Í Î4Ï>Ð Ñ - polytechniquedubos/Men/Omrani09.pdfÍ Î4Ï>Ð Ñ ÒÓÑÔ Õ Î.ÖX× ØÙÑ5Ô...

30

Transcript of Í Î4Ï>Ð Ñ - polytechniquedubos/Men/Omrani09.pdfÍ Î4Ï>Ð Ñ ÒÓÑÔ Õ Î.ÖX× ØÙÑ5Ô...

Page 1: Í Î4Ï>Ð Ñ - polytechniquedubos/Men/Omrani09.pdfÍ Î4Ï>Ð Ñ ÒÓÑÔ Õ Î.ÖX× ØÙÑ5Ô Ú>Û Ü[Ý Þ ß à á!ârÝ8ã©ßZÜ ä åçæ èfà!Þ é à áÌÝ Þ èrêfèsã©ë

! "#$%&'(")*',+-**./010 -+32-4"*567 8'9":;<$%&'("=>+-8<?@%

ACBED&FHGIKJ,LMCN

OQPSR8T BVUXWZY[Y P

Page 2: Í Î4Ï>Ð Ñ - polytechniquedubos/Men/Omrani09.pdfÍ Î4Ï>Ð Ñ ÒÓÑÔ Õ Î.ÖX× ØÙÑ5Ô Ú>Û Ü[Ý Þ ß à á!ârÝ8ã©ßZÜ ä åçæ èfà!Þ é à áÌÝ Þ èrêfèsã©ë

\]-^_`]

ab8cSdfehgjikb8l<monpgrqscStrl<moufb8vwvpb8cyxQulzstrqf|rb8~bqmSl<vpnp~tjmonufbeh8trvwnc8b8cSdstremhnweyxQb8cy*gQxQvpb8cyxQb/nwel<ufvtmhnwgqqf8ehtrvwb vwgstjv$vpnw#tjmob#*gQxQbvE91xf8l<ehnprbqmxQb8clzfbvpvwbccodstjmontjvpb8cxQb~6treontjsnwvpnmh xQb~vE gehxfeobxQbufb8vpufbcl<bqmtjnpqfb8cxfbynpvwg~<moehb8c8b8c 8lzfb8vwvpb8c~cogrqm~moehgrdHnp~dgemtjqmobcdgufe~db8eo~b<momoehbSusqfbeob8dfeocb8qmhtmhnwgqeh8tjvpnckmhbyxQbvE np~dstrl<m~xfb8cdseogQl<bcocouscxQbdZb<monwmob;lzfbvpvpb<mob8vufbvV grehgr|ehtrdfzfnpb%cusevpb8c-conp|rqstruQCltrvplufvwcQabdseob8npbegQikb8l<monwxQbvt eh|nwgqstjvpncotjmonpgrqbckm;xQgqsl(xQbeh8xQufnpehb9vwbc8lzfb8vwvpb8cxQb8cdfzf8qfgr~8qfb8ceocgvwucdstrevpb l8tjvl<ufv0xQb8clzstrqf|rb8b8qmhcl<vpnw#tmhnpufbc%b8quQmhnwvpnpchtjqmxfb8c-~gQxQvpbb gjpmh8cxQbzstruQmob;eh8cogrvpuQmhnwgq tjqclb%mheht6trnwvErqfgusc<mhusxQnpgrqscvV b<¡[b<mxQuS|rufnxftr|rb-mhb~dgeob8v[nwqsxfn¡[ehbqslnwcoufevpb8c|ehtrqsxQb8c$b<mvwbc$dZb<mhnmhb8clzfbvpvwbccoufeusq*gQxQ8vwb9#trnwehbvwnp~nmhbf¢£vc8 tr|rnwm%x¤ usqy~gQxQ8vwb9ustcnw¥E|gckmheogdfzfnusb E¦ ;xfbuQ*l<guslzfb8c-xstjqscvE trdfdfehg6Qnw#tmhnwgqxfuCdfvtjq,Z§<mt;¨gehl%tjuQ#Zgrexfc3dtjexfb8cxQgrqfqsb8c©trnwfvpb%eh8cogrvpuQmhnwgq#npchcusb8cxQbcht;rb8ehconpgrq#|rvpgrtjvpbt6rb8l%xQb8cl<gqsxQnwmonpgrqscvpnpnwmobc-dZehnwgQxQnufb8c8b;xQnpcodZgconmhnnp~b;vpt~gQxQvpncotjmonpgrqCeh|nwgqstjvpbrª3q*l<g~dstjetjncgqCt6rbl%vpb8cmheht6truQCxQb9«Qtjvtj~b8zbm,tjvE E¬j­r­® (usngqmmousxQnpyvV b<Qnckmhbqslbyx¤ ufqHmob~dsc#xfbS|ufnpxstj|rbSgrdQmhnw#trvufn~nwqsnw~ncb*vt¨grnc vE beheob8ufe coufe~vpt|ehtrqsxQbb<m-vtdbmonwmob9lzfbvpvwb;xftrqscufq~gxfvpb;vwnpqf8trnwehbtjqstrvw¯monufbrQqsgrusc-t6grqscuQmonpvpnpco;ufq~gxfvpblgr~dfvp<mhb~bqm-qfgrqvpnpqf8trnwehb;usndb8eo~bmufqfbgqfqfb;ehbdfeh8cobqmtmonpgrqyxQu*ltrehtl1mo8eoblzstjgrmonufb;xQu*c¯Qcmo~b;tmh~gcodfzfehnusbr°&tremtjqmxQbvE z¯dgrmozfcbufb(qfgrmoehb(~gQxQvpbb8cmdstre©trnmufqsb dZb<mhnmhbb8eoehbusexftrqsc%vpb8cl<gqsxQnwmonpgrqscnpqfnmhnptrvwbcl<grql<bqmheo8bbcocobqmhnwb8vwvpb~bqm3coufe0vwbc!|ehtrqsxQbc!lzfbvpvpb8c0ltcol8trxQbikuscou l<gqmtj~npqfbevwbc0dbmonwmobc08lzfb8vwvpb8c86vwnp~nmtjqm3tjnpqscnQvtdfeo8npconwsn¥vpnwmoxfu#qfgjmheob-~gQxQvpbr°0grufe3~bcufehbe$vpt;dseo8nconwfnpvpnmhxQbqfgrmoehb-gQxQ8vwb ¦ rqfgusc3t6rgqsc&ltrvplufvpvwbmhtjufxQb%leognpchchtjqslb ª&QdgcotrqmxQba¯tjdfufqsgQxQb(vtCxQnw¡[ehbqslb(npqfnmhnptrvwb8~bqmdZb<mhnmhb b8qmheobxQbufmhtjmhcxQb vE tmo~gcdfzsehbra!t,dfehncnpfnpvwnwmoxfuC~gQxQvpb|rvpgrtjvsbckmdfehnwqslnwdtjvpb~bqmlgrqmoehgrvpb-dstre$vE b¡ZbmZ§<mtfdstjel<grqmheob%xstjqscvwb%gQxQ8vwb9tjnpehbvwnp~nmhb anp~nmhb8x± eobt~*gQxQbvEsa ± vt dseo8nconwfnpvpnmh9tjuf|b8qmobutjqsxyvt(mtjnpvwvpb;xQu*xQgr#trnwqfb;xQu*a ± .xQnpnpqufbrª$qscusnmhbvE b¡ZbmxQuC|rufnxftj|b-|retjqsxfb8lzfb8vwvpb%t;mob<ftj~npqfbqCuQmonpvwnchtjqm$vt9~<mhzfgQxQbxQuCgQxQ8vwb%dstjeo©tjnwm8°gufelb©trnwehbrqsgrusc;t6grqsc%<mhtrfvpn3xQbuQbQd8eonpbqslb8c² k³ vE b<¡[b<m(xQu|rufnxftj|btS<mh mousxfnw#coufeufq~gQxQ8vwb ¦ ´yxQbufl<guslzfb8cxQbztjuQmhbCeh8cogrvpuQmonpgrq/|ufnpxfCdstre xQbc xQgqfqf8b8c(©tjnpfvpbCeh8cogrvpuQmonpgrq V¬ ;vt*~§~bCbdZehnpbqslbSt*mo,©trnmhbyt6b8l~vpbC~§~b~gQxQ8vwbxftjqsccht#rb8ehconwgq*tjnpeob9vpnpnwmo8b a ± µt6b8l;ufqqfgrufb8truydtjetj~<mheob(xfb l<gqmoeh¶rvpb#²vt~mhtrnwvpvpb xQuxfgr#tjnpqfbrZ trqscvpb8cxQb8uQb<QdZehnwb8qsl<bcvt,~bnpvwvpbuselgreheo8vptjmonpgrq*cdtmontjvpb(b8qmoehb(vpb8ccnp(ufvtmhnwgqsc|ufnxQbc%bmvt,eh<¨ehbql<b(tC<mogrfmobqufbt6b8lufqSmob8dcxQb;|rusnpxftr|rb

τxQbvV grexQehbxfuSmhb~dsc-xQb;dseo8nconwfnpvpnmh

Tpτ = 0, 5Tp

<

·¸ ^¤¹º0»9¼$¹

¢£qSmozsnpc½greh[½b;l<grqcnxQbemozsbb¡Zbl1m-gjnwqxQnpchl<ehnpnpqstjmobquxQ|rnpqf| monpbgq,mozsbvtjeh|rbtjqxCco~trvwv¤chltrvwbcgr!tjqynxQb8trvwnp¾bxvpnpnwmobxCtreobtgQxQb8vZconp(ufvtmhnwgq¿zfb%vwnp~nmhb8xCtreobt~gxfbvZnpctmk½g(vpt6¯be$ustcnw¥E|bgckmheogdfzfnl~gQxQb8vsgrq,mozfbbmht¥Àdfvtjqfbxfeonprb8qtjm%nwmhc-Zgrufqxftjehnwbc-¯CnwmhcÁ Â|rvpgrtjvÁÃybecnpgrqy½-nwmoz*dZbehnwgQxQnlgufqsxftreo¯yl<gqsxQnwmonpgrqs¿zsnpccbmoufd~nw~nlcmozfblgrqQÄ|rufetmhnwgqCuscob8xS¨gre-ehb|nwgqstjv¤lvwnp#tmob~gQxQbvpvpnwqf|g~dstjehb8xymhgCt#dfehbnpgruscckmhusxQ¯¯«Qtrvptr~bz*b<mtjvE E¬j­r­® ½-zfgCnwqrbckmhnw|tmhb8xymozsb(b<Qnckmhbqslb9gj3trqgrdQmhnw#trvqusxQ|nwqf|mhnw~b~nwqsnw~np¾npqf|mozfbbehehgre$gqSZgjmhzSvtjeh|rbtjqsxSc#trvwv¤chltjvpbnwqyt(vpnwqsb8tje~gQxQbvEQ½bzfb8eobuscobt(¨ufvpvw¯CqfgrqQ¥Àvpnwqfbtje~gxfbv½-zsnplztrvwvpg½cuscmhg#eob8dfeobcb8qmmhzfb(lzstrgjmhnplqstjmoufehb;gjmozfb tmo~gcdfzsbehb²s|rnprbqSmozsb9db8e¨bl1mustcnw¥À|rbgckmheogdfzfnl~gQxQbvEb8eoehgrec3npq#mhzfbnpqfnwmontjv¤l<gqsxQnwmonpgrqsc8l<grql<bqmhehtjmob8x~#trnwqfvp¯~nwqCmozfbvtjeh|rbecol8tjvpb8c3gr~gjmhnwgqtr~dfvwnw¨¯#tjqsxSltcol8trxQbnpqmog(mozfbco#tjvpvwb8ecol8tjvpb8c8Qbbqmoustrvwvp¯#ehb8coufvmhnwqs|npq*tdfeobxQnl1monpgrqy½-nwmozvwg½KcnpvwvE¿!g,ustjqmhn¨¯#mhzfb;dfehb8xQnl1mtjfnpvpnmk¯,grgufe-ustrcon¥À|rb8gcmoehgrdfzsnpl%gQxQb8vVs½b~b8tcufehbmozfb;etmhbgr0xfnwbeh|rbql<bgjmozfb9co¯Qckmhbmhehtjikb8l<mogrehnpb8cnwqÅdfztrcob,cdstl<b a¯trdfufqfg*bdZgrqsbqm¨ehgrÆtycob<m9grconp(ufvtmhnwgqscnwqsnmhnptjmob8x:½-nwmoz/tdZbeomoufehstmhnwgqgrtehb<¨b8eob8qsl<b(nwqfnwmontjvckmtmhbrs°$eobxQnpl<mhtrfnwvpnwmk¯SgrmhzfbSÇ<|rvpgrstrvÀÇjQdb8eonpgQxQnpl9gQxQb8vnc~gcmovp¯Slgrqmoehgrvpvpb8xy¯Cmhzfb9Zb<mhtCb<¡[b8l<m8s¢£qmhzfb9a ± /fdfehb8xfnpl<mhtjsnwvpnmk¯SxQb8leobtrcob8ctrcmhzfb9xQgr#trnwqcnp¾bnpqsl<ehb8tcbc¿zsbq6mhzfbb¡Zbl1m&gjvptreo|b<¥£col8tjvpb&quxQ|rnpqf|%nc0ckmhusxQnpb8x ¯;uscnpqf|%mozfb!È8dZbeo¨b8l<m0~gQxQb8vÈtjdsdfeogtrlz6¿!½gcbmhcgjsbQdb8eonp~bqmhc½behb;dZbeo¨grehbx² k³ mozfb9b¡Zbl1mgr0qusxf|rnpqf|~npc-npqrbckmhnw|tmobxC½-nwmozt#Á |rvpgrstrvÁÃSzfnp|rzehb8cogrvpuQmonpgrqSmk½g#vt6¯rb8e-utrcon¥|bgckmheogdfzfnl,~gQxQbvxQehnprbqH¯tvpg½Éeobcgvwufmonpgrq/mk½gvt6¯rb8eustcnw¥À|rbgckmheogdfzfnl~~gxfbvE V¬ conw~npvptre~conp(ufvtmhnwgqsc treoblgrqsxfusl1mhb8xŽ-nwmozÅmozsb,mk½g*vpt6¯be9ustrconw¥E|bgcmoehgrdszfnpl a ± ½-zfbehb#mozsbCcnp¾b#grmozfbCa ± ÊxQg#tjnpqÅlgr~b8c9nwqmogdfvpt6¯nwqtxfxQnwmonpgrqHmogmozfbyÄseckm~cbmgrconw ufvtmonpgrqc¢£qmhzfbCmk½gÅcbmhc gj%bdZbehnpb8qmhc8!mhzfbbckm#codstmhnptrvlgreheob8vptjmonpgrq/Zb<mk½bbqmhzfbqusxQ|rb*conw ufvtmonpgrqËtjqsxmozfbehb<¨b8eob8qsl<bynpcgscb8eob8x/½-nwmozÌt:qusxQ|rnpqf|monp~blvwgcbSmogmozsbydseobxQnpl<mhtrfnwvpnwmk¯Åmonpb

Tpτ = 0, 5Tp

1

³

Page 3: Í Î4Ï>Ð Ñ - polytechniquedubos/Men/Omrani09.pdfÍ Î4Ï>Ð Ñ ÒÓÑÔ Õ Î.ÖX× ØÙÑ5Ô Ú>Û Ü[Ý Þ ß à á!ârÝ8ã©ßZÜ ä åçæ èfà!Þ é à áÌÝ Þ èrêfèsã©ë

Í Î4Ï>ÐÑ ÒÓÑÔ Õ Î.ÖX×ØÙÑ5Ô

Ú>ÛÜ[ÝÞßàá!ârÝ8ã©ßZÜ äåçæ èfà!ÞéàáÌÝÞèrêfèsã©ë ì¬ ³ *gQxQvpb;utrcon¥À|r8gcmoehgrdfzfnufb À¦ ; ;;;;;;;;;;;;;;;;;;;; í

¬ ³ ³ î ustmhnwgqsc ;;;;;;;;;;;;;;;;;;;;;;;;; í¬ ³ ¬ ï greh~b;trxQnp~bqsconwgqfqfb8vwvpb;xQb8c-ustmhnwgqsc#;;;;;;;;;;;;;;;;; ð

¬ ¬ °3eo8npconwsnwvpnmhñ;;;;;;;;;;;;;;;;;;;;;;;;;;; ®¬ ¬ ³ grqslbdQm%xQb;trcobò;;;;;;;;;;;;;;;;;;;;;;; ®¬ ¬ ¬ tjvl<ufvxfbvE b<Qdgcotrqm-xfb9a¯trdfufqfg ;;;;;;;;;;;;;;;;; ³8­

¬ ó ufnpxstj|rbô;;;;;;;;;;;;;;;;;;;;;;;;;;;; ³8­õ=ö÷ßà!øëVé*ùyëVßsú0èfë ÚQåós ³ °0trehtr~<moehb8cdfz¯Qconpufbcxfuy~gQxQ8vwbû;;;;;;;;;;;;;;;;;;;; ³¬

óf ³ ³ °0trehtr~<moehbβ

;;;;;;;;;;;;;;;;;;;;;;;; ³¬óf ³ ¬ °0trehtr~<moehb

κ;;;;;;;;;;;;;;;;;;;;;;;; ³ ó

ós ¬ °3eo8npconwsnwvpnmhñ;;;;;;;;;;;;;;;;;;;;;;;;;;; ³üóf ¬ ³ tjvl<ufvpb9xQbvV b<QdZgchtjqmxQb9a¯tjdfusqfg#b<m-trvwustjmonpgrqxQb;vE npqsl<b8emhnmhusxQbý;;;;;;;; ³üóf ¬ ¬ ¢£qQþsufb8qsl<b9xQbc-dstrehtrmoehb8cdfz¯Qcnufb8c-coufe-vt dseo8nconwfnpvpnmhû;;;;;;;;;;; ³ü

ós ó %bcolbqmobbqlzfbvpvwbbm|rufnxftj|b ;;;;;;;;;;;;;;;;;;;;; ³ÿóf óf ³ ufnpxstj|rbdstrexQb8c%xQgrqfqsb8cdtjeo©tjnwmob8cÆ;;;;;;;;;;;;;;;;; ³óf óf ¬ ufnpxstj|rbdstrexQb8c%xQgrqfqsb8c-xf|retrxfb8c;;;;;;;;;;;;;;;;; ³ íóf óf ó -8coufvwmhtmc4;;;;;;;;;;;;;;;;;;;;;;;;; ³ ðóf óf ü -b8lgrqscmonwmouQmhnwgq*xQb8cdZb<mhnmhb8clzfbvpvpb8cdstrevpb;~gQxQvpbÄsvc ;;;;;;;;;;;; ¬j­

