch la o v c f v f 6 v aum ha h coffcn af h follown conon If h conon c f 4 f c f c f 4 f or roo From 6 w oban A a rul whn af hn E 6 ha pov roo hrfor E hav purly manary f c c f 4 f f arcco h characrc uaon hav a par of purly manary roo h proof compl Lmma h ranvraly conon R or af Proof By ffrnan boh of E 3 wh rar o an olvn hav hn c R c R Pa 76

4 can c f by conon co cn c f or hu R h proof compl Lmma 3 For E 3 f all of h roo hav nav ral par Pov ulbrum aympocally [ abl an h pov ulbrum prouc opf bfurcaon n III E DIRECION OF E OPF BIFURCAION AND E SABILIY OF PERIODIC SOLUIONS In w oban h conon of opf bfurcaon In h con w cu h rcon of opf bfurcaon an h ably of h bfurcan proc oluon ba on h normal form nrouc by Chn al [3] an h cnr manfol hory nrouc by aar al [4] In h follown par w aum ha h ym 3 unro opf bfurcaon a h ulbrum E for an w l h corrponn purly manary roo of h characrc uaon a h ulbrum E S R clarly h opf bfurcaon valu of ym 3 S u u for convnnc w connu o u u a u Dffrnal Euaon FDE ym n C C[ ] R hn h ym 3 uvaln o h follown Funconal u L F u 3 u u u L : C R F : RC R ar vn rpcvly by L B B F L 为 C[ ] R p B m p k B By h R rprnaon horm hr x a marx funcon who componn ar boun varaon funcon :[ ] R uch ha In fac w ak L C 3 Pa 77

5 B B 33 a Dla funcon For C [ ] R w fn opraor A an R a follow [ A 34 F hn h ym can b wrn a h follown form: [ R 35 u A u R u 36 Sn C [] R h aon opraor A an a blnar nnr prouc vn by A of A fn a ] From h cuon n Scon w know ha ar nvalu of A hu hy ar alo nvalu of w calcula h nvcor of A blonn o an nvcor By h fnon of A w hav A an In aon of A Nx A blonn o h nvalu B B A 39 L E39 bcom p k m p p k By h fnon of A w hav A an hn Pa 78

6 Pa 79 A B B 3 L D E3 bcom p m k p p ] [ D ] [ B D ] [ k D o ] [ k D Nx w uy h pcfc paramr of h rcon an for bfurca proc oluon Un h am noaon a n aar al [4] w fr compu h coorna o crb h cnr manfol C a Dfn } R{ u u 3 On h cnr manfol C w hav 3 hr In fac an ar local coorna for cnr manfol C n h rcon of an No ha ral f u ral conr only ral oluon For h oluon of E36 n cnr manfol C w hav R A * R A R A F 33

7 L from 33 an 35 w hav By 3 w hav hn F F V V F F 36 F u u u 37 u u D[ [ ] Comparn h coffcn wh 36 follow ha D D D D[ ] By E35 36 an fnon 3 of w hav Pa 8

8 A R u A R F F F [ A 38 L A hr From h fnon 3 of w hav 39 3 By an 3 w hav A A 3 For [] by an 39 w R[ F ] Comparn h coffcn wh 39 w v ha 3 33 I follow from 3 ha A 34 Solvn for an w oban Pa 8

9 3 E E 35 By 34 an 3 w 36 From 37 an 38 w Accorn o E35 36 an 37 w hn E E E3 3 E E E 37 I E E E E 38 3 p E E p k m p m p Accorn o h abov Propoon an [7] w can compu h follown paramr: C R{ C} R{ } R{ C } 3 Im{ C} Im{ } IV NUERICAL SIULAION Numrcal mulaon how from abl o unabl complx ranformaon proc h ym 3 v a concr xampl o how h ynamc bhavor of comprhnv naonal rnh mol ak 3 k 8 m 9 n h ym 3 So w can oban Pa 8

10 ulbrum E 893 hrouh calculan of h ahmacal ofwar w oban 6 88 By lmma an abov concluon w can oban ha ulbrum E abl f 7 Fur By conra h ulbrum E unabl f Fur B whn 886 h proc oluon occur from h ulbrum E Fur 3-4 owvr our analy nca ha h ynamc of h comprhnv naonal rnh y mol wh m lay can b much mor complca han w may y hav xpc I ll nrn an nprn o rarch x x y FIGURE E EQUILIBRIU E FIGURE E EQUILIBRIU E OF SYSE 3 IS SABLE I 7 OF SYSE 3 IS SABLE I x FIGURE 3 SYSE 3 PRODUCE E PERIODIC SOLUIONS I 886 Pa 83

11 Inrnaonal Journal x of Ennrn Rarch & Scnc IJOER ISSN: [ ] [Vol- Iu-3 arch- 6] y FIGURE 4 IE SEQUENCE DIAGRA OF FIGURE 3 V CONCLUSION h papr apply lay n h comprhnv naonal rnh mol whch how rch ynamc bhavor Dffrn from prvou u w a h nflunc of m lay fback n h ym 3 Dynamc bhavor of comprhnv naonal rnh mol wh m lay analy by un h mho of uanav hn lay $\au$ acro a r of crcal valu nonlnar ynamc ym nra h opf bfurcaon In aon by mployn h normal form hory an cnr manfol mho w oban om abl rul on h u u normav hory an cnr of popular horm oban h calculan mho of h rcon of opf bfurcaon an ably of proc oluon Fnally h abov horcal analy vrf by numrcal mulaon h ynamc bhavor of h comprhnv naonal rnh mol rch any apc no mnn y o b Rfrnc CONFLIC OF INERESS h auhor clar ha hr no conflc of nr rarn h publcaon of h papr ACKNOLEDGEENS h auhor ar raful o h rfr for hr hlpful commn an conrucv uon AUOR CONRIBUIONS Concv: Xaohon an Drawn raphc: Xaohon an Calcula: Xaohon an of: Yanhu Zha ro h papr: Xaohon an REFERENCES [] S an Dffrnal uaon mol an chao Journal of Chna Scnc an chnoloy Unvry pp [] X Xn opf Bfurcaon Analy n Comprhnv Naonal Powr ol wh wo Dlay[J]Journal of Yl Normal UnvryNaural Scnc Eo vol no3 pp 4 7 [3] BS Chn XX Lao YQ Lu Normal form an bfurcaon for h ffrnc-albrac ym Aca ah Appl Sn n Chn [4] B aar D Kaarnoff Y an hory an Applcaon of opf Bfurcaon Cambr Unvry Pr Cambr 98 [5] L an Z Lu opf bfurcaon of prc Rylh uaon wh m lay Journal of Jln Unvry vol47 no3 9 [6] GD Zhan LL Zhu BS Chn opf bfurcaon an ably for a ffrnal albrac bolocal conomc ym Appl ah Compu [7] Y an an Y Z Sably an opf bfurcaon of ffrnal uaon mol of prc wh m lay hlh of Scncpapr Onln vol4 no [8] Y Z Y B Y Xon an XN opf bfurcaon analy for h mof Raylh prc mol wh m lay Abrac an Appl Analy vol3arcl ID pa 3 [9] D D aar N D Kaarnoff an Y an hory an Applcaon of opf Bfurcaon Cambr Unvry Pr Cambr UK 98 [] J a YQ Chn LX Luopf bfurcaon an chao of fnancal ym on conon of pcfc combnaon of paramr Journal of Sym Scnc an Complxy 8 Vol no pp [] J al hory of Funconal Dffrnal Euaon Sprnr Nw York NY USA n on 977 Pa 84