çö÷ßà!øëVéSÞZã©ßZÜ!èfë:èfã¨ÞéëVãã©Ýé ååü ³ *gQxQvpbeo8|rnpgrqstrv[~trnwehbvwnp~nmhbñ;;;;;;;;;;;;;;;;;;;;; ¬r¬ü ¬ *<mhzfgQxQgrvpgr|nwb;;;;;;;;;;;;;;;;;;;;;;;;;; ¬r¬ü ó ª&¡ZbmxQbvt(mtjnpvwvpb9xQu*xQg~trnwqsb9;;;;;;;;;;;;;;;;;;;;;; ¬ óü ü %bcolbqmobbqlzfbvpvwbbmufnpxstj|rbò;;;;;;;;;;;;;;;;;;;;; ¬ü

ü ü ³ ª&¡ZbmxQb;Zgrexfc dszfqfg~qfb9xQb9npfsc ;;;;;;;;;;;;;;;; ¬üü ü ¬ ufnpxstj|rb ;;;;;;;;;;;;;;;;;;;;;;;;;; ¬ü

äçæ ßZÜ!âë©á 8ãVßÜ éÝ$éÞ$éâjÝ8ã©êé å ã¨ú0ë©ãVßÞè!ãVé å

¬

Page 4: Í Î4Ï>Ð Ñ - polytechniquedubos/Men/Omrani09.pdfÍ Î4Ï>Ð Ñ ÒÓÑÔ Õ Î.ÖX× ØÙÑ5Ô Ú>Û Ü[Ý Þ ß à á!ârÝ8ã©ßZÜ ä åçæ èfà!Þ é à áÌÝ Þ èrêfèsã©ë

Í Î4Ï>ÐÑ ÒÓÑÔ ÓÙXÑÔ

¬ ³ ª&Qb~dfvpb9xQb9lzstj~dxQbrgemhnplnmhxfgrqfqfdstre-vwb;~gQxQ8vwb ¦ ;;;;;;;;;;;;; ®¬ ¬ ª&Qb~dfvpb9xQb9l<gufeoZb;x¤ qfb8eo|nwb²fvt dZbqmhb9xQb;vptdstremhnwbvpnwqs8tjnpehblgreheobcdZgrqxC

λ;;;;;;; ³8­

ós ³ *gQxQvpb;#xQbuQyl<grulzfb8c ;;;;;;;;;;;;;;;;;;;;;;; ³ óós ¬ tjehnptjmonpgrqSxfb

λbqy¨grql1monpgrqxQb

κb<mβ;;;;;;;;;;;;;;;;;;; ³ÿ

ós ó %ntj|ehtr~bb<Qdfvpnpl8tmhn&xQb;vt ~mozfgQxQb9xQu ~gxfvpbdtjeo©tjnwm"!@;;;;;;;;;;;; ³ós ü ª3qfbeh|rnpb;xQb;vE ltjeom-xQb;vpt ¨gqsl1mhnwgqxfb9l<grusehtrqmdstre-ehtrdfdgem-tjulzstrdc-xQb;eo¨ehbqslb ;;;;; ³ós ÿ ª3qfbeh|rnpbCxQbSvV 8l8tjeomxQbSvpt*¨gl<monpgrqHxfbSl<gufetjqm(dstre(etjdfdZgjm~*vpteo¨ehbqslb,dZgrufe~xQnw¡Z8eob8qmc;mhb~dsc~xQb

|rufnxftj|bqsgreh~trvwncodstre-vwbmhb~dscxfbdseo8nconwfnpvpnmhrfvt~l<gruseoZbbq<mhgrnpvwbc-l<grehehb8codgqsx,τ = 0, 5Tp

;; ³ íós ï npvwmoehbdtrchcbstc-xftjqscvE b8codstrlbcodZb8l1mhehtrv ;;;;;;;;;;;;;;;;;;; ³ ðós í zstj~dscxfb9rgemhnplnmhdZgjmhbqmonpbvpvpb9dgufevt#l<guslzfb9coufd8eonpbuseob9t6b8l

α = 1/3Qvt~dfeob8~nw8eob;vpnw|qfb tsf

l8l<geoehb8codZgrqsxS~vV <mtm%xQb;eh<¨8eob8qsl<bQvpt#xQb8uQQnw8~b x¤fbQh-truQylztj~dscÄsvwmoeh8c-bmvpt mheognpconw8~b |sfzntjuQ,lzstr~dsc$|ufnxQ8c8vpb8clgrvpgrqfqfbc$lgreheobcdZgrqsxfbqm$ehb8codbl1monprb8~bqm

τ = 0, 01Tpτ = 0, 4T p

τ = Tp

³ ðós ð zstj~dxQb,greomonl<nwmo#dZgrufe(vptlgruslzsb,cufdZehnpbufehb

α = 1/3τ = 0, 4T p

vwbc9dstrqfqfbtjuQ t;b<m l<grehehb8codgqsxQbqmtruQlzstrdcxQbeh<¨ehbql<brZvpb8cdstjqsqfb8truQ l8bm xstjuQlzstr~dsc%|ufnxQ bm bb<m ~vpt,xfn¡[ehbqslb;bqmoehb9vpb8cxfbuQ*lzstj~dsc8fvpb8c%dstjqfqsb8tjuf t1 l8-b<m b-lgreheobcdZgrqxQbqmtjuQlzstj~dscxQbdbmonwmob;8lzsbvpvwbb<m 1 xsbm htjuQSlztj~dsc-xQb;vt|retjqsxQb8lzsbvpvwb.;;;;;;;;;;; ³8®

ós ® # ustr|rb;xQbdgnwqmdZgrufe-vpb9lzstrdxQb;greomonl<nwmo α = 1/3

τ = 0, 5Tp

;;;;;;;;;; ¬j­ós ³­ at#dZbqmob

abm%vpb l<gb%$,lnwb8qmxQb(l<geoehvtmhnwgq

rb8q*¨grqsl<monpgrqxQumob~dscxQb(|ufnxftj|b

τn = τTp

qfgeo#tjvpncdgufe-vptdZb<mhnmhb tb<mZb<mvt |ehtrqsxQb8lzfb8vwvpb lbm%xsdgufe%xQnw¡[ehbqmobc6trvwb8ufecxfb

α;;;;;; ¬Q³

ü ³ *gQxQvpb;#tjnpeobvwnp~nmhb;np(fehnusxstjqscufq~gQxQvpb;|rvpgrtjvÆ;;;;;;;;;;;;; ¬r¬ü ¬ %ntj|ehtr~bb<Qdfvpnpl8tmhn&xQb;vt ~mozfgQxQb9xQu~gQxQvpbdstjeo©tjnwmtjdfdsvwnufb;tju*a ± ;;;;;;; ¬ óü ó ª&Qb~dfvpb9xQb;vV b<¡[b<m%xfb(nwfc-tjugehxsc-xQuya ± ~vE nwqmo8eonpbusexQu*ltrxfeobqfgrnpe1 ;;;;;;; ¬üü ü zstj~dSxfbgreomonl<nwmo%dZgrusevt(lgruslzsbcoufdZehnwb8ufeob

α = 1/2τn = 0, 4

%vpt(dfeob8~nw8eobvwnp|rqsbl<geoehb8codZgrqsxtjuQClzstr~dscxfbeh<¨8eob8qsl<b tsQQvtcbl<grqxQbtruQ,lzstj~dsc|ufnpxf ljfxsb<mvt xQb8eoqsnw8eob(vtxQn¡[ehbql<bbqmoehbvpb8cxQb8uQClzstrdc brh<fvpt dfehb~npehbl<gvwgqfqfbl<geoehb8codgqsx# vpt dbmonwmob8lzsbvpvwbb<m-vt cob8lgrqsxQb(vt|retjqsxQb8lzsbvpvwb ;;;;;;;;;;;;;;;;;;;;;;;;;; ¬rÿ

ü ÿ # ustr|rbCxQbSdgnwqmc dZgrnpqmhc qfgrnpechbmxQehgrnwmobyx¤ tikuscmob8b8qm b8qeoguf|rb6dZgrufe vptdbmonwmob tf0ljb9b<m vt|retjqsxQbS8lzfb8vwvpb 3x¤ht6rblCxQnw¡[ehbqmhc mob8~dsc~xQbS|ufnxftj|b

τ = 0.01Tp ts01

τ = 0.4Tp lr&xs

b<mτ = Tp

brfh>;;;;;;;;;;;;;;;;;;;;;;;;;; ¬&ü at~dZbqmob

ab<mvwb(lgb%$,l<npbqm%xQb(lgreheo8vptjmonpgrq

rdgufe%vt~dbmonwmob tfbmvt~|retjqsxQb;lzfbvpvpb lj[xsdZgruse

xQn¡[ehbqmhb8cα;;;;;;;;;;;;;;;;;;;;;;;;;;; ¬ í

ó

Page 5: Í Î4Ï>Ð Ñ - polytechniquedubos/Men/Omrani09.pdfÍ Î4Ï>Ð Ñ ÒÓÑÔ Õ Î.ÖX× ØÙÑ5Ô Ú>Û Ü[Ý Þ ß à á!ârÝ8ã©ßZÜ ä åçæ èfà!Þ é à áÌÝ Þ èrêfèsã©ë

' ×-ÔHÖÑ ÒÓÑ5Ô ÖXÎûÏ>ÐÑÎ)(

ós ³ tjvpbufecdgcoconpfvwbcdgufe-vpbdtjetj~<mheobβ;;;;;;;;;;;;;;;;;;; ³¬

ós ¬ tjvpbufecdgcoconpfvwbc-xQuydstjetj~<mheobκ;;;;;;;;;;;;;;;;;;;; ³ü

ós ó ¢£qsl<b8emhnmhusxQb;ehbvtmonprbvpttjehntmonpgrqxQbvE bdZgchtjqmxQb9a¯tjdsufqfg#bqS¨grql1monpgrq*xQbβb<mκ;;;;; ³ü

ós ü tjehnptjmonpgrqSxfb;vV b<QdZgchtjqmxQb;a¯trdfufqfg#b8qy¨grqsl<monpgrqxQbβ;;;;;;;;;;;;; ³ü

ós ÿ tjehnptjmonpgrqSxfb;vV b<QdZgchtjqmxQb;a¯trdfufqfg#b8qy¨grqsl<monpgrqxQbκ;;;;;;;;;;;;; ³ÿ

ü ³ tjehnptjmonpgrqSxfuSmhb~dscxfb;dfeo8npconwsnwvpnmhTpbqS¨grqsl<monpgrq*xQbvptmtjnpvwvpb;xQu*a ± ;;;;;;;; ¬ ó

ü

Page 6: Í Î4Ï>Ð Ñ - polytechniquedubos/Men/Omrani09.pdfÍ Î4Ï>Ð Ñ ÒÓÑÔ Õ Î.ÖX× ØÙÑ5Ô Ú>Û Ü[Ý Þ ß à á!ârÝ8ã©ßZÜ ä åçæ èfà!Þ é à áÌÝ Þ èrêfèsã©ë

* +-,/.10325476 8

9;: ÖÙ=< Ò>>?ôÖX×@< :

a!t;l<g ufqstruQmo-chl<npbqmonwÄufbeobl<gqfqst6m&vwb-eh¶rvpb#tikbuse3xQb8c&~gQxQ8vwbc&xQbl<npehlufvtmonpgrq|r8qfetjvpb vpgrstrvvwnp#tmobgxfbv93tju~qfnprbtju#xQb-qfgrmoehbl<gdseo8zfbqsconpgrq#xQb8c3dfeogQlb8chcusc0dsz¯Qconpufbc&ehb8codgqscotrfvpb8c&xQb-vE 8rgvwuQmhnwgq~b<mxQb%vpttjehntjfnpvwnwmoxfuco¯Qckmh~blvwnp#tmhnpufb # 8trqf~grnpqsc86ltruscobxQbvpbufe&lgr~dfvpb<Qnmhbm3xQb8cvpgrqf|ufb8cdZehnwgQxQbc!qs8l<bcochtjnpehb8c%vtconp(ufvtmhnwgqxfbco¯Qckmh~b8c0lvwnp#tmonufb8c8l<bc0gQxQ8vwbc0cgqmmoeh8cbnp|rbtjqmhcb8q(ehb8chcgufehlb8c!nwqf¨greh~tjmonufb8c88°0gufe&l<b<momobetjncgq6vwbc09xfgrnprbqmb~dfvpg¯rbexQb8c#tjnpvpvwbc3mheoc$|ehtrqsxQb8c xQbvE gehxQehb%xQbdfvpusconwb8ufehcl<b8qmhtjnpqfbc$xQb#<Ae8vV <mhusxQbxQb8cnpdtrl1mcxfb8clztjqf|b~bqmhcl<vpnp~tjmonufb8ctjdsdfeo8zfbqxQ8ccusevE bqnwehgrqfqsb~bqm8vt cogQl<np<mob<m-vE l<grqsgr~nwbqf8lb8chcnwmob8qmxQbvE npqQ¨greh#tmhnwgqdsvwusc%xQ<mtjnpvwvpb9ufqfblzfbvpvwb;dsvwuscÄqfb9ufb;qfbvwb;dZbeh~b<momobqmvpb8c9/ab8c~gQxQvpb8c;xfu:lvwnp#tm;eo8|rnpgrqstrv -b8|rnpgrqstrv3vpnp~tjmob#*gQxQb8vB-yzstruQmob~eh8cogrvpuQmonpgrq:l<gqsckmhnmhufbqm9xQgqslufqguQmonpvbcocobqmhnwb8vdZgrusevt~gQxQ8vwncotjmonpgrqxQu lvwnp#tm8bc~gQxQ8vwbc!dZbusrbqm0§<mheobcgnm0|rvpgrtjuQ¤6eh8cogrvpuQmonpgrq9treontjfvpb$dZbeh~b<momhtjqmxfb~npbuQÅxQ8leonpehb(ufqfb#eh|nwgqdstremhnplufvwnpehb~xQb~vptydfvtjqfmobcgnm;xQgr#tjnpqfbvwnp~nmh anw~nwmob8x ± eobtS*gQxQb8v&bqÅtjqf|vptrnpca ± yxQgqmyvpb8cconw ufvptjmonpgrqscydfeh8cobqmhbqmSvt cogrvpuQmhnwgqXx¤ ufqKdfeogfvp~b:truQ̨eogqmonpehb8c Zgrufqsxstjeh¯¥Àtjvpufb:dfeogfvpb,1 tjqclb(ltcvwbcgrqxQnmhnwgqsc%truQ*anpnwmobca!tmhetjvpb8c a!a!%cgqmxQehnwbc#dstremhnwexQb8clzstj~dscxQb;¨greDCtr|rb9~©trnwsvwbeh8cogrvpuQmhnwgqydseogbqstrqmcogrnwmx¤ tjqstrvw¯Qcob8c-~mogeogvwg|rnufb8cgruxfb(cnp(usvptjmonpgrqscxfb 9/a$ trdfdfehgQlzfb;vpt,dfvwuc%l<g~(ufqfbxfu,moetjnwmob8~bqmxQbca!ab8cmvt;ehbvtftmhnwgq,xQb8c$tjehnptrfvwbc$xQuE- rbec3vpb8clzstrdc$xQb%¨greDCtj|bxftjqc3ufqfb¾gqfbvpb%vpgrqf|xfb~cob8c¨ehgrqmhnw8eobc%vtmhetjvpb8c8Zchtjqsceh<mheogtrl1mhnwgqscxQu~gQxQ8vwb~rb8ehcvpb8c;lzstrdcxfb ¨geFC8tj|b gqfb½t6¯*qfbckmhnwqs| t6nwbc³® í 1&b<momobStrdfdfehgQlzfb,b8cmtruschcntjdfdZbvpbSvptxfb8chl<bqmhbCbqH8lzfb8vwvpbSxQ¯qstr~npufb,dtjel<bCufbSvwbcG7cogrqm coufdfdZgco8c§moehbltrdstjsvwbcCxQb|qf8eob8eCxQbvt/tjehntjfnpvwnwmoxQbdZb<monwmob:8lzsbvpvwbufnq b8cmCdstrcCeocgvwusb*dstjeSvwbcClzstrdcCxQb*¨geFC8tj|b atrdfeoncobCb<m#tjvEp ¬r­r­r­ IH %byª3vpnptb<m#tjvEp ¬r­r­¬ 1a3 grQikbl1monwx¤ b<¡[b8l<moufb8e vt:xQb8chl<b8qmhbCb8q8lzfb8vwvpbxQ¯qstj~nusbCdstre vpb8c5bckmxfbl<ehb8e&ufqfb6treontjsnwvpnmhcdtmonpgj¥Emob8~dgeob8vwvpbxQb-Äsqsb8lzfb8vwvpb%ufnq bQnpcmobdtrc3xftrqsc3vpb8c$lzstj~dscxQb-¨greDCtr|rb-xQb©trnwsvwb eocgvwuQmhnwgqs¿!gruQmhb<¨gnpc8snpvbckm%snwb8ql<grqsquu ufqco¯Qckmh~b9xQ¯qstrnufb;mhbv0usb9vwb c¯Qcmo~b(lvwnp#tmhnpufbseh|rndstjexfb8c 8ustmhnwgqsc9qsgrqQ¥Àvwnpqftjnpeobc9xQ<mhbeh~nwqfncmob8c8dZbufm dfeh8cobqmob8e(ufq l<gr~dZgreomob8b8qm(behehtjmonufb#grulzstrgjmhnpufb!x¤ nwqfÄs~b8clztjqf|b~bqmhc$xftjqsc$vV <mtmnwqfnwmontjvZxQuCco¯Qckmh~b-bqmoet6 qstrqm3dtje$vt9coufnwmob%xQbc$lzstrqf|rb8~bqmhc&np~dgemtjqmhc$b<mtrdfdstreob8~b8qmnp~dfehnpconpfvwbc~xftjqsc~cogrq rgvwufmonpgrq/mhb~dgeob8vwvpbrKJ%qHmhbvl<g~dgemhb~bqmlztjgjmhnpufbSdb8uQm,vpnw~nwmobe~vtdseo8nconwfnpvpnmhyxQuco¯Qckmh~b9l<vpnp~tjmonufbb<m~b<mb8qufbckmhnwgqSvE ufqfnl<nwmo9xQb;vt~cgvwuQmhnwgqyx ufqsb9cnp(ufvtmhnwgqSeh|nwgqstjvpbr°$vwusconpbufec-<mhusxQbc-xQb;dfehnpconpfnwvpnwmo;xftjqc-vwbc-a ± ~gqmheob8qm-ufbvpb8c-xQnw¡[ehbqslb8c-b8qmheob;xQb8uQSconw ufvtmonpgrqcdfnwvpgjmhbcdstjevpb8c(~§~b8c a!a b<m(dstjeomonpe9xQb8c l<gqsxQnwmonpgrqsc;npqfnwmontjvpb8c ¢vp|ehb~bqm;db8emhufeh8b8cmhbqsxQb8qm xQbc;tjvpbufec;Zb8tjul<grusddsvwuscdZb<monwmobcufb l<bvpvpb8c%ufnltrehtl1mo8eoncobqmvwbc9/¤b8c%6trvwb8ufec%xQ8db8qsxQbqmxQb9vt~mhtrnwvpvwb xQuxQg~trnwqsbr[bl<nt#~bqf± qmozfbcb<mtjvE k³8® ð ÿ ,Ël<grql<vpufeob:ufbvpb8ca ± cgqm*ltrehtl1mhehnpco8cCdstjeyufqfb Ç1dseo8nconwfnpvpnmh<mhbqsxfufb Ç b<mob8qsxQb8xdseobxQnpl<mhtrfnwvpnwmk¯Q<& trqscdfvwucnpbufec <mouxQb8c83l<bSdfzfqsgr~qfbyb8cm#tmmheonpfufvV trxQb8l<monpgrqHxQbyvE npqQ¨greh#tmhnwgqHxQb8c|ehtrqsxQb8clzfbvpvpb8c3np~dgc8btruQ ¨ehgrqmhnw8eobc&rb8ehc$vV nwqmhehnwb8ufexQuSxQgr#tjnpqfb ufnplbnltjqsx,°0trb|vwb ³8® ð ® 1 ufnplbnpltjqsx~ª$eoehnplgs³®r®­ 1sb8db8qsxftjqmQvV tjqstrvw¯QcobxQbvt~l<ehgrncochtjqslbxQbvE b8eoehbusecoufe-vt(|ehtrqsxQb8lzfb8vwvpbxstjqscvpb8c-a ± gqmoehbufbvE b8eoehbufelgrqmhtr~nwqfb ehtrdfnxQb~bqmvwbc%dZb<mhnmhb8c8lzsbvpvwbc atjdseoncb b<m;tjvEw ¬j­­r­ 1 %b~ª3vpntCbmtrvVp ¬r­r­¬ 1¤b8lnnp~dfvpnpufb~ufb(vt:Çdseo8nconwfnpvpnmh;<mob8qsxQufbÇdb8uQm§<mheobvt dseogdfeonp<mhbfl<vpuscnprb;xQbc|retjqsxfb8c8lzfb8vwvpb8c8¢£v¤©tjufm%tjuschcn[qfgrmob8eusbvt~l<gqsl<vpusconwgq~b8qmhnwgqfqf8bl<nw¥ÀxQbcocouscqfbc8 trdfdfvpnpufbdstc~qf8lb8chcotrnwehb~b8qm#tjuf ckmtmhnpcmonufb8c#chtjncgqfqfnpehb8cxQb8c,6treontjsvwbcxfb8cL/ bqsnpc(b<m(tjvE E¬j­r­¬ kgrqm~gqmheo,u ufqfb,conp(ufvtmhnwgqÅxfb -yzstruQmob,eh8cogrvpuQmhnwgq¨gehlb~dstje;usqfb,tjuQmheob,conp(ufvtmhnwgqdseogbqstrqm&xQu~~§~b-~gQxQvpbcuse3ufq#xQgr#trnwqfbdfvpusc&|ehtrqsx¤jxQgqm0vpb8c3dbmonwmob8c38lzfb8vwvpb8c0mhtjnpbqm&Äsvwmoehbcjt6trnm3fnpbq~ehb8leo8vpb8ccmhtmhnpcmonufb8c$~bqscoufbvpvpb8c3gu,xQb%dsvwusc$vwgqf|rbc3lzfbvpvwbmob~dZgrehbvpvwbxQb%vt9conw ufvptjmonpgrq#uQmhnwvpnc8bdgufe$vpb%dsnwvpgjmtj|rb l<g~bvt;~g¯rb8qfqfb-~bqcufb8vwvpbbm$vpttreontjqslbmob8dZgrehbvpvpb1attjehntjfnpvwnwmo-npqmob8eoqfbxftjqsc3vwbc$a ± t9l<g~b8qsl<§moehb<mhusxQnpbeh8lb~~bqm8Qnpgreh|rnsbmtrvV V¬r­r­r­ £gqm3b¡Zbl1mhufxQbcconp(ufvtmhnwgqsc3nxQbqmhnpufbcxQba ± /dZgruselzstl<ufqsb%xQb8custmheobcotrnpcogrqscxfb,vV tjqsqfba3 bqscob fvpb~tmo#|qf8eo~bqÅnp~dgcotrqm(xfb8c;dZbeomoufehstjmonpgrqsc;xQbc9xQnw¡Z8eob8qmob8c(tj~dfvpnmhusxQb8c(tjuQ¢guÅtruQa!a3at;tjehntjfnpvwnwmo-npqmob8eoqfbx¤ usqfb%conw ufvptjmonpgrq#dZbeomoufehZb%tmo86trvwusb%lgr~~bvpt~g¯bqfqfbcodstjmontjvpb%xQbvt;xQnw¡Z8eob8qsl<bustxQetmonufb-bqmheoblb<mmhbxQb8eoqsnw8eob-bm$vpt;conp(ufvtmhnwgq~xfbstrcob%lgreheobcdZgrqsxstjqmobufnsq <mtjnwm$dstc3dZbeomoufeh8brab8c3eocufvwmhtjmhcxfbl<bmmhb%<mhusxQbgqmoehbqm$ufb%vt;tjehnptrfnpvwnwmo-npqmobehqfbqfb%xQ8db8qsx#qfn[xQbvt(cogrufel<b-qsnxQbvV tj~dfvpnmhusxQbxQb8c$db8emhufeotmonpgrqcl8tjexftrqscmoguscvpb8cl8trcbvpvpbt6trnm-vt ~§8b;8rgvwuQmhnwgqCmhb~dZgrehbvpvwbb<mvt~§~b;nwqmhbqsconmhr

ÿ

Page 7: Í Î4Ï>Ð Ñ - polytechniquedubos/Men/Omrani09.pdfÍ Î4Ï>Ð Ñ ÒÓÑÔ Õ Î.ÖX× ØÙÑ5Ô Ú>Û Ü[Ý Þ ß à á!ârÝ8ã©ßZÜ ä åçæ èfà!Þ é à áÌÝ Þ èrêfèsã©ë

MNPOPQBRTSUWVYX[ZER]\GSU_^@`EaMWS RT^@\

a!tmtjnpvwvpbSxfu xQg~trnwqsbSxQbca ± ònwqQþufbqslbytruschcnvwb8ufe6treontjsnwvpnmhCnpqmobehqfb ± vwbQtrqsxQehub<m#tjvE V¬r­r­ írgrqm<mouxQnwvpb8cb8qscob(svwbcSxQbcnp(ufvtmhnwgqscCx ufqXa ± b¡Zbl1mhufbc,dZgrufeydfvpuscnpbusehcSxQgr#tjnpqfbcCxQbcCxQnw¡Z8eob8qmob8cCmhtrnwvpvpb8c8l<gufehtrqmCvptl¶jmhb*ª$ckmCxfb*vV ± ~ehnusbxQu # grex¤$ab8c#lzstrqf|rb8~bqmhc#xQb*vt:mhtrnwvpvpbxfu xQgr#tjnpqfbgrqm,lzstrqf|r*l<gqsconpxQ8ehtrfvpb~bqm#vptxfnpcmoehnwsuQmonpgrq9codstmhnptrvwb0bmvV tj~dfvpnwmousxQbxQb&vpttreontjfnpvpnmh&nwqmob8eoqsbrTJ%qfb$tjus|r~bqmhtjmonpgrqxQb&vtmhtjnpvpvwb$xfuxfgr#tjnpqfb3t-tjus|r~bqmo|qf8ehtrvwb8~bqm%vt,tjehnptrfnpvwnwmo9npqmob8eoqfb[bmmob<mhusxQbt,~grqmoeh(truscoconvtCxfdZbqsxftrqsl<b xQb chtCxQncmoehnwfufmonpgrq|gr|ehtrdfzfnufbxfb;6treontjsvwb;conw ufvw8bra$ z¯dZgjmozs8cob~xfb~lbmoet66trnwv0bckm9ufb~vwb~¨geFC8tj|b(b<¡[b8l<mouf#usqfnpufb8~bqm9cufe;vpb8c¨ehgrqmhnw8eobc%vtmhetjvpb8cq b8cmdtrc;cou$,cotrqmdZgruse$dfehgQxQufnpeob-xQbc$conw ufvptjmonpgrqsc0eh8trvwncmob8c0dtje3vpb8cB l8tje&dfvpusconwb8ufehc3<mouxQb8c$grqm3~grqmoeh-usbvV 8lgrufvpb~b8qm$xfb|retjqsxfblzfbvpvpbxQb8c5-c#db8uQmyxfnwbeh|rbe#xQbl<bvpufnnw~dZgco*dstje,vpb¨greDCtr|rb*tjuÌZgruQmyx¤ usql<b8emtjnpqHmhb~dscCtru l<gufehc#xfbvptconp(ufvtmhnwgq ± vpb<ftjqsxfeouËb<m,tjvEp ¬j­r­ íj< # grusc~dseogdgcgqsc xfgrqslyufq˨greDCtr|rbycufe~mhgruQm#vpb*xQgr#tjnpqfb*xQuÌa ± t6b8lSvptmhb8lzfqsnpufb;xQb|rufnxftr|rbrª3qyb<¡[b<mflj b8cmufqsb%mhb8lzfqfnufb;ufn¤l<grqcnckmhb(ehtrdfdZbvpbevwb~gQxQvpbbec$vpb8clzstr~dscxQb%¨geFC8tj|bdZgruselbeomhtrnwqscdstrehtrmoehb8c-dZgruseu npvqsb c8 ltjeomob dstrc%moehgrdxQb8clzstrdc%np~dgcctruQ¨eogqmhnw8eobcxfuxQg~trnwqsb xQbvpteh|nwgqCconw ufvpbr[«Qtjvtj~b8zb<mtjvE E¬j­r­® gqm<mhusxQnp(vE np~dstrl<mxQu|rufnxftj|b;|retjqsxfb98lzfb8vwvpb coufevptCeob8dcb8qmtmhnwgqxfb8cdZb<mhnmhb8c~lzfbvpvwbc vV tjnxQbx¤ ufqH~gQxQvpbyvwnpqf8trnwehbStrqstjvp¯monufb qfbSehbdseocb8qmhb,dstc(vpbyl8tjetrl<moehbClzstrgjmhnpufbyxQbSco¯mo~blvwnp#tmhnpufb6y$ab*dfehnpqsl<npdstjvpbeocusvmtm,xQbl<bmmhb*<mouxQb*b8cm,vptÅxQmob8eo~npqstmhnwgqËx¤ ufqËmob8~dsc,xQb*|ufnpxstj|rbgdQmonp#tjvufn~npqfnpncob;vV beheob8ufexQbc-lzstj~dsc|ehtrqsxQb8lzfb8vwvpbb<mdZbehbmvptdfehgQxQusl<monpgrqxQb9lzstj~dscÄqfb;8lzfb8vwvpbdfvpusceotjvpnpcmobcab~dfeocb8qmmheht6trnwv&tjdsdfeogr¨grqsxfnmvV <mhusxQb#xQb~vE b¡Zbm(xQb~|rufnxftj|bcoufevtStreontjfnpvpnmh npqmobehqfb#xftrqscvwbcb-ýa$ npqmoeh§<mdseonpqsl<npdstrv&b8cmvtyltjdtrl<nwmo~xQbc_-ûxQb~dfehgxfufnwehb~xQb8cdZb<mhnmhb8c;lzfbvpvwbcl<grehehb8l1mhb8cbq¨grqsl<monpgrq:xQumhb~dsc9xfb|ufnxftj|btrdfdfvpnpufa3 <mhusxQb,bckm9trcob#cuse9ufqÅbqscob fvpb,xQb,cnp(usvptjmonpgrqsc8¤b<¡[b8l<mouf8bCt6rbl(ufqsb,~gxfvpb,utrcon¥À|r8gcmoehgrdfzfnufb xfbuQ#l<guslzfb8c8ufq#~gQxQvpb%conw~dfvpb-#tjnc3cou$,cotr~b8qm3qsgrq#vpnwqftjnpeobdZgrufe$eob8dfeocb8qmobe&usqfbxQ¯qstj~nufb%tmh~gcodfzfehnusbeh8trvwncmobr

Page 8: Í Î4Ï>Ð Ñ - polytechniquedubos/Men/Omrani09.pdfÍ Î4Ï>Ð Ñ ÒÓÑÔ Õ Î.ÖX× ØÙÑ5Ô Ú>Û Ü[Ý Þ ß à á!ârÝ8ã©ßZÜ ä åçæ èfà!Þ é à áÌÝ Þ èrêfèsã©ë

* +-,/.10325476 c

d ÎûÒÓÙXÑ Ò> ÖÙXÎfeÎ ×-Ð

a$ tjmo~gcodfzf8eobb8cmb<moeh§~b~b8qmtjehnptrfvpb xftrqscvV b8codstrlb bm9xftjqscvpb(mhb~dsc8!bdZbqsxstjqm8¤cogruscl<bmmhbtreontjfnpvpnmhrZbvpvwbehvpb%ufqsbcmhtrfnwvpnwmoxQb;l<gr~dZgreomob8b8qmufn¤cb#tjqfnw¨b8cmobdstrevE bQnpcmob8qsl<bxQb8cl<vpnp~tjmhc8Qab8c|ehtrqsxfc&mhehtrnmcdb8eo#tjqsbqmhcxfuCl<gdZgreomob8~bqmxQbvE tmo~gcdfzsehbrjb8q,dstjeomonl<usvwnpbe$xfb8c3~gufrb8b8qmhc$tjmo~gcodfzf8eonufb8c8rlgrqscmonwmoufb8qml<b%u grq,trdfdZbvpvwbvt#l<npel<ufvtmhnwgqC|r8qfetjvpbxQb;vV tmh~gcodfzfehbrb8vwvpb;l<nxfgrnwm§moehb;xQ8leonwmob;bmb<Qdfvpnpuf8br b8cm(°3zsnwvpvwnpdsc k³8®ÿ& kufn&tdfehgxfufnmb8q ³®ÿ[ vtdfehb~nw8eob#conw ufvtmonpgrqÅvpgrqf|ufb#xQufehb,xQb#vt*l<npel<ufvtmhnwgq:|qf8ehtrvwb~xQbvE tmo~gcdfzsehbrQ¢£v¤uQmhnwvpnpcob;dZgrufelb©tjnpeobufq~gxfvpb9utrcon¥À|r8gcmoehgrdfzfnufb%~xQb8uQSl<guslzfbcfxftrqscufq §mht¥Àdfvtjqyltjqtjvg!t6b8lufqsbdfz¯Qcnufb-eouxQnw~b8qmtjnpeobj°3zfnpvpvwnpdsc3bftj~nwqsbvwbc&8lztjqf|b8c&x¤ qfb8eo|nwbxfbvV grqxQbbq,xfrb8vwgdfdZb~bqm3b<ml<grqckmtmobusqyZgrq*tll<gehxSustjvpnmtmhn&t6rbl%vpb8cgscb8eotjmonpgrqscxQbc-c¯Qcmo~bcstjehgQl<npvwnpqfb8c-xstjqscvE tjmo~gcodfzf8eobabC~gQxQvpbCustcnw¥E|gckmheogdfzfnusblgrqscmonwmoufbytjnpqsconV!ufqZgrq/grufmonpvdZgrufe vwbc9mousxQbc(xQbycb8qsconwfnpvpnmhCltre(npvt*vE t6trqmtj|bx §moehb9cnp~dfvwbmhgruQm-b8qymhtrqm-eh8trvwnckmhbr

hLiTj klLmonWp%qsr _S»9^utFvw9] l ^¹º lyxoz t r _ q|~fs tjqc!lb&moet6tjnpvrqfguscmousxQnpgrqcvtxQ¯qstj~nufb$tmh~gcodfzfehnusb$xftjqcvt¨greh(ufvtmhnwgq;utrcon¥À|r8gcmoehgrdfzfnufbr<a!tcmoehusl1mhufeobbeomonltrvwbxQb-vE tjmo~gcodfzf8eobb8cm&gQxQ8vwncobbq#xQb8uQ l<guslzfbcabc08ustjmonpgrqsc0xQb-vtxQ¯qstj~nusbcogrqm08leonwmobc0bq~cbstcotrqmcoufe-vE tremhnplvwb_tjnxQrg|rbv¤trqsx7bvx k³® ð ­ 1

WK @ ~]a!t¨greh~b%xQnp~bqsconwgqfqfb8vwvpbxQb8c$8ustjmonpgrqsc3xfb%~grufb~bqm3dZgrufeufq#~gQxQvpb%ustrconw¥E|gcmoehgrdszfnpufb9xQb8uQ#l<grulzfb8c3xstjqscvpb;dfvtjq

βc 8leonwm;²

∂Q1

∂t+ J(Ψ1, Q1) = −υ56 Ψ1

V¬ ³

∂Q2

∂t+ J(Ψ2, Q2) = −υ56 Ψ2 − κ56 Ψ2

V¬ ¬ ab8cnpqsxQnl<bc ³ bm ¬ ehbdfeh8cobqmob8qmehb8codbl1monprb8~bqmvtyl<grulzfbcoufdZehnwb8ufehb bmvtyl<grulzfb npqQ¨8eonpbufehb~xQu~gQxQ8vwb

ΨiQicogrqm#vt¨gqsl1mhnwgqËxfbl<grusehtrqm~b<m#vt:rgreomonl<nwmodZgjmhbqmonpbv-dZgrufe#vtÅl<guslzfb

iJvwbtrlgrfnpbq zfgrehnp¾grqmtjv

J(Ψi, Qi) =

(∂Ψi

∂x∂Qi

∂y − ∂Ψi

∂y∂Qi

∂x )b<m

∇2vwb vptrdfvtrl<npbq*zsgrehnw¾8grqmhtrv

∇2Ψi = ∆Ψi = ∂2Ψi

∂x2 + ∂2Ψi

∂y2

[abcxfbuQ*l<guslzfb8c%grqmvt#~§~bztjuQmhbufeWñtjuSeob8dgc

ν<mhtrqmvt xfn¡[usconwgqCb<m

κvpt(¨ehnpl<monpgrqxQbcoufeo©trlbrQabcnwmobcocob8c

(ui, vi)cogrqmeob8vwnpbc vpt(¨grql1monpgrq

xfb9l<grusehtrqmdstre-vVfustmhnwgqyxQntj|qfgcmonufb V¬ ÿ <

Q1 = ∇2Ψ1 + βy +1

2R2

d

(Ψ2 − Ψ1)V¬ ó

Q2 = ∇2Ψ2 + βy +1

2R2

d

(Ψ1 − Ψ2)V¬ ü

í

Page 9: Í Î4Ï>Ð Ñ - polytechniquedubos/Men/Omrani09.pdfÍ Î4Ï>Ð Ñ ÒÓÑÔ Õ Î.ÖX× ØÙÑ5Ô Ú>Û Ü[Ý Þ ß à á!ârÝ8ã©ßZÜ ä åçæ èfà!Þ é à áÌÝ Þ èrêfèsã©ë

ZXZE^@`PVYauOR ¡@¢^WSUb^Q"NRTaV¤£¥b¡¦ MNOQBRTSUV Z§MOP`¨UWV©`5a=SUPOªOR]

(ui, vi) = (−∂Ψi

∂y,∂Ψi

∂x), i = 1, 2

E¬ ÿ

Rd = (g′ H2f2

0

)1/2mhtjqmvwb eht6¯grqxQb xQ¨greh#tmonpgrqxfbG-gcoco¯r

Hvt,ztjuQmhbufetjuehbdZgcxQb(lzstusb(l<guslzfbr

g′ = g4θθ0vt~|ret6nmheh8xfufnmhbr

θvt mob~dZetmhufeobdZgjmhbqmonpbvpvwb

θ0vtmob8dZetmhufehbdZgjmhbqmonpbvpvpb9xQb;eo¨ehbqslbb<m

f0vpb;dstjetj~<mheobxQb

geonpgrvpncfabxfdfvtrlb~bqmxQb;vE npqmobeo©trlbbqmoehbvwbc-xQbuQl<guslzfb8cηb8cm%xQ<Äsqsndstje f0η

H = 1

2R2

d

(Ψ2 − Ψ1)

ab8c8ustjmonpgrqsc E¬ ó$b<m V¬ ü $~grqmoehbqm-ufbvt rgreomonl<nwmodgrmob8qmhnwb8vwvpbb8cmlgrqscobehr8b%b8qScoufnwtrqmvpb%þsufCzfgrehnp¾grqmtjvchtjufdZgruse9vwbc9dfehgQl<bcocousc;xQncoconpdstmhn©c8l<bc;dfehgQl<b8chcousc;tj|npchcb8qm;coufe9vtrgemhnplnmh~eob8vptjmonprb

∆Ψi, i = 1, 2 n¬« ³ ¬ tymheht6becvt

xfn¡[usconwgqυb<mvt(¨ehnl1monpgrqxQb9coufeo©trl<b

κ

«Qufnwtrqm0vE tremhnplvwb %trnpxfrgr|bvtjqsxy%bvx¤ ³8® ð ­ <<qfgusc!lgrqsconxQehgrqscvwbxQbvpgrdfdZb~b8qm0xQb8c!dZbeomoufehstmhnwgqsc!dstje!nwqscmhtrfnpvwnwmotjehglvwnpqfbdstreetjdsdgem&ufq~8l<gufvpb~bqm&xQ<Äsqsnfdstje

U2, V 2, V 1 = 0b<mU1 = U

t6b8lU i, V i

vtnwmobcocob¾gqstjvpb~g¯bqfqfbbmvpt~nwmobcocob8eonxQnpgrqstrvwb~g¯rb8qfqfbr ± nwqcnE

Ψ1(x, y, t) = −Uy + ψ1(x, y, t)E¬

Ψ2(x, y, t) = ψ2(x, y, t)E¬ íj

ψi, i = 1, 2b8cm0vptxQntmhnwgq(dstreetjdsdgem0vtg¯bqfqsb3mob8~dgeob8vwvpbxfbvt%¨grqsl<monpgrq xfblgrufetjqm

Ψi(x, y, t) = Ψi(x, y)+ψi(x, y, t)J%qfb;qfgrmhtmhnwgqyconpnpvtjnpeobb8cmuQmhnwvpnpcobdZgrufe%vptrgemhnplnmhdZgjmhbqmonpbvpvpbr

WK­ ®WB¯°²±³K´Pµ°²±uP]±¶µ¶]±=´P±¸·uP P°0gufeltrehtl1mo8eoncobevV grexQehbxQb;|ehtrqsxQb8ufe-xQb8cdfehnpqsl<npdstjvpb8ctreontjfvpb8cxQ¯qstr~npufbc-xQugQxQ8vwbQvwbctjehnptrfvwbc

xytψ

cogrqmtrxfnw~bqcnpgrqfqsb8cdtje RdRd Rd

U

URd

xbmycgqmvpb8clzfbvpvpb8c-xQb;vpgrqf|bufebqvpgrqf|nmhusxQbb<mb8qvptjmonwmousxQb

± nwqcnEfvpb8c-utmonpgrqc E¬ ³ bm V¬ ¬ xQb8nwb8qfqfbqm;²

∂q1

∂t+ J(ψ1, q1) = −υ56 ψ1 + F1

V¬ ð

∂q2

∂t+ J(ψ2, q2) = −υ56 ψ2 − κ56 ψ2 + F2

V¬ ® ± rblvpt~greomonl<nwmodgrmobqmhnwb8vwvpb

q1 = ∇2ψ1 +1

2(ψ2 − ψ1)

E¬ ³8­

q2 = ∇2ψ2 +1

2(ψ1 − ψ2)

E¬ ³r³

F1 = −∂q1∂x − (β + 1

2)∂ψ1

∂x

b<mF2 = −(β + 1

2)∂ψ2

∂x

mhtjqm-vpb8cmhbehbc-xQb¨greDCtr|rbab8c-dtjetj~<mheobc$qfgeo#trvwnccusntjdfdtjetjncocobqmxstjqsclb8cutmonpgrqcogrqm

β = βR2

d

U

κ = κRd

U

b<mυ = υ

R3

dU°&tre-bcuseob;xQb9conw~dfvpnl<nwmorxftjqscvt~cusnmhb;vwbc6treontjsvwbc-trxQnp~bqschcnpgrqsqfbccobehgrqmqfgjmhbc-cotrqscvwb

ð

Page 10: Í Î4Ï>Ð Ñ - polytechniquedubos/Men/Omrani09.pdfÍ Î4Ï>Ð Ñ ÒÓÑÔ Õ Î.ÖX× ØÙÑ5Ô Ú>Û Ü[Ý Þ ß à á!ârÝ8ã©ßZÜ ä åçæ èfà!Þ é à áÌÝ Þ èrêfèsã©ë

Z Z5Q"UW¢_ªuRR]¹"R]RTS ¢ MNOQBRTSUV Z§MOP`¨UWV©`5a=SUPOªOR]

2 4 6 8 10 12 14 16 18 20 22 24

2

4

6

8

10

12

14

16

18

20

22

24

x

y

−10

−8

−6

−4

−2

0

2

4

6

8

10

º@»¬¼½ ¬ ³¾ ª&Qb~dfvpb9xQb9lzstrdxQbrgemhnplnmh;xQgrqfqs;dstjevpb;gQxQ8vwb ¦ «Qufe$vtÄs|ufeob E¬ ³ 0qsgrusct6rgrqc0eob8dfeocb8qmo-ufq#bb8~dfvwbxQblzstr~d~xfb%rgemhnplnmh-dfehgxfufnmdstje3vpb~gQxQvpbrqfgusc&dZgrufgrqscxfgrqsl rgnweufb vpbgQxQ8vwb~l<ehb~xQbclzstr~dscl<gr~dstrehtrfvpb8c%yl<b~ufn&cbdstcocob xftrqscvptSeotjvpnmh t6b8l9vpb8cmhgrufehfnpvwvpgrqscxQbgreomonl<nwmodZgjmob8qmonpbvpvwb;qf8|tjmonprbb<mdZgconmhnwbxfbmhtjnpvpvwb;l<g~dstjetjfvpbtjueht6¯grqCxfby-gchc¯

hLiFh ¿ º0]@ÀLt<^t ¸ t p t¹!]

W­@ ÁÂÃI±ÅÄÆ´±/ÇPKȱ± l8tjuscobxQbvptqstmhufehbyqsgrqQ¥ÀvwnpqftjnpeobyxQuËco¯Qckmh~b*l<vpnw#tjmonusbr0vwbc~dbmonwmobc,xQnw¡Z8eob8qsl<bcxstjqscvpb8c#l<gqsxQnwmonpgrqscnpqfnwmontjvpb8cc8 tj~dfvpnÄsb8qm~xftrqsc vwbCmob~dscvpgrec(xQbyvE npqmo|ehtjmonpgrqxQu ~gxfvpbytrudgnwqm~xQbylgrqmhtr~nwqfb8e lgr~dfvp<mob8~bqmvtcogrvpuQmhnwgqtru/guQm xQbSusbvusb8cikgufehc xQb,dfehncnpgrq!b<mmhbCvpnw~nwmob,bckmtjdfdZbvpb#vt*dfehncnpfnpvwnwmo,xQu c¯Qcmo~b,lvwnp#tmonufbr!°0grufe~bcufehbe vptdfeo8npconwsnwvpnmhrgrql8tjvl<ufvpb,vpb,mhtruQ/xQbSleognpchcotrqsl<b,xfb,vptxQnw¡Z8eob8qsl<b,npqfnwmontjvpb~bqmdbmonwmobybqmoehbCxQb8uQ/<mhtjmhc ageob8qf¾r ³®[ ó1# gruscdgufrgqscxQ8leonpeobvE 8rgvwuQmhnwgqCmhb~dZgrehbvpvwb;xQb;vE tjmo~gcodfzf8eobdstre-vV 8ustjmonpgrqxQn¡[ehbqmhnwb8vwvpb_É~ ï etjbxQeonlz ³8® ðíj

dt= f (ψ)

V¬ ³6¬ t6b8l

ψufqgrQikbm%ufnxQ8leonwm%l<g~dfvwmob8b8qmvE <mhtjm%xQb;vE tjmo~gcodfzf8eob

«Qn!#tjnpqmob8qstjqmqsgruscl<grqcnxQehgrqc-vV grvpuQmonpgrqxstjqsc%vwb;mhb~dscxQb xfbuQ*<mtmcufqdfehb~npbe%mhtjmxQb eo¨ehbqslbψ0(t)

b<musq:xQb8uQQnw8~bmhtjm

ψ(t)t6rb8l(nwqsnmhnptrvwb8b8qmusqfb dZb<mhnmhb~db8emhufeotmonpgrqdstjeetjdfdZgreom

ψ0(t)a!tSxfn¡[ehbqslb b8qmoehblb8c

xfbuQ#<mtmc&grvpufbxftjqc3vpb-mob8dcδψ = ψ(t) − ψ

0(t)b<mmhtrqm$moeh8c&dZb<mhnmhbrqfgrusc$dgufrgqsc&vpnwqs8tjehncb8e3vV 8ustjmonpgrq V¬ ³6¬

trurgrnconwqstr|rbxQbψ0

mobv!usb²

d

dtδψ =

df

ψ0

δψV¬ ³ ó

®

Page 11: Í Î4Ï>Ð Ñ - polytechniquedubos/Men/Omrani09.pdfÍ Î4Ï>Ð Ñ ÒÓÑÔ Õ Î.ÖX× ØÙÑ5Ô Ú>Û Ü[Ý Þ ß à á!ârÝ8ã©ßZÜ ä åçæ èfà!Þ é à áÌÝ Þ èrêfèsã©ë

Z Ê~Z¡auR]`O_¡@V MNOQBRTSUV Z§MOP`¨UWV©`5a=SUPOªOR]

lb<momob9xQnw¡Z8eob8qsl<b9leogjpm-guyxQl<ehgm-b<Qdgqfbqmhnwb8vwvpb~~bqmt6rblvpbmob8dctcobvpgrqyvE utmonpgrq V¬ ³8ü ²

δψ (t) = δψ0 exp(λt)V¬ ³8ü

λbckm&tjdsdb8vwvV b<QdZgchtjqml8tjetrl<moehnckmhnpufbxQba¯trdfufqfgb<m&ufnfltrehtl1mo8eoncob&vpt~b8coufeobxQbvptdfehncnpfnpvwnwmo # grusc!dZgrufgrqsc

trnwqcnxfnwehb;u ufq*co¯Qckmh~bb8cmdseo8nconwfvpbikuschu ~ufqsblzf8trqsl<bTp = 1

λ

Tp<mhtrqmvwbmob8~dscxQb;dfehncnpfnpvwnwmo

W­@­ ÁÂK¶Ã˶G´P±Ì¶%ͬ±ËÎWÄ_ÈK"´±ÌÏyÐÄPPÑ°0gufe%bckmhnw~bevV b<QdZgchtjqm%xQb(a¯trdfufqfg,qsgrusct6rgrqc-ltjvl<usvw9vE qfb8eo|nwb9xQb(vpt,xfn¡[ehbqslb9bqmheob(xfbuQ<mtmhc8susq*<mhtjmxQbeh<¨8eob8qsl<b;bm%ufq*mhtmdb8emhufeoZrZ trqsc-qfgrmoehb(ltc-qfguscl<gqsconpxQ8eogqscvt¨grqsl<monpgrqxQulgrufetjqm

ψdgufelzstufb;l<grulzfb8c³ ¬ 3qfguscdstjeomogqscxQbxfbuQ,mhtmcnwqsnmhnptruQ,vwbdfehb~npbe

ψ0r1ψ0r2

dfeoncdgufeeo¨ehbqslb%bmvpbxQbuQQnp~bvw8|rehb~b8qmdZbeomouseoZ

ψ0p1ψ0p2

mob8vufbψ0pi = ψ0ri + εi, i = 1, 2

t6b8lεiufq*lzstj~dsctrvwtmognwehbx¤ tr~dfvwnwmouxQb

10−3

# grusc©trnpcogrqcmogruseoqfb8evwb~gQxQvpb;b<m-qfgusc-ltrvplufvwgqscvV qsbeh|rnpbxfbvt~xQnw¡Z8eob8qsl<bbqmheobvpb8c-xQb8uQSlzstrdc-xQbvpt ¨grql1monpgrqxfulgrufetjqm

ψr1 (t)ψp1 (t)

b<m ψr2 (t)

ψp2 (t)

<

E(t) =

‖∇ (ψr1 − ψp1)‖2 + ‖∇ (ψr2 − ψp2)‖

2

2+

(

(ψr1 − ψp1) − (ψr2 − ψp2)

2

)2∥

dxdyV¬ ³6ÿ

°&trevtcusnmhb$qfgusc¤mhehtC<grqc¤vpt%l<gruseoZb3bq;lzfbvpvpb3vwg|treonwmozf~nufbrat-dZbqmobxQb$vt-dstremhnwb3vpnpqf8trnwehb$xQb3vtlgrufehbl<grehehb8codgqsx#xfbuQ,¨gnpcvV b<QdZgchtjqmxfb9a¯trdfufqfg[

0 10 20 30 40 50 60 70 80 90 100

100

t

E(δψ

)

º@»¬¼"½ ¬ ¬y¾ ª&Qb~dfvpb9xQb9lgrufehb;x¤ qsbeh|rnpb²svptdZbqmob9xQbvt~dstjeomonpbvwnpqftjnpeobl<geoehb8codgqsxCλ

hLiÒ _ot m »Ów qa!ttjehntjfnpvwnwmonwqmob8eoqsbx¤ ufq~gQxQ8vwbeo8|rnpgrqstrv~b<mSb8qXufbckmhnwgq vV ufqfnl<nwmoÅxQbchtcogrvpuQmonpgrqJ%qfbdZb<mhnmhb:xQnw¡[ehbqslbrtruQdZb<mhnmhb8c;lzfbvpvpb8c;xQbc9cdZb8l<moehb8c;xQbc9lzstrdc;cnp(ufvp8c8cob#dfeogdstj|b becvwbc;8lzfb8vwvpb8cdsvwusc9|retjqsxQbc[bm(tikgrufmob~ufqfb

³8­

Page 12: Í Î4Ï>Ð Ñ - polytechniquedubos/Men/Omrani09.pdfÍ Î4Ï>Ð Ñ ÒÓÑÔ Õ Î.ÖX× ØÙÑ5Ô Ú>Û Ü[Ý Þ ß à á!ârÝ8ã©ßZÜ ä åçæ èfà!Þ é à áÌÝ Þ èrêfèsã©ë

Z Ê~Z¡auR]`O_¡@V MNOQBRTSUV Z§MOP`¨UWV©`5a=SUPOªOR]

lgr~dZgchtjqmobtjvp8tjmogrnpehbtjucnp|rqtjv¨grel<ab|ufnxftj|b$dZbuQm3§moehbufqsbcogrvpuQmhnwgq(dZgrufe&l<bmmhbvpnpnwmhtjmonpgrq6ª3q~b¡Zbm8vpb|ufnxftj|bbckmufqfbmoblzfqfnusbufn[l<grqcnckmhb%(tjikuscmobevwbc3tjehntjfvpb8cxQ¯qstj~nusbtjuf~trqstjvp¯Qcbc3~<mhgrehgrvpgr|npufbc0gruCtruQ#cgemhnwbc$xQb9°0gufe~ufq lzstr~d xQbydstjetj~moehbSxQgqfqfr0dstre bQb~dfvpbyvt*¨gqsl1mhnwgq xQb*lgrufetjqm

ψ&qfgrusc~lgrqsconpxfbehgrqsc(ufq lzstr~d xQb

eh<¨8eob8qsl<bψref

b<musqlztj~dS|ufnxQψnudge

mob8vufb~²

∂ψnudge∂t

= f (ψnudge) −1

τ(ψnudge − ψref )

E¬ ³ t6b8l

τb8cmvwbmob8~dscxQb;|ufnpxstj|rb

ª$q#b<¡[b<m8cnτbckm&mheoc0dbmonwm$vwbmhbeh~b

1

τ (ψnudge − ψref )xQbnpbqm&moeh8c0|ehtrqsx xfbtjqm3vpbmobeh~b

f (ψnudge)b<m3vE ustmhnwgqE¬ ³ xfbnwb8qm9²

∂ψnudge∂t

= −1

τ(ψnudge − ψref )

E¬ ³ íjlr bckmo¥Àj¥Àxfnwehbqsgrusc©trnpcogrqscusqfb9cnp~dfvpb;eob8vptjQtjmonpgrqyxQuylztj~d|rufnxQrb8ehcvpb9lzstj~dxQb;eo¨ehbqslb «Qnτb8cmmoeh8c|retjqsx¤vpbmhbeh~b

1

τ (ψnudge − ψref )xQbnpbqmmoeh8cdbmonwmxQbtjqmvwbmob8eo~b

f (ψnudge)bm-vV 8ustjmonpgrq E¬ ³

xfbnwb8qm9²

∂ψnudge∂t

= f (ψnudge)E¬ ³ ð

b8cm¥£¥£xQnpeob;ufb;vpb9lzstj~d|rusnpxQ9xfbnwb8qm%l<gr~dfvp<mhb~bqmnpqsxQ8db8qsxftjqmxfulztj~d*xQb;eo¨ehbqslbr b8cml<g~b9congqqsb;|rufnxQbdstrc8 tjqc;l<bmoet6tjnpvV[qfgusc;tjqstrvw¯QcogrqscvE b¡Zbm(xQumhb~dsc9xfb|ufnxftj|bcoufevptytjehntjfnpvwnwmo~npqmobehqfb~xQuÅqfgjmheob~gQxQvpb ¦ ´bqdseogQl<xftjqmufqfb9coehnwb;x¤ b<QdZehnwb8qsl<bcquf~ehnpufbct6b8lxQnw¡Z8eob8qmhcmob8~dscxQb|rusnpxftr|rb

³r³

Page 13: Í Î4Ï>Ð Ñ - polytechniquedubos/Men/Omrani09.pdfÍ Î4Ï>Ð Ñ ÒÓÑÔ Õ Î.ÖX× ØÙÑ5Ô Ú>Û Ü[Ý Þ ß à á!ârÝ8ã©ßZÜ ä åçæ èfà!Þ é à áÌÝ Þ èrêfèsã©ë

* +-,/.10325476 Ô

< ÒÓØ5ÐÑ Õ Ð@<ûÏ>Î Ð

tjqcCl<blzstjdfnwmoehb*qfgusc#mhehtrnmhgrqsc#vpb~gQxQvpbxftrqscCchtÅrb8ehconpgrqË|rvpgrstrvwb # guscCtrvwvpgrqsc,<mouxQnwb8eCvpb8cCdstjetj~<mheonchtmonpgrqcdsz¯Qconpufbcbmvpbufec-npqQþsufb8qsl<bccoufevt#dfehncnpfnpvwnwmo(xfu~gxfvpbr # grusctjvpvwgqsc%truschcnbQdfvwgeob8evE b¡ZbmxQu|ufnxftj|b9cufevptconp(ufvtmhnwgqyxQbc-dZb<monwmobc-8lzsbvpvwbcbqvwnpbqt6rbl%vpbmob8~dscxQb;dfehncnpfnpvwnwmo

Ò5iTj ¿ »;º» ` n ¹º q ^ x7zÖ ^t r _ q ^ m _ ` lLmonWp%qab8cdstrehtrmoehb8cusntjdfdstrehtrnpchcobqm$xftjqcvpb8cutmonpgrqctxQnw~b8qscnpgrqsqfbcxQbqfgrmoehb~gQxQvpbutrcon¥À|r8gcmoehgrdfzfnufb-b<m-ufncogrqm-trchcgQlnwc vptxQbcoleonpdQmonpgrqCdfz¯conusbxQuydfehgrfvp~bcgqm

βb<mκf tjqsclbufn[t~cufnpeobqfgrucqsgruscdfehgrdZgcogrqscxQb

lzsbelzfbeufqfb;|tj~~b;xQb;tjehnptjmonpgrqSxfb9l<b8c-xfbuQCdtjetj~<mheobc

×WK Ø7¯~K°²Ùů±β

ab;dstjetj~moehbβb8cmtrxQnp~bqsconpgrqfqf;coufnp6trqmvV 8ustjmonpgrq óf ³ ²

β = βR2d

U

óf ³ t6b8l

β = 2Ω

R cosφRvpbeht6¯grq xQbvtÅmhbeheob

Ωvptnwmobcocobtrqf|rufvtjnpehbxfbvt/eogrmhtjmonpgrq xQbvtÅmob8eoehbr

φvpt/vtmhnmhusxQb

Rd =(

g′H2f2

0

)1/2 vwbSeht6¯grqxQbyxQ<¨geo#tjmonpgrqxQbo-gchc¯f0vwbydstjetj~moehb,xQbgeonpgrvpnpc

f0 = 2Ω cosφ0

g′vpt|ret6nmh

eh8xfufnmhbg′ = g∆θ

θ0

θvt mob~dZetmhufeobdZgjmhbqmonpbvpvwb

trvwb8ufec-quf~ehnpufbc²Ω = 7, 25.10−5rd.s−1

R = 6380 km

cosφ0 = 0, 1

∆θ = 40K

θ0 = 350K

g = 9, 8m.s−1bm

H = 20km

tjqc-vwbmtjfvpb8tru óf ³ qfgusc-t6rgqscltjvl<usvwvpb8ctjvpbufecdZgchcnpfvpb8c-xQbβbqy¨grql1monpgrq*xQbvtvptjmonwmousxQb

φ

Ú"ÛÜ ½Zóf ³P¾ trvwb8ufehcdZgchconwfvpb8cdZgruse-vwbdstrehtrmoehbβ

φí ÿ r­ üÿ ó ­

β(10−11)­ ÿj®s³ü ³ ³8ü¬& ³ f³ÿr® ³ ® í ®f³

f0(10−4)s−1³ ü­ ðró ³ ¬[¬ í ³ ­ ó ³­ ­ í ¬j®r­

Rd kmí ÿQ³ üÈ ó ðróðf ³ü ð ³­¬& ÿf³ í ³üÿQ³ í ³6ÿ

β(U = 30ms−1)­ ³r³³ ó ­ ¬[ í ÿ ­ ÿ& í ÿ ³ ó ®r­ ó

β(U = 20ms−1)­ ³ í ­ ­ ü­f³ ó ­ ð ÿQ³ ó ¬ ­ ð ÿü

ab#lzfgrnwxQb~vptSvptjmonwmousxfbbckmnpdZgco~dstjevE trdfdfehg6np#tmhnwgqustrcon¥À|r8gcmoehgrdfzsnpufb9ufn&bckmtjvtjfvpbxstjqscvpb8c~g¯rb8qfqfb8cvtmhnmhusxQbc ± nwqcnqsgrusc-t6grqscqfgrmoehb|tj~~b;xQb6treontmhnwgqyxfuydtjetj~<mheob

βusn¤b8cmbqmoehbÝ ­ ³ ¬ßÞ

³¬

Page 14: Í Î4Ï>Ð Ñ - polytechniquedubos/Men/Omrani09.pdfÍ Î4Ï>Ð Ñ ÒÓÑÔ Õ Î.ÖX× ØÙÑ5Ô Ú>Û Ü[Ý Þ ß à á!ârÝ8ã©ßZÜ ä åçæ èfà!Þ é à áÌÝ Þ èrêfèsã©ë

Ê~ZXZ5QËOUOP¨SUWV"7QNàBRTauV"7`Ea¸á^@`P"ÅV MNOQBRTSUWVâÊ~ZE^@`PVY¡W"^@¹@OP

×WK­ Ø7¯~K°²Ùů±κ

ab;dstjetj~moehbκtrxfnw~bqcnpgrqfqscobvpgrqyvE utmonpgrq ós ¬ ²

κ = κRdU2

óf ¬

κmhtrqmvpblgbT$,l<npbqmxfb ¨ehnl1monpgrq:xQb~vptycoufe©tl<busnehbdseocb8qmhb(vpb~dgdtj|rbx¤ ª3~trqbm

U2

vwbrb8qm¾gqstjv&xftrqscvptlgruslzsb~xfb,cufeo©trlbr¤°0gruse(xQ<mhbeh~nwqfb8e;vV grexQehb#xQb#|retjqsxfbufe;xQb

κqfgusc<mhusxQnpgrqsc;vpb#fnpvptrq/x¤ 8qfbeh|rnpb~x¤ usq/gQxQ8vwb,

xfbuQl<grulzfb8cehbdfeh8cobqmhdstrevptÄs|ufehb óf ³

º@»¬¼"½Zóf ³P¾ *gQxQ8vwb9~xQb8uQSlgruslzfbc

Hb8cm;vV dtjncocobufe;xQb~vtCdseob8npehb~l<grulzfb vptystcocob moehgrdZgcodfzfehb<

δb8cm;vV dstrnpchcobufe;xQb~vtSlgruslzfb~x¤ ª3#tjq

Ktvwb

lgb%$,l<npbqm%xQbvt#xQn¡[usconwgqCmhufehfufvpbqmob9tcocogQl<nptruydZgr~dstr|rbx¤ ª3~trqa$ bQdfeobcoconpgrqxQb

κbckm~xQ<mhbehnpqfybqHxfeobcochtjqm(vpbCfnpvtjqHx¤ qsbeh|rnpbrª$qHb<¡[b<m0bqetjncgqfqstrqm9b8qmhbehbc(x¤ grexQehbCxQb

|ehtrqsxQb8ufe8vE 8qfbeh|rnpbnwqmh|rehb;coufe-vtlgruslzsbx 8dstjncocobuseHc8 l<ehnm;²

E = U2

2 .H óf ó

a!t~xQnpchconwdstjmonpgrqcusemogrufmob;vt ztjuQmhbufeHc b<Qdfehnw~b²

dE

dt≈ −κE ≈ −κU2

2H óf ü

ufnb8cm|trvwbvpt#xfnpchcnpdstjmonpgrqSxstjqsc-vt~l<guslzfb;x¤ ª3~trq

dE

dt≈ −

(

U2

δ

)2

Ktδ óf ÿ

geδ '

Kt

f0

qfgrucgrQmhnwb8qsxQehgrqsc²

κ ≈δKt

Hδ2≈ f

δ

H

óf x gã

³ ó

Page 15: Í Î4Ï>Ð Ñ - polytechniquedubos/Men/Omrani09.pdfÍ Î4Ï>Ð Ñ ÒÓÑÔ Õ Î.ÖX× ØÙÑ5Ô Ú>Û Ü[Ý Þ ß à á!ârÝ8ã©ßZÜ ä åçæ èfà!Þ é à áÌÝ Þ èrêfèsã©ë

Ê~Z Z5Q"UW¢_ªuRR]¹"R]RTS ¢ MNOQBRTSUWVâÊ~ZE^@`PVY¡W"^@¹@OP

κ = κRdU2

=f0δ

U2H

(

g′H

2f20

)1/2

U2

(

g′

2H

)1/2 ófíj

Ú"ÛÜ ½ós ¬_¾5 tjvpbusehcdZgchcnpfvpb8c-xQudstrehtrmoehbκ

U2(m.s−1)

¬j­ ó ­ ü­ ÿr­κ

­ ÿ¬ ­ ó ÿ ­ ¬& ­ ¬f³

abdtjetj~<mheobκxQ8db8qsx,usqfnpufb8~bqmxQbvt9nwmobcocob

U2

cn[qfgusc3Äsgqscvwbc|ehtrqsxQb8ufehcHbmδÄsb

κ6treonpb%b8qmoehb

0.5bm

0.2mhtrfvpb8tju ós ¬

Ò5iFh ¿ º0]@ÀLt<^t ¸ t p t¹!]

×W­@ ÁÂK¶Ã˶]±³´±Ì¶%ͬ±ÅÎ@Ä_"K"´P±/ÏyÐÄuÑä±Ë·ËÑ K¶µ ~]f´±Ì¶TÍÃI±Å¯~¥~P´P±°0gufebckmhnw~b8e%vE npqsl<b8emhnmhusxQbcufevt#tjvpbufexQb(vE bQdgcotrqmxQb a¯tjdsufqfgCqsgrusct6rgrqc©tjnwmxfb8c%conp(ufvtmhnwgqscb8qdfehbqstrqmlzstrufb~¨gnpc(ufqlzstj~d/npqfnwmontjvtrvwtmhgrnpeob#xQb#¨gqsl1mhnwgqxQbClgrufetjqm ó ­ cnp(usvptjmonpgrqsc;dZgruse lzstusb,l<grusdfvwb7å

βκ æ 1ª$qscusnmhb9qfgruct6rgqscltjvl<usvwvt~g¯bqfqsbcoufevpb8c-ó ­ 6trvwb8ufecmoehgrufbcbmvwb9lgb%$,l<npbqm%xQb;tjehntmonpgrqxQÄsqfn!lgr~~bvwb

etjdsdgem-bqmheobvE ltjeom¥Emk¯dbσb<m-vt~g¯bqfqsb

µ

cv =σ

µ

tjqcvpbmhtjsvwbtju ós óqfgruct6rgqsceob8dfeh8cobqmovpb8cxQnw¡[ehbqmobc-tjvpbufec-grfmobqufbc-dZgrufe%xfn¡[ehbqmhc3ikbuQxQb9dstrehtr~<moehb8c(β, κ)

Qvt#xQnpcodZbecnpgrqxQb8ctrvwb8ufehc-xfbvE b<Qdgcotrqm-xfb9a¯trdfufqfg,lgreheobcdZgrqxftjqmcob;l<gqQÄsqfb;b8qmheob ³­ bm ³ÿç

ÚÛÜ ½[óf ó ¾ ¢£qsl<b8emhnmhusxQbeob8vptjmonprb;vpttjehntmonpgrqyxQb;vE bdZgchtjqmxQb;a¯tjdfusqfg#bqy¨grql1monpgrq*xQbβb<mκ

cv β = 0, 15 β = 0, 25 β = 0, 35

κ = 0.4­ ­® í ­ ­ ­[rü¬ ­ ­r® í ü

κ = 0.5­ ­ ð ­s³ ­ ­ÿ í ­ ­ ­[r­s³

κ = 0.6­ ­® ó ¬ ­ ­ ð ü­ ­ ³ óí

×W­@­ èé¨P±PÃI±Ì´±u¸ÄP¯~K°êÙ˯~±¸Äë"Ðȵ@±ÈP¯ì¶]Yį~·ËÑ]Ǩ¶µ¥·± Äsq~xQbxf<mob8eo~npqfbevE npqQþsufb8qsl<b-xQbc!xfbuQ9dstrehtr~<moehb8c

βbmκcoufe!vtdseo8nconwfnpvpnmhxQu ~gxfvpbqfgusc0t6rgqsc¤©trnm0ufqfb<mhusxQb

dtjetj~<mheonufb9bqÄfftjqmSlzstrufb(¨grnc%usqxQbc%dstrehtr~<moehb8c%b<mbq©trnpcogrqc%tjehnpbe%vE tjuQmheob°0gruselzstufb(l<gufdfvpb β,κ

qsgrusc;t6grqsc%©tjnwm(ó ­ cnp(usvptjmonpgrqsct6rbl xfn¡[ehbqmobcdb8emhufehstmhnwgqscnwqfnwmontjvpb8c;x trdsvwnwmousxfb10−3

¤°0grufevpb,xQntj|rqsgcmonusbqsgrusct6rgrqcuQmonpvwnco9vpt,g¯bqfqsb;b<m%vpb(l<gbT$#lnwb8qm%xQb(tjehntmonpgrqQvwbceh8coufvmtmccgqm%xQgrqfqs8c%xstjqsc-vpb8cmhtrfvwbtjuQ ós ü b<m óf ÿ <

ÚÛuÜu½[ós üÓ¾5 tjehnptjmonpgrqyxQb;vV b<QdZgchtjqm-xQb9a¯tjdfufqsg#bqy¨grqsl<monpgrqxQbβ

β(κ = 0, 5 ­ ³6ÿ ­ ¬ÿ ­ ó ÿ ­ üÿ ­ ÿÿ

µ­ óóró ü ­ ¬ í ¬ÿ ­ ¬¬ ó ³ ­ ¬f³ ðó ­ ¬¬ü­

cv­ ­ ð ­s³ ­ ­ ð ü­ ­ ­r­s³ ­ ­®ÿ ó ­ ­ íó ®

ª$q(©trnpchtjqmtruf|r~b8qmhbeβ mtjfvpb8trusóf ü qfgrucehb#tjeufgrqcfnpbq ufqfbmobqxftjqslbvpt%stjnchcbxQbvE b<QdgcotrqmxQba¯tjdfufqsgZ

bm%vt,xQncdZbecnpgrq*xQbc-tjvpbufec-ehb8cmob;npqQ¨ehnpbufehb9 ³8­ç l<bl<n!b8cml<gzfehbqmt6rblvwbc-eocusvmtmhcgrQmhbqusc-dstre tjvpvpnpc k³8® ðróufnb<Qdfvpnpufb,lb<momobCxQ8db8qsxftrqsl<b#dstre;vwb,©tjnwm(u b8qtjuf|b8qmhtjqm

βgrqÅetjvpbqmon&vt*ltcol8trxQb#x¤ qsbeh|rnpb#rbecvpb8c;dZb<mhnmhb8c

lzfbvpvpb8cb<mdtjel<gqsco8ufbql<bgrq*tjuf|~bqmobvpbvtdfehnpconpfnwvpnwmo9xQu*co¯cmo8~br

³ü

Page 16: Í Î4Ï>Ð Ñ - polytechniquedubos/Men/Omrani09.pdfÍ Î4Ï>Ð Ñ ÒÓÑÔ Õ Î.ÖX× ØÙÑ5Ô Ú>Û Ü[Ý Þ ß à á!ârÝ8ã©ßZÜ ä åçæ èfà!Þ é à áÌÝ Þ èrêfèsã©ë

Ê~Z Ê~ZE`V"ÅM V"\GS V©V\/¢¨MNVuÅVìVSâ¡aR]`O_¡@V MNOQBRTSUWVâÊ~ZE^@`PVY¡W"^@¹@OP

Ú"ÛÜ ½ós ÿy¾ tjehntmonpgrqxQb;vE bQdgcotrqm-xQb9a¯tjdsufqfg#bqS¨grql1monpgrq*xQbκ

κ(β = 0, 25)­ ó ­ ü ­ ÿ ­ ­ í

µ­ ¬r®f³³ ­ ¬j®[ ó ­ ¬ í ¬ÿ ­ ¬ íó ® ­ ¬&jü­

cv­ ­ í ¬f³ ­ ­[jü¬ ­ ­ÿ í ­ ­ ­ ð ü­ ­ ­ ð ÿ[

± rblκqfgusc$t6grqsc&truschcnufqfb-mhbqsxftrqsl<b;vpt9stjncocobrrfnwb8qCu bvpvwbehb8cmob-ehbvtmhnwb~bqm&©trnwfvpb mhtrfvwbtjusós ÿ 1[J%q,eh8coufvmtm

tjmmhbqsxQu*l8tje-x¤ ufqdgnwqm%xQbufbdfufehb~bqm-dfz¯Qcnufbdfvpusc-grqtjus|r~bqmobκlj b8cm%~xQnwehbdfvwuc-grqtjuf|~bqmobvpt ¨ehnpl<monpgrq

xfb9cufeo©trlb;usn[mobqxy~vpncocobevpb8clzstr~dscQdsvwuscqsgjmoehb;c¯Qcmo8bbckmdfehncnpfvpbrJ%qfb truQmoehb;©tC<grq*dsvwuscnpcoufb8vwvpb9dZgrufeeob8|tjexQb8e#vpt#¨grnc%vt,xQ8db8qsxftjql<b(xQb

λκb<mβl<gqsconpcmob(,eob8dfeocb8qmobecufevpt

~§8bÄ|rufehbvpttjehnptjmonpgrqxQbλbqy¨gqsl1mhnwgqxfb

βbmκ

0.14

0.14

0.16

0.16

0.16

0.18

0.18

0.18

0.2

0.2

0.2

0.2

0.22

0.22

0.220.22

0.24

0.24

0.24

0.26

0.26

0.26

0.28

0.28

0.3

0.3

0.320.34

β

κ

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.550.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

º@»¬¼"½ós ¬y¾ tjehntmonpgrqxQbλb8qC¨gqsl1mhnwgq*xQb

κb<mβ

«Qufe,vtÄ|rufehb óf ¬ qfgusc~r8eonwÄsgqscvpt©tjnpfvpb*xQdZbqsxstjqslbxQbλtjuÌl<gb%$,lnwb8qm,xQb¨ehnl1monpgrq

κtjvpgrecusb*dgufe

βvpt

xfdZbqsxftrqsl<b;bckmdsvwusc-conp|rqfnwÄltjmonprbª$q~lgrqslvwucnpgrq6vpblgb%$,l<npbqm3xQb¨eonl1mhnwgq q b8cm&dstrcb8q dfetmhnpufbufqfbtjehnptrfvpblvwbdZgrufe0vE mousxQb-xQbvt%dseo8nconwfnpvpnmhxstjqscqsgjmoehbgQxQ8vwb trqsc!mogrufmobvptcusnmhbqsgrusctjvpvwgqsc!ÄfQvpb8ctjvpbusehcxQb

βbmκeobcdZb8l<monprb8b8qm0

0.25bm

0.5l<grqf¨greh8~bqm

~vtvwnwmmhetmhufeob trvwvpnpc ³8® ðró$b<mPtjnxQrg|rb8v[trqsx7b8vpx k³® ð ­ 1

Ò5iÒ í-q ^¼ qWî ¹ qsqWî ]-¼ zoqWp%p%qïq ¹ðw(_Et m »yw q tjqc&l<b<momob-dstremhnwbqfgusc0b<Qdfvpgrehgrqsc!vE b¡Zbm$xQu |ufnxftj|bcoufe&qfgrmoehb~gQxQvpbdstje&xQb8uQ trdfdfehglzsb8c0xQnw¡Z8eob8qmhb8c8jatdfehb~nw8eobt6b8lufq#|ufnxftj|b-dstjeo©tjnwm8lj b8cm;xQnpeobb8q#|rufnxftrqmdstjeuvteotjvpnmhñ!-gru#vpb8c$lztj~dsc3xQb%eo¨ehbqslbra!t;cbl<gqsxQbl<gqsconpcmob%|ufnxQbedtjeoxQb8cgscb8eotjmonpgrqsc]!gru*xQbcxQgqfqfbc-xQ|ehtxQbc©tjnpfvwb;eh8cogrvpuQmhnwgqb<momob;~<mhzfgQxQb(c8 trdfdb8vwvpbvV tjdsdfeogQlzfbxfu~gQxQvpb(dstre©trnmgru Âmhzfb(dZbeo¨b8l<m~gQxQb8vtrdfdfehgtrlzZà Äs|ruseobóf ó %bª3vpnptCb<mtrvVp ¬j­r­¬ <[ b8cmufqfb ~<mozsgxfb ufnlgrqsconckmhby|r8qfehbeufqfb#conw ufvtmonpgrqzstjufmobeocgvwuQmhnwgqcoufeufq:vptreo|b xQg~trnwqsb npln0mogrufmvpb#xQgr#trnwqfb|rvpgrtjv 1[tjdfdZbvpbxfgr#tjnpqfbdZehb gu#fnp|9fehgjmozsbe ­òò o1ª3qscoufnmhbrxQb%xQ8|retrxQb8e&vwblzstj~d~grfmobqu,;vE tjnxQb%x¤ ufq#Äsvwmoehbdstcocob<¥Àstc0nw~nwmhtjqm

³ÿ

Page 17: Í Î4Ï>Ð Ñ - polytechniquedubos/Men/Omrani09.pdfÍ Î4Ï>Ð Ñ ÒÓÑÔ Õ Î.ÖX× ØÙÑ5Ô Ú>Û Ü[Ý Þ ß à á!ârÝ8ã©ßZÜ ä åçæ èfà!Þ é à áÌÝ Þ èrêfèsã©ë

Ê~Z Ê~ZE`V"ÅM V"\GS V©V\/¢¨MNVuÅVìVSâ¡aR]`O_¡@V MNOQBRTSUWVâÊ~ZE^@`PVY¡W"^@¹@OP

vpb8c%grscobehtmhnwgqsc©tjnpfvwb(eocgvwuQmhnwgq Äs|ruseob9ós 1sab9lzstj~dyÄsvwmoeh9b8cmbqQÄq*uQmonpvwnco9dgufe-¨gehlbexQbccnp(usvptjmonpgrqsccoufeusqxfgr#tjnpqfbdfvwucdbmonwm%tjdfdZbvpvwb;xQg~trnwqsb%Äsvc guyanmomovpb ò eogrmozfb8e a ò b8qSuQmhnwvpncotrqm-vpb§8~b~gQxQ8vwbf trqscqfgjmheobl8trcvpb9xQgr#trnwqfbdZehbb<mvpb9xQgr#tjnpqfbÄsvcgrqmvt ~§8bmtjnpvwvpbb<mvpb;lzstj~dyÄsvwmoehcobeom|rufnxQbe-vpb9lzstrdconw ufvw

º@»¬¼"½[óf ó ¾ nptr|retj~~bbQdfvwnltjmonw0xfbvt~~<mhzfgQxQb9xQu k~gQxQ8vwbdstre©trnm!

×W­×@ ó©P´PôB±âįõ´P±á´PPP·Å±ÄP¯~öDK¥±a!t#dfehb~npehb;b<Qd8eonpbql<b9l<gqsconpcmob9~|ufnpxfbe%qsgjmoehb;~gQxQvpb9dstrevwbcxQgqfqfbcxQb9eh<¨8eob8qsl<b9t6b8lufqsb(l<gqsxQnwmonpgrqnwqsnmhnptrvwbvp|ehb~bqm%dZbeomouseoZbZab(lzfgrnw*xQu*mob8~dsc%xQu|rufnxftj|b

τb8cmdseonc%truxQfufm|tjv

TpdZgrufeufq~ikbuxQb(dstrehtr~<moehb8c

κ = 0.5

β = 0.25

1«usevpt#Äs|rufehb óf ü qfgusc-ehbdfeh8cobqmhgrqscvE ltrem%xQb(vV qfb8eo|nwb;b8qmoehb;vpt#eh<¨8eob8qsl<b;bm%vpb9~gxfvpb(tjulgrufec-xQb;vt~cnp(ufvtmhnwgq

0 10 20 30 40 50 60 70 80 90 10010−4

10−3

10−2

10−1

100

t

E(δψ

)

τn=1

º@»¬¼"½ós üG¾ ª3qfb8eo|nwb;xQb;vE 8ltremxQb;vt ¨grqsl<monpgrqxQb9l<gufetjqmdtje-etjdfdZgreomtru*lzstj~dsc-xQbeh<¨8eob8qsl<b± uxQ8dstjeom%xQb9vt#cnp(usvptjmonpgrqvV 8l8tjeom%xQb9vE 8qfbeh|rnpb;l<geoehb8codgqsxy#vpt~dZbeomouseostjmonpgrqnpqfnmhnptrvwb usneobckmhb;©tjnpfvpb;bqscoufnwmob(lb<mltrem~xQnw~npqufbSdZbqxftjqm~ufqHnpqmobehtjvpvwbxQbCmob~dsc~xQgqfqf8byt6tjqm~u npv-tjuf|b8qmobSx ufqsbS#tjqsnw8eobSb<Qdgqfbqmhnwb8vwvpbr0ab

³

Page 18: Í Î4Ï>Ð Ñ - polytechniquedubos/Men/Omrani09.pdfÍ Î4Ï>Ð Ñ ÒÓÑÔ Õ Î.ÖX× ØÙÑ5Ô Ú>Û Ü[Ý Þ ß à á!ârÝ8ã©ßZÜ ä åçæ èfà!Þ é à áÌÝ Þ èrêfèsã©ë

Ê~Z Ê~ZE`V"ÅM V"\GS V©V\/¢¨MNVuÅVìVSâ¡aR]`O_¡@V MNOQBRTSUWVâÊ~ZE^@`PVY¡W"^@¹@OP

~gQxQ8vwbxQnwbeh|rb9dsvwuscgrugnwqcetjdfnxQb~b8qml<bl<nvptrnpchcb(db8qscb8eu ufq|ufnxftj|b9xQb(vE gehxfeob(xfb9mob8~dscxfb dfehncnpfnpvwnwmoxfuy~gQxQ8vwb;q b8cmdZbuQm§moehbdtrc-cou$,cotrqm°0gufedZgrufgrnpe-b8cmonpb8eufqmhb~dsc%xfb;|rufnxftj|b;grdQmhnw#trvVfnpvbckm%qs8l<bcochtjnpehbx bdsvwgeob8e-ufqvptreo|bnpqmob8eotjvpvpb9xQbmob8dc%xQb|ufnxftj|bb8qC¨gqsl1mhnwgq*xQbmob8dc-xQb;dfehnpconpfnwvpnwmob<m%xQmob8eo~npqfbecogrqynwqfþsufbql<b9coufe-vwb9lgr~dZgreomob~b8qmxQu~gQxQvpbr

0 10 20 30 40 50 60 70 80 90 10010−16

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100

t

E(δψ

)

τ=0.01τ=0.02τ=0.04τ=0.1τ=0.2τ=0.4τ=0.5τ=0.6τ=0.7τ=0.9τ=1

º@»¬¼"½fós ÿ¾ ª3qfb8eo|nwb%xfb%vE ltjeomxfb%vt¨gQl1monpgrqSxQblgrufetjqm$dstre$etjdfdZgjm9vt;eo¨ehbqslbdZgrufexQnw¡Z8eob8qmc&mhb~dscxQb%|ufnxftj|bqsgreh~trvwncodtjevwbmhb~dscxfb;dfeo8npconwsnwvpnmhrQvt~l<gruseoZbbq<mhgrnpvwbc-l<grehehb8codgqsxC

τ = 0, 5Tp

«Qufe9vtCÄ|rufehb óf ÿ qsgruscehb#tjeufgrqcu npv3¯txQbuf©tj~npvwvpb8c;xfb,l<gruseoZb8c;t6b8lxQbuQ:db8qmob8c9nwqrb8ehcobc%bm(ufb#vptmhehtrqsconmhnwgqcob(©trnm;

τ = 0, 5TpZª$qb¡Zbm8¤dZgrufe;xQbc%mhb~dsc;xQb|rusnpxftr|rb(moeh8cdbmonwmhcvpblztj~d|ufnpxftreoehnwb(ycufnpeob

dsvwusctju~~grnpqsc3lgreheobl1mob8~bqm&vwb%lztj~dsc3xQb-eh<¨8eob8qsl<bjdstre3l<grqmheobxQc3u grq#c tjdsdfeogQlzfb-xQu~mhb~dsc3xfbdfehnpconpfnwvpnwmorrvwblztj~dS|ufnxQbckm-Zb8trusl<gufdC~grnpqsc-lgrqmoetjnpqmcufnpeobvpblztj~dSxQbeh<¨8eob8qsl<bxfbdsvwuscvpb|rufnxftj|bxfbnwb8qmvpufnw¥E~§~busqfb9cgufel<b;xQbdb8emhufehstmhnwgqyufnlgrqsxfufnm%ufqsbxfnwbeh|rbql<bdstje-etjdfdZgreomvV <mtm%xQbeh<¨ehbql<br

×W­×@­ ó©P´PôB±âįõ´P±á´PPP·Å±Æ´·Åô¯~K´·Å±°0gufe-xQ|ehtxQbevwblzstj~dCxfbeh<¨8eob8qsl<bqfgusct6rgqsc3uQmhnwvpncufq,Ävmhehtr|rbcodZb8l1mhehtrvdtrchcbstrcufbvE grq,Äsb9vE tjnxQbx ufqlgb%$,l<npbqm

αufbqfgusc-tjvpvwgqsc-tjdsdb8vwb8e-ehtrdfdgemxQbeh8cogrvpuQmonpgrqy~gQxQvpbÄsvpc |rusnpxQ6µ÷gQxQ8vwbdZehb ufn|ufnxQb<

ª$qSb¡Zbm8vwb~gxfvpb ¦ ñufbqfgruc$uQmhnwvpncgqsc$bckmufqSgQxQ8vwbbqyxQn¡[ehbql<bÄqfnwbxQgqsldZgrufedZgrufgrnpe$Äsvwmoehbevpb8clzstj~dscxfb9cgemhnwbcqsgrusc-t6grqsctjdsdfvwnufvpt mhehtrqsck¨geo~8bxQb ï grufehehnwb8etruQSlzstr~dsc-xQbvpt ¨gqsl1mhnwgqxfblgrufetjqm

ψ # grusc-t6grqsc

gQmob8qutrnwqcn¤ufq*lzstr~dycogrusc¨geo~b;x¤ ufqfb9cogr~~b;x¤ ztjehgqfnusb8c8ustmhnwgq óf ð²

ψ(x, y, t) =∑

n,m

ψnm(t) exp 2iπ(mx

L+ny

L)

óf ð

t6b8lLvpt mtjnpvwvpb;xQu*xQgr#tjnpqfb vpb9xQg~trnwqsbmhtrqm%l8tjeheo63bm

ψnmvpb8c-l<gbT$#lnwb8qmhc-xQb ï grufeheonpbe

± nwqcnxstjqscvE bcdtrl<b;codbl1mhehtrv¤qfgjmheobÄsvwmoehb9cb8ehtxQÄsqfnlgr~~bcoufnwm9²

ψobsnm

= ψrelnm, si(

mL

)2+

(

nL

)2≤ k2

coupure

= 0, si(

mL

)2+

(

nL

)2≥ k2

coupure

t6b8lkcoupure = 1

2

2πL αetjdszfnpufb8~bqm8grq~db8uQmvpb-ehbdfeh8cobqmob8e3l<g~b%coufe3vtÄs|ufeob óf <r¿gusc3vpb8c$l<gb%$,l<npbqmc3xQbvt;dstremhnwb-zstlzufeo8b-cgqm

eh8xfufnmc-¾ehgb<m-grqyqfb;l<gqscnxQ8eobufbvwbc-l<gb%$,l<npbqmcufncobmoehgrufbqm-xftjqscvpb;l<b8ehlvwbfvtjqslrfabeht6¯grqCxfuylbel<vpbbckmvpbKcoupure

³ í

Page 19: Í Î4Ï>Ð Ñ - polytechniquedubos/Men/Omrani09.pdfÍ Î4Ï>Ð Ñ ÒÓÑÔ Õ Î.ÖX× ØÙÑ5Ô Ú>Û Ü[Ý Þ ß à á!ârÝ8ã©ßZÜ ä åçæ èfà!Þ é à áÌÝ Þ èrêfèsã©ë

Ê~Z Ê~ZE`V"ÅM V"\GS V©V\/¢¨MNVuÅVìVSâ¡aR]`O_¡@V MNOQBRTSUWVâÊ~ZE^@`PVY¡W"^@¹@OP

º@»¬¼½Zóf Ó¾Cï npvwmoehb;dstrchcbstc-xftjqcvV b8codstrlb;cdZb8l<moetjv

×W­×@­× øõ·P¶] ~°0gufeusq,ikbu:xQbdstjetj~<mheobc

β = 0, 25, κ = 0, 5ufn&l<geoehb8codZgrqsx*Cufqmhb~dscxQb(dseo8nconwfnpvpnmh

Tpqsgrusct6rgqsc-©trnm

mhgrufehqfb8evpb%~gQxQvpbdgufexfn¡[ehbqmhc$mob8~dscxQb%|ufnpxstj|rbτufn[6treonpbqmbqmoehb

0, 01Tpb<mTpbmufqSehtrdfdZgreomxQb%eh8cogrvpuQmhnwgq

αbqmoehb

1

2

b<m1

8

º@»¬¼"½ós í ¾ zstj~dscxQb rgreomonl<nwmo(dgrmobqmhnwb8vwvpb(dZgrufevtCl<guslzfb(coufdZehnwb8ufehb(t6rblα = 1/3

fvt,dfehb~npehb9vpnw|qfb tfZZllgreheobcdZgrqxyvV <mtm9xQb~eh<¨ehbql<br[vtxQbuQQnp~b x¤br[htruQlzstrdc%Ävmheocb<m;vptSmheognpconw8~b |s¤zn tjuQ:lzstj~dsc|ufnxQ8c8fvpb8c-l<gvwgqfqfbclgreheobcdZgrqxQbqmeobcdZb8l<monprb~b8qm

τ = 0, 01Tpτ = 0, 4T p

τ = Tp

«Qufevt;Ä|rufehb ós ír&qfgusceob8~trehufgqsc&ufb%dZgruseufq,mob~dscxQb|rufnxftr|rb-mheoc3dZb<monwmτ = 0, 01Tp

lj b8cm¥£¥£xQnwehbxftrqsc$vpbl8trcguqfgusc¨geFCgrqsc%vwb ~gxfvpb Ccob lgrvpvwb8etruQ*grscobehtmonpgrqcfvpb(~gQxQ8vwb |%ehbdfehgQxQufnwm%dstjeo©tjnwmob8~bqmvt,|ehtrqsxQb9lzfbvpvwb

³ ð

Page 20: Í Î4Ï>Ð Ñ - polytechniquedubos/Men/Omrani09.pdfÍ Î4Ï>Ð Ñ ÒÓÑÔ Õ Î.ÖX× ØÙÑ5Ô Ú>Û Ü[Ý Þ ß à á!ârÝ8ã©ßZÜ ä åçæ èfà!Þ é à áÌÝ Þ èrêfèsã©ë

Ê~Z Ê~ZE`V"ÅM V"\GS V©V\/¢¨MNVuÅVìVSâ¡aR]`O_¡@V MNOQBRTSUWVâÊ~ZE^@`PVY¡W"^@¹@OP

xs#tjncdstrc-xfuSmhgruQm-vpb8cdZb<mhnmhb8c-lzfbvpvwbcQdstjelgrqmoehbdgufe-ufqSmob~dsc%xQb|rufnxftj|b%mheoc|retjqsxτ = Tp

vpb~gQxQ8vwb nc8 8ltremhb;l<gr~dfvp<mhb~bqm-b<mxQu*lzstr~dyxQbvt|retjqsxQb8lzsbvpvwb b<mxQu*lzstrdxQbeh<¨ehbql<b l1fabmhb~dsc-xQb;|ufnxftj|bufn&lgreheobcdZgrqsx

τ = 0, 5T pcob fvwb §<mheob vwb(mob~dscgdQmonp#tjvdgufeufb vwb ~gxfvpb zZ%treoehnwb(Ccob(etjdsdfeogQlzfb8evpb(dfvpusc

xfb;vpteh8tjvpnwmo <# gruscehb#tjeufgrqsc-ufb vwb(gQxQ8vwb b8cmmoeh8ccb8qsconwfvpb(tru*mob8dc%xQb |rufnxftj|brf#trnpc%dgufedZgrusrgrnpe%xQncmonpqf|rufb8e%vE b¡ZbmxQu|ufnxftj|bcoufe-vt|retjqsxQbb<mvtdbmonwmob9lzfbvpvpbcodtjeh~bqmqfgrusc%t6rgrqc3mhehtl<vwbc-xQbufS8lzsbvpvwbc óf ð1

º@»¬¼"½-óf ð ¾ zstrd xfbrgemhnplnmhydZgruseCvt/l<grulzfbcusd8eonpbufehb α = 1/3

τ = 0, 4T p

~vpb8cCdstjqfqsb8tjuf t~b<m lgreheobcdZgrqxQbqmtruQlzstj~dscxQb(eh<¨8eob8qsl<bvpb8cdstjqsqfb8truQ l8%b<m xstjuQlzstr~dsc|ufnpxf(b<m b6bm h,vt,xQnw¡Z8eob8qsl<bb8qmoehb%vpb8cxQbuf#lztj~dsc8vpb8cdstrqfqfb8truQ t1 l8$b<m b6$lgreheobcdZgrqxQbqmtruQ,lzstj~dscxQbdbmonwmob8lzfb8vwvpb%bm 1 xs3bm htruQylztj~dscxQb;vt |ehtrqsxQb8lzsbvpvwb

ab~gQxQ8vwbeob8dfeogQxQusnm-l<geoehb8l<mob~b8qmvpt(|retjqsxQb%8lzfb8vwvpbrufq,eh8coufvwmhtjmtmomob8qsxQuCuyusbvpb%|ufnpxstj|rbb8cmtrdfdfvpnpufufqfnusb<¥~b8qmdstjevt|retjqsxfblzfbvpvpb;dstje%lgrqmoehbdgufevt~dbmonwmob9lzfbvpvpb9lj b8cmgnwqc-nxQbqml8tje-qfgusc-rg¯grqscmheoc-fnpbq*u npvehucoconm~ehbdfehgQxQufnpeobvpb8cZgrqfqfbccmoehusl1mhufeobc~trnpcnpvcogruscbckmhnw~b;vE tr~dfvwnwmouxQb9xQu*lzstj~d

³8®

Page 21: Í Î4Ï>Ð Ñ - polytechniquedubos/Men/Omrani09.pdfÍ Î4Ï>Ð Ñ ÒÓÑÔ Õ Î.ÖX× ØÙÑ5Ô Ú>Û Ü[Ý Þ ß à á!ârÝ8ã©ßZÜ ä åçæ èfà!Þ é à áÌÝ Þ èrêfèsã©ë

Ê~Z Ê~ZE`V"ÅM V"\GS V©V\/¢¨MNVuÅVìVSâ¡aR]`O_¡@V MNOQBRTSUWVâÊ~ZE^@`PVY¡W"^@¹@OP

×W­×@¬ù øõ±ÃI[~]~]ú´±áÄ_±Ë~¥±§·Ã뱶µ¶]±§Ä ¯ì¶±³°êW´Ù¶±©ûb¶uschufbnl<nsqsgrusc3t6grqsc08tjvpufbe$x ufqsb%#tjqfnpehbustjvpnwmhtmhnwb-vpt;l8tjdstl<nwmoxQu#~gQxQ8vwb%9eobl<grqckmheoufnpehbvpb8c3dbmonwmob8c8lzsbvpvwbcJ%qfby~trqfnpehbCdfvpusc~ustjqmonwmhtjmonprbCc8 nw~dZgcobSlbycmhtxQbrqfgusc t6grqsc(xQgrqslSuQmhnwvpnpcoSxfb8cgruQmhnwvcckmtmoncmonusb8c(mobv-ufbSvwblgb%$,l<npbqm%xQb;l<geoehvtmhnwgqCbmvpt~<mhzfgQxQb9x¤ tikuscmob8b8qm8°0gufe$l<b©trnwehbrrqfgrusc$t6rgrqcmhehtl<vpb-|retjdfzfb-xQbxQnpcodZbecnpgrqgru~vpb-qutj|rb-xQb%dgnwqmhc$xQb

qb8q~¨grqsl<monpgrq#xQb

qrefrª$qb¡Zbm8vwb

qustr|rb%xfbdgnwqmc$bckm$usq,gq,npqsxQnltmhbufedgufe$ehnÄbe3ufqsb%l<geoehvtmhnwgqb8qmoehbvwbc$lzstr~dscqref

b<mq«nvwbc$dZgrnpqmhccgqm

cogrucvt#¨greh~b(x¤ ufqqustr|rbfnwvb8cm%¨gem#dstreonpbeusb(vpb8c%xfbuQlzstrdcqfbcgqmdstrcl<geoehvp8c8«[ nwvccb8(fvpbqmxQbcoconpqfbeusqfb9l<gufeoZbrQqsgrusc-lzfb8ehlzsbehgrqsc~xQ<mhbeh~nwqfb8e-vptqstjmoufehb;xQb;vtlgrufehZbb8qdfeogQl8xftrqm-ufq*tikuscmob8~bqm8# grusc-t6grqscxQgrqllgrqsconxQeh%vpb;lzstj~dSxfbgreomonl<nwmol<gr~~bufqfb;c8eonpbckmtmoncmonusb~xQb8uQ,tjehnptrfvwbc

qrefb<mqbmqsgrusc

t6grqscehbdfeh8cobqmo(vwb~qustj|b xQbdZgrnpqm;trchcoglnwMx,y(qref (x, y), q(x, y))

Äs|ruseob ós ® 1¤abcdgnwqmhccob fvwb8qmcoufnpehb(usqfbxfeognmhbr # grusc-trvwvpgrqscxQgqslmob8qmobe-usqtjikuscmob~b8qmt&$~qsbr# grusc$xQ<mhbehnpqfgqsc0vptxQehgrnwmob-x¤ tikuckmhb~bqm3|reDürlbufqsb-eo8|rehb8chcnpgrq vwnpqf8trnwehbra!t;xQeognmhb-x¤ tjikuscmob~b8qm3tdZgrufe&ustmhnwgqq = a× qref + b

t6b8l

a =cov(qref , q)

σ2

qrefbmb = q − aqref# grusc%xQ<Äsqsnpchcgqsctjuschcon¤vwb9lgbT$,l<npbqmxQb9l<geoehvtmhnwgqr²

r =cov(qref , q)

σqrefσq

t6b8lcov(qref , q) = 1

nm

nm∑

i=1

(qiref − qref )(qi − q)

vt-l<gtjehnptrqsl<bσqref =

1

nm

∑nmi=1

(qiref − qref )2bmσq =

1

nm

∑nmi=1

(qi − q)2

vpb8c8ltremcmk¯dZb8c%trchcgQl<np8cqref

bmqfbm

qrefqvpb8ctjvpbufec~g¯rb8qfqfbxQb

qrefbmqcoufemoguQmob;vt|rehnwvpvwb;xQu~gQxQvpbr

−8 −6 −4 −2 0 2 4 6 8 10

−8

−6

−4

−2

0

2

4

6

8

qref

q

r=0.9571a=0.9208

º@»¬¼½[ós ®Ó¾5# utj|rb;xQbdZgrnpqmdgufe-vwb9lztj~dyxfb;rgreomonl<nwmo α = 1/3

τ = 0, 5Tp

«Qufevpt(Äs|ruseob ós ® 3vt xfeognmhbxQbeh|rehb8chconwgq~t dZgrufe8ustjmonpgrqq = 0, 92× qref − 9, 1

bmvpbl<gb%$,l<npbqm-xQbl<grehehvtmonpgrqrbckmdfehtjmonufb~bqm%8|trv

1r = 0; 96

< # gusc-dZgrufgrqsc%xQgrqsl;t&$#eo~becotrqscmoehgrd*x¤ behehbufe%ufb;vpb8cxQbuflzstr~dsccogrqmmheocfnwb8qlgreheo8vwc9¤ %bdfvpuscvptSdZbqmob

ab8cmmheoc%dfehgQlzfb xfb

1l<busn0conp|rqfnwÄsb~usb vpb(~gQxQvpbeob8dfeogQxQusnmmoeh8cfnpbqvpb

lztj~dxQb~eh<¨8eob8qsl<b bqÅtj~dfvpnmhusxQbr[¢£v&bckm;Cqsgjmo~ufb#l<bmmob Äs|ufeob~l<geoehb8codZgrqsxufq:npqsckmtjqm;xQgrqfqsxQb~vtSconp(ufvtmhnwgqdZgrusevpt~lgruslzfbzstruQmobmheogdgcdfzsehbbmdgufe-ufqymob8dcxQb;|rusnpxftr|rb

τ = 0, 5T p

°0gufeufq*xQntj|qfgcmonufbdsvwusc|rqsetjvpbqsgrusc-t6grqscltrvplufvpvpb9l<gbT$#lnwb8qm%xQb9l<geoehvtmhnwgq,b<mvt~db8qmob9~lzstrufbnpqscmhtjqmb8qscoufnmhb$qfgusct6grqscdfeonc¤vt-~g¯rb8qfqfb$coufevpb8c ¬j­ xQb8eoqsnw8eobcufqfnwmoxQb3mhb~dsc88b8qQÄsq;qfgusct6grqsc[mhehtl<3vpb8c!xQbufdstrehtr~<moehb8crabqy¨grqsl<monpgrqxQuymob8dcxQb;|rusnpxftr|rbqfgreh#tjvpnpco

τn = τTp

dgufe-vpt|ehtrqsxQbb<mvtdbmonwmob;8lzsbvpvwb ós ³­ <

¬j­

Page 22: Í Î4Ï>Ð Ñ - polytechniquedubos/Men/Omrani09.pdfÍ Î4Ï>Ð Ñ ÒÓÑÔ Õ Î.ÖX× ØÙÑ5Ô Ú>Û Ü[Ý Þ ß à á!ârÝ8ã©ßZÜ ä åçæ èfà!Þ é à áÌÝ Þ èrêfèsã©ë

Ê~Z Ê~ZE`V"ÅM V"\GS V©V\/¢¨MNVuÅVìVSâ¡aR]`O_¡@V MNOQBRTSUWVâÊ~ZE^@`PVY¡W"^@¹@OP

0 0.25 0.5 0.75 1 1.25 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

τn

a

α=1/2α=1/4α=1/6α=1/8

0 0.25 0.5 0.75 1 1.25 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

τn

r

α=1/2α=1/4α=1/6α=1/8

0 0.25 0.5 0.75 1 1.25 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

τn

a

α=1/2α=1/4α=1/6α=1/8

0 0.25 0.5 0.75 1 1.25 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

τnr

α=1/2α=1/4α=1/6α=1/8

(a) (b)

(c) (d)

º@»¬¼"½Qóf ³8­¾ atdZbqmobab<mvwb%lgb%$,l<npbqmxQbl<grehehvtmonpgrq

rbq~¨grql1monpgrq,xfu#mob8dc$xQb|ufnxftj|b

τn = τTp

qfgreh#tjvpnpco-dZgrufevt~dZb<monwmob t~b<m-bmvpt~|ehtrqsxQb8lzfb8vwvpb lb<m%xdgufexQnw¡Z8eob8qmhb8ctjvpbusehc-xQb

α

«Qufe;vptS|retjqxQb lzfbvpvpbr[qfgrucehb#trehufgqscufb~dZgrufe;xQbcτnmheocdbmonwm(xQb vE grexQeob~xQb

0, 01bm

0, 03vwb#lgbT$,l<npbqmxfb

lgreheo8vptjmonpgrqrehb8cmobmheocdfehgQlzfb;xQb

1ltre%xftrqscl<b ltrcqsgrusctjdfdfvpnusgrqscufq*|ufnxftj|bmoeh8c¨gemdstrevt~|retjqsxQb;lzfbvpvpbr

vpbgQxQ8vwbb8cm;xQgrqsl(¨grel<,ehbdfehgQxQufnpeob(vwb~lzstj~d:xQb eh<¨8eob8qsl<b°&trelgrqmoehb(xQcufbvpb |ufnpxstj|rb xQbnpbqmdsvwuscvýürlzfbτn > 0, 5

Qvpb;~gxfvpblgr~~bqslb;xfnwbeh|rbe!ikuschu #lb;u npv¤qfb;dstreonpbqmdsvwusc% ehbdseogQxQufnpehb%vpb9lzstj~dxQbeo¨ehbqslbrab lgr~dgemhb~bqmxQb(vt#dZb<monwmob8lzsbvpvwb b8cmxQnw¡Z8eob8qm8Zª3qb<¡[b<m[bq|ufnpxstjqmmoeh8c%¨greomqsgjmoehb9~gQxQvpb(qsb(tCleo8be%ufbvt*|ehtrqsxQb#8lzsbvpvwb,dtje(lgrqsco8ufb8qm qfgrusc(q t6rgrqcdfehtjmonufb~bqm(dtrc(xfb,dbmonwmobS8lzfb8vwvpbCl<bSusn$b<Qdfvpnusb,vwbc9tjvpbusehc©trnwsvwbc(xQb

r0tjnc;cnqfgruc9tjdfdsvwnufgrqscusq/|rufnxftj|b~gnwqc¨greom(bqÅvtjncochtjqm;dfvpusc(xfb#vwnpb8emhCtjuÅ~gQxQvpb#dZgrufel<ehbe

vpb8cdZb<mhnmhb8c;dZbeomouseostjmonpgrqsc8[vwbc;xQbuflzstj~dsclgr~~bqslbqm9SxQb8rb8qfnwe~npbuQlgreheo8vwc[°0grusevpb8c|ehtrqsxQbc%tjvpbusehcxQbτnqfguscdgufrgqscl<gqscnxQ8eob8eufbvpb~gQxQ8vwbq b8cmdsvwusc-xfuSmhgruQm-|ufnpxfrb<m-qfgusceob8#tjeusgrqscu npv¤b8cml<gr~dfvp<mhb~bqm

xf8l<geoehvpdstje-etjdfdZgreomvpteh<¨ehbql<br

¬Q³

Page 23: Í Î4Ï>Ð Ñ - polytechniquedubos/Men/Omrani09.pdfÍ Î4Ï>Ð Ñ ÒÓÑÔ Õ Î.ÖX× ØÙÑ5Ô Ú>Û Ü[Ý Þ ß à á!ârÝ8ã©ßZÜ ä åçæ èfà!Þ é à áÌÝ Þ èrêfèsã©ë

* +-,/.10325476 þ

< ÒÓØ5ÐÑ Ù=ÿä ×@< : Î Ð Î ×ÙÑ Ð-×-Õ ×$Ö=ÿÑ

a$ ufmonpvwncotjmonpgrqxQbc~gQxQvpb8c%,tjnpeob;vpnpnwmo8b(xQulvwnp#tm%dZbeh~b<mxQb xQnw~npqufbevpb(lgQm%quf~8eonufb9xQb8ccnp(usvptjmonpgrqsc8 ± rb8lvt9~§8bdfufncochtjql<b%xQbltrvplufvErgrqSt9xQgrqlvt9dZgchcnpfnpvwnwmoxQb©trnwehb%xQbcconw ufvtmonpgrqc$9dsvwusczstruQmobeocgvwuQmhnwgq#cdstjmontjvpb%guxfb8ccnp(usvptjmonpgrqscdfvpuscvwgqf|rufbc%gudfvpuscqfg(fehbuscob8c8IAqdb8uQm;|tjvpb~bqmufmonpvwncb8e;xQn¡[ehbqmc¨geFC8tj|rbc%truQgehxscxQuxfgr#tjnpqfb%lgrufbeom0dtje3vpb~gQxQvpbrr tjqc$l<blztjdfnwmoehb-qfgruc3tjvpvwgqsc&mousxQnpbevptbecnpgrq~tjnpehb-vwnp~nmhb%xQb%qfgjmheob-~gQxQvpb ¦ a!t#mtjnpvwvpb xQuxQgr#tjnpqfb(lgrqscmonwmousb(ufqqsgrufb8tju*dtjetj~<mheob9xQbl<gqmheo¶vwb # grusctjvpvwgqsc%xQgqsl9ehb|tjexQbe-vE nwqQþufbqslb xQbl<bdtjetj~<mheobcufevt9dfehncnpfnpvwnwmoxQuSa ± bmehdZ<mhbevV b<Qb8ehlnplbcoufevE b<¡[b<mxQu,mhb~dscxQb|rufnxftj|bcoufevpt(eob8dfeocb8qmhtmhnwgqxfb;vptdZb<monwmob(8lzfb8vwvpb;xftjqscvpb9a ± /

iTj klLmonWp%q º]WwÓt lyî » p »yt1º q p t` t¹!] qa$ npqmo8|retmonpgrqyufqfnxQnpeobl1mhnwgqfqfbvpvpbxfb;gQxQ8vwbc~tjnpeobvpnw~nwmo8brslgrqQÄs|ufeh8cxQb©t[Cgrqtjdfdseogdfeonpbl<g~b9gxfvpb8ceh|rnpgj¥qtjuQCxQuyl<vpnw#tjm -1Q(vE nwqmo8eonpbusexQb*gQxQ8vwbcxQbl<npel<ufvtmhnwgq,|r8qfetjvpb 91Qb8cm-xQbbqufb%usqfb%mhb8lzfqsnpufbmoeh8cufmonpvwb;xQbeoxQusl<monpgrqCx¤ 8lzsbvpvwbdgufecnp(ufvpbevpblvwnp#tm-cuseufqyxQgr#trnwqfbvpnw~nwmorsbmk¯dbxQbgQxQ8vwbdZbehbmx b¡Zbl1mousbexfb8c-npqmo|ehtjmonpgrqscxQbcdfehglb8chcucdsz¯Qconpufbc-vwgqf|mobeh~bb<m%t6rbl%usqfb;|retjqsxfbeh8cogrvpuQmonpgrqcodstmhnptrvwbmogrufmbqyeoxQufncotrqm|ehtrqsxQb8b8qmvpbmob8~dsc-xQb9ltrvplufv¤ehb8ufncAquQmhnwvpnpcob#tjnpqmhbqstrqmvwbcW-ÆxftrqscxQbqfg(fehbuQyl<bqmheobc-xQbeoblzfbelzfb8clvwnp#tmhgrvpgr|rnufb8cxftrqscvpb;gqsxQb tjqs|#b<m%tjvEp ¬j­r­rü <

º@»¬¼"½ ü ³P¾ *gQxQ8vwb;~tjnpehbvpnpnwmo8b;nw fehnpuf;xftrqscufq~gxfvpb;|rvpgrstrva!tÄs|rufehb ü ³ ehbdfeh8cobqmhbSufqÌ~gQxQvpb*trnwehbyvwnp~nmhb*zstruQmobeh8cogrvpuQmonpgrq nw feonuf*xftjqc usqË~gQxQ8vwb|rvpgrtjv-©trnwsvwbeh8cogrvpuQmhnwgqSbm-¨grel<;tjuQC¨eogqmhnw8eobcvptjmoetjvpb8cdstre-vwbc-xQgrqsqfbc|rvpgrstrvwbc

iFh k ]¹ zElLmolyp%l wÓt q tjqcqfgjmheob#ltc¤qsgruscuQmhnwvpnpcogrqcvptyrb8ehconpgrqeh|nwgqstjvpb xfu:~gQxQ8vwb#ustrconw¥E|gcmoehgrdszfnpufb t6rb8l vtS~§8b#trdfdfehglzsbxQu~gQxQ8vwbdstre©trnmª$qSb¡Zbmqsgrusc©tjncogrqscmogufehqfbevwb~gQxQvpbxftjqsclb8cxQbufCrb8ehconwgq |vwgstjvpb%bmtjnpeobvwnp~nmhb6truC~§~b

¬r¬

Page 24: Í Î4Ï>Ð Ñ - polytechniquedubos/Men/Omrani09.pdfÍ Î4Ï>Ð Ñ ÒÓÑÔ Õ Î.ÖX× ØÙÑ5Ô Ú>Û Ü[Ý Þ ß à á!ârÝ8ã©ßZÜ ä åçæ èfà!Þ é à áÌÝ Þ èrêfèsã©ë

Z Ê~Z5V VSÆ`VìO SOR]Vì`5aá`^@OPR]\V MNPOQRTSUWV ZEá^@`ÅV©UW¢¡@RT^@\POPÌOPR]UWV©ÅR]RTS ¢BV

mhb~dsc8vt;dfehb~npehb-dfehnpcob%l<g~b%eo¨ehbqslb-b<mgq~¨grel<b-vt9cob8l<gqsxQb%t6b8lvpb8c$xfgrqfqf8b8cxQ|ehtxQbc ©trnwsvwbeh8cogrvpuQmhnwgq1a!t~xQbehqfnpehbmhtrdb9lgrqsconpcmob;~l<g~dstjehbevpb8c-xQb8uQSlzstrdc Äs|rufehb ü ¬ <

º@»¬¼½ ü ¬_¾ %ntj|ehtr~bb<Qdfvpnltmhn&xQbvt ~mozfgQxQb9xQu~gQxQvpb;dstre©trnmtjdsdfvwnufb;truya ±

iÒ Âq ¹ moq1p » ¹»yt p%p%qïm _ mol `»yt îEq# gruscqsgruscdfehgrdZgcogrqscxftrqscl<bmmobdstremhnwb;x¤ <mhusxQnpbe-vE b¡ZbmxQbvpt mhtrnwvpvpbxQu*xQg#tjnpqfbxfuya ± Æcufecht dfeo8npconwsnwvpnmhrQª3qb¡Zbm8vptymtjnpvwvpbCxQuxfgr#tjnpqfb,xQ8db8qsxxQb,xQb8uQÅ6treontjsvwbcÅHvpb#qfgr fehb,xQb,et6¯rgrqcxfb5-gchco¯

n×Rdb<m vwb#qfg(fehb,xQb

dZgrnpqm%xQb;|rehnpvwvpbuQmonpvwncoddcoufe-ufqxQgr#trnwqfb;xQb;vtjeh|rb8ufe

L = n×Rd°$vwusconpbufecmousxQbc-grqmmo;©trnmhb8cmheht6becvV ª3ufehgrdZb;dgufe86trvwusbevE b¡ZbmxQb9vtmhtrnwvpvpb9xQuxQgr#trnwqfb9x ufq/rgqfb8cbm(tjvE k³®r®ÿ £gqml<gqsl<vpu:ufb~vtSmhtjnpvpvwb#xQuÅxQgr#trnwqfb#x¤ ufq§ûxQgnm9§moehbcou$,chtj~~bqm;dbmonwmob#dZgrufe9ufb~vpb~gQxQ8vwbqsbc8 8ltremhbCdstc moehgrd xfb8c~lzstj~dsc~xQbvt|ehtrqsxQbS8lzfb8vwvpbSnp~dgccdstje~vpb9ýb<m,cou$,cotr~b8qm~|retjqxQbCdZgrufe~vpufndZbeh~b<momoehb9xQb9l<ehb8echtdfehgrdfehbxf¯qtj~npufbbmvwbc-dbmonwmobc-8lzfb8vwvpb8c-tcocogQl<npbcufbvpb(9Æqfb¨grufehqfncocobqm-dstrc8# grusc0qsgruscdfehgrdZgcogrqsctjnpqsconQxQblzsbelzfbevpttjehnptrfnpvwnwmoxQu mob~dsc&xfbdfehncnpfnpvwnwmob8q9¨grql1monpgrq~xQbvpt%mhtrnwvpvpbxfuxQg#tjnpqfbrab8c-eh8coufvwmhtjmhccgqm-xfgrqfqf8b8cdstre-vwbmtjfvpb8tru ©ü ³

Ú"ÛÜ ½ ü ³¾ tjehntmonpgrqxQuymob8~dscxQbdfehncnpfnpvwnwmoTpbqy¨gqsl1mhnwgqyxQb;vt mhtrnwvpvwb9xfua ±

dd 64 × 64 80× 80 96 × 96 112× 112LRd

12 15 18 21

Tp 18, 3574 14, 7006 13, 5084 10, 0816

ª$qStjus|r~bqmhtrqm$vpt9mhtrnwvpvwbxQuSxQg#tjnpqfbvt9dfehncnpfnpvwnwmo%xfnw~npqusbrsb8lnsdZbuQm-c b<Qdfvpnusbe$dtjexfbuQ#etjncgqscrª3qCb¡Zbm8vptdseob8npehb(bckmvpt,mhtrnwvpvwbltrehtl1mo8eoncmonusb9x¤ usqmogufehfnwvpvpgrqxQb rgemhnplnmh9dgrmob8qmhnwb8vwvpb~usnxQb vV grexQehb(xQueht6¯grq*xQbGgchc¯Rd

± vpgrec-dfvpusc-vtmhtrnwvpvwb xQu*xQgr#tjnpqfb9truf|r~bqmhb;dfvwuc-vwb(qfgr feob;xQbc-mhgrufehfnpvwvpgrqscbckm|retjqsxSb<mdfvpusc%qfgrusc%t6rgrqcxfb8cnpqmob8ehtl1monpgrqybqmheobvpb8c-xQnw¡Z8eob8qmhcmhgrufehfnpvwvpgrqscxQgqslgrqc8 tjmmob8qsx~usbvt~dfehncnpfnpvwnwmo;xQu~gQxQvpb;xQnw~npqufbra!tSxfbuQQnp~b~ehtrnpcogrqbckmvpb(mhb~dsc;x¤ trxQb8l<monpgrq¤lj b8cmvwb mob8dcqf8lb8chcotrnwehb Sufqmhgrufehfnpvwvpgrq:xQb~rgemhnplnmh(dZgjmob8qmonpbvpvwbdZgruse-moet6rb8ehcobevwb9xQg#tjnpqfb; vtnmhb8chcb;ltrehtl1mhehnpcmonufb

UQ¢£vxQ<Äqfndstre

Tadv = L/U # gruscdZgrufgrqscxQgqsl8leonpehb ²

Tadv = LRd

Rd

U = LRd

× T ⇔ Tadv

T = LRd

Tadv

T

b8cmvpbmob8~dscx¤ trxQb8l<monpgrqStxQnpb8qmonpgrqfqf¬ ó

Page 25: Í Î4Ï>Ð Ñ - polytechniquedubos/Men/Omrani09.pdfÍ Î4Ï>Ð Ñ ÒÓÑÔ Õ Î.ÖX× ØÙÑ5Ô Ú>Û Ü[Ý Þ ß à á!ârÝ8ã©ßZÜ ä åçæ èfà!Þ é à áÌÝ Þ èrêfèsã©ë

Z ZE`V"ÅM V"\GS V©V\/¢¨MNVuÅVìVSâ¡aR]`O_¡@VsMNPOQRTSUWV ZEá^@`ÅV©UW¢¡@RT^@\POPÌOPR]UWV©ÅR]RTS ¢BV

± vwgehcdfvpuscvwbxQgr#tjnpqfb9bckmdbmonwmdsvwuscvpb9mhb~dscx¤ txQrbl1mhnwgq*b8cmdZb<monwm;xQbtjqmvwb(mob~dscxQbdfehnpconpfnwvpnwmoTpb<m;xQgrqsl

lr bckml<g~b%consqsgrusct6rgrqc&|rufnxQvt9conw ufvtmonpgrqrvpb%a ± q t9xQgrql-dstrc$vwbmob~dscxQb%xfnwbeh|rbe$b<mxQbl<ehb8e$l<bc3dfehgrdfehb8clztj~dsc8s°&tjel<grqmheob9convpb(xQg~trnwqsb9b8cm%|retjqx¤

TadvxQbnpbqm%|retjqx*xQbtjqm

Tpb<m%vpb a ± xQnprb8eo|b;dstjeehtrdfdgem%truQ

lztj~dsc-xQb;eh<¨8eob8qsl<b tjqc%mhgruQmhb(vtCcoufnwmob(qsgrusct6rgqsclzfgnpconxfb9moet6tjnpvwvpbet6rbl;ufqfb(mhtjnpvpvwbxQua ±

L = 18Rdxstjqscufqxfgr#tjnpqfb9|vwgstjv

xfb24Rd

i í-q ^¼ qWî ¹ qsqWî ]-¼ zoqWp%p%qïq ¹ _ot m »Ów qùP¬ù y±Ë´P±³Ç_¯´PĨë·P°êٱ̴±²ó©]ÇÇagrecxfb8ccnp(usvptjmonpgrqscqfgruct6grqsc$eob8~trehufufqCb¡ZbmxQbZgrexfcnw~dZgreomhtrqm-ufn¤t9ufqsbtrdsvwnwmousxfb|ehtrqsxQbdtjeehtrdfdgemtru:lztj~d:xfb#strcob bm(ufn0©tjuschcob~xfb#~trqfnpehbconw|qfnwÄltjmonprbvwbc6trvwb8ufec;xQb~rgreomonl<nwmo~dfehnpqsl<npdstjvpb~b8qmdZgrufe;vtydbmonwmoblzfbvpvpb Äs|ufeob ü ó1

2 4 6 8 10 12 14 16 18 20 22 24

2

4

6

8

10

12

14

16

18

20

22

24

x

y

−10

−8

−6

−4

−2

0

2

4

6

8

10

º»¬¼"½ ü ó ¾ ª&Qb~dfvpb9xQb;vV b<¡[b<m%xfb(nwfc-tjugehxsc-xQuya ± ~vE nwqmo8eonpbusexQu*ltrxfeobqfgrnpe1

ª$q,b<¡[b<m%vwgehc$xQbvE mousxQbxQb8c$c8eonpb8c3xfb ï gufeonpbe$b<mxQbc&mhehtrqsck¨geo~8b8c&xQb ï grufehnpbejnwvZtjdfdtjet6m0dtjeo¨grnc&ufqfb%xf<¨greh#tmhnwgqxfuÌconp|rqstrvV&l<gqfqufb*cguscvwbqfg4xQb*dfzsqfg8qfb*xQbnwsscb*dszfqfg~qfbb8cm#ufqËb<¡[b<mCxQbZgrex ufncobdfeogQxQusnm:dfehg6Qnw~nwmox¤ ufqfb*xQncolgrqmonpqufnmhr& tjqsc~qfgrmoehbyl8trcvtxQnchl<grqmhnwqufnwmocbymheogufrbSb<¡[b8l1mhnwb~bqm,tjufgehxfcb8qmheobvpb8cxfgrqfqf8b8cÄsvwmoehbc ©tjnpfvpbeh8cogrvpuQmhnwgqSbmvwb;lzstrdydfehgQxQufnwmdstje-vpb;a ± .~ufqsbeh8cogrvpuQmhnwgqSdfvpusc8vwb8rr

ùP¬ù­ ó©P´PôB±°0gufemousxfnwb8evE b¡Zbmxfu,|rufnxftr|rb%coufevwba ± qsgrusc$uQmonpvwncogrqscvpt(~§~b%~<mhzfgQxQgrvpgr|nwbusb%dZgrufevwb~gxfvpb|vwgstjvErª3qb¡Zbm8jqsgrusc¨geFCgrqscvpba ± 5tjuf;¨eogqmhnw8eobcvptjmoetjvpb8cdstre&xQb8c&xQgqfqf8b8cÄsvwmoehbc&t6rb8lxQnw¡Z8eob8qmc0ehtrdfdZgreom0xQb-eh8cogrvpuQmhnwgq

αmhgruQmbq#|rusnpxftrqm0dtje$lb-§8~b%lzstj~d#xQbxQgqfqf8b8c&#tjnc3coufe&mhgruQmhbvpt|eonpvpvwb-xQu,a ± /ab8c3eocufvwmhtjmhc&cogrqm$l<g~dstjetjfvpb8ctru*ltrc|vwgstjvE

Ÿ

Page 26: Í Î4Ï>Ð Ñ - polytechniquedubos/Men/Omrani09.pdfÍ Î4Ï>Ð Ñ ÒÓÑÔ Õ Î.ÖX× ØÙÑ5Ô Ú>Û Ü[Ý Þ ß à á!ârÝ8ã©ßZÜ ä åçæ èfà!Þ é à áÌÝ Þ èrêfèsã©ë

Z ZE`V"ÅM V"\GS V©V\/¢¨MNVuÅVìVSâ¡aR]`O_¡@VsMNPOQRTSUWV ZEá^@`ÅV©UW¢¡@RT^@\POPÌOPR]UWV©ÅR]RTS ¢BV

º@»¬¼"½ ü üE¾ ztj~d:xfb#rgreomonl<nwmo~dZgrufe;vtl<guslzfb#cusd8eonpbufehb α = 1/2

τn = 0, 4

vtSdseob8npehb~vwnp|rqsb~lgreheobcdZgrqxtruQClzstrdcxfbeh<¨8eob8qsl<b tsZ%vt cob8lgrqsxfbtruQClzstj~dsc|ufnpxf ljQxsb<m-vt xQb8eoqsnw8eob(vt xfn¡[ehbqslbb8qmoehbvpb8cxQb8uQlztj~dsc bQh<svt dseob8npehb;l<gvwgqfqfb;l<geoehb8codgqsxCvtdbmonwmob;8lzsbvpvwb;bmvt#cob8l<gqsxQb~vpt|ehtrqsxQb8lzfb8vwvpb

«Qufevt;Ä|rufehb ©ü ü qfgusc$ehb#trehufgqsc3ufb%dZgruseufq,mob~dscxQb|rufnxftr|rb%xQbvV grexQehb%xQbτ = 0, 4Tp

vpbgQxQ8vwbeob8dfehgxfufnmdsvwusc&tru9~grnpqsc0l<geoehb8l<mob~b8qmvpt%|retjqsxfb$8lzsbvpvwbt6rbl&ufqfbbehehbufetjuQmhgrufe0xQb

10%°&tje0l<grqmheob8npvft%xQu#tjvQehbdseocb8qmhbe

vt dbmonwmob:8lzfb8vwvpbsnwb8qXufbvpb8cydfehnpqsl<npdstjvpb8cSckmheoul1mouseobcScogrqmSdseocb8qmhb8c8ab~gQxQ8vwb:cb8(fvpb:coufeomoguQmycogruscSb8cmonp~bevE nwqmob8qsconmh9xQu*lzstj~d# gruscehb#tjeufgrqc$tjucoconZufbvE b8eoehbufevt(dfvpuscnpdZgreomhtrqmobcobl<grql<bqmheobcufevpb8cgehxscxfuSxQg~trnwqsbufn¤dZbuQm§<mheobxQufbvtxQncolgrqmonpqufnmh-bqmoehbvwbc3lzstrdc&xQb-rgemhnplnmhtru~gehxfc0b8qmoehbvpb8c$xQgrqfqsb8cÄvmheo8b8c3©tjnpfvpb-eocgvwufmonpgrqb<m3vpblzstj~dscdseogQxQufnwmhcdstje-vpbgQxQ8vwb9lb;usb;qfgrusc-t6grqscxQÄsqfndseol<8xfb~~bqm-dstje-vE b¡Zbm%xQb(npfsc8°0gufe-ufqfb~bnpvwvpbuseobehbdfeh8cobqmhtjmonpgrqCxfb8clzstj~dsc-xfbgreomonl<nwmoqsgrusct6rgqscbqfvpbmhgruscvpb8c-xQgqfqf8b8c-coufevwbcgehxfcxQuxfgr#tjnpqfbdgufecoufdfdfehnw~b8e-vV b<¡[b<mxQb(npfsc-ufnlgr~~bqfgruscvE t6grqscuS©tjuschcb8qfgreh~~bqmvV tj~dfvpnmhusxQb9xQu*lzstr~dxfb;rgjmhnplnmhr

¬rÿ

Page 27: Í Î4Ï>Ð Ñ - polytechniquedubos/Men/Omrani09.pdfÍ Î4Ï>Ð Ñ ÒÓÑÔ Õ Î.ÖX× ØÙÑ5Ô Ú>Û Ü[Ý Þ ß à á!ârÝ8ã©ßZÜ ä åçæ èfà!Þ é à áÌÝ Þ èrêfèsã©ë

Z ZE`V"ÅM V"\GS V©V\/¢¨MNVuÅVìVSâ¡aR]`O_¡@VsMNPOQRTSUWV ZEá^@`ÅV©UW¢¡@RT^@\POPÌOPR]UWV©ÅR]RTS ¢BV

−2 −1 0 1 2−2

−1

0

1

2

qref

q lam

−10 −5 0 5 10−10

−5

0

5

10

qref

q lam

−2 −1 0 1 2−2

−1

0

1

2

qref

q lam

−10 −5 0 5 10−10

−5

0

5

10

qref

q lam

−2 −1 0 1 2−2

−1

0

1

2

qref

q lam

−10 −5 0 5 10−10

−5

0

5

10

qref

q lam

r=0,4390a=0,0874

(a) (b)

(c) (d)

(e) (f)

r=0,9997a=0,9978

r=0,9477a=0,7723

r=0,9969a=0,9965

r=0,0131a=0,0094

r=0,6195a=0,6771

º@»¬¼"½ ü ÿb¾ # ustr|rb;xQbdgnwqmc dZgrnpqmcqfgrnpech$b<mxQehgrnwmobx¤ tikuscmob8b8qm bqyeoguf|rb6&dZgrufe-vt dbmonwmob tsQljQbbmvt|retjqsxfblzfbvpvpb sxQht6rblxfn¡[ehbqmhc-mob~dsc%xQb|rufnxftj|b

τ = 0.01Tp tff<

τ = 0.4Tp ljsxb<m

τ = Tp brfh

°0gufeufqmhb~dscxfb&|rufnxftj|bmoeh8c¤dZb<mhnmτ = 0.01Tp

Äs|ruseob ü ÿ 1dstjqfqsb8tju<vt|retjqsxfblzfbvpvpb3l<grehehb8codgqsx%dtjeo©tjnwmob~b8qmvteo¨ehbqslbbmgqSehb<mheogufrbufqfb;l<geoehvtmhnwgq,dfehb8chufbdtjeo©tjnwmobrfa!t(dZb<mhnmhb;8lzfb8vwvpbr dstjqsqfb8truSt<Qb8cml<gr~dfvp<mhb~bqmxf<¥£l<grehehvpbrfagrecoufb

τxQb8nwb8qm|ehtrqsx

τ = TpvV beheob8ufecoufevt~|retjqsxQb;lzfbvpvpbr dstjqfqsb8tjuyh<Zxfbnwb8qm%nw~dZgreomhtrqmhb;b<m

b8qmoet6 qfbdstjelgrqsco8ufb8qsl<bufqfb;|ehtrqsxQbbeheob8ufe-cuse-vptdZb<monwmob;lzfbvpvpbr dstjqfqsb8tjuyb<ab;mob8~dscxQb9|ufnxftj|bufnnpqfnp~npcob(vpt¨gnpcvV beheob8ufecoufevt~dbmonwmob9bm%vpt~|ehtrqsxQb8lzsbvpvwb dstrqfqfb8truQyl9b<m%xs<sb8cm%xQbvE grexQeob;xQb|ehtrqsxQbusexfbmob~dsc%xQb;dfehnpconpfnwvpnwmo

τ = 0, 4Tp

J%qfb~#tjqfnpehb dsvwusc|qf8ehtrvwbx¤ tjqstrvw¯QcobevV b<¡[b<m(xQu:|rusnpxftr|rbcufevtyeob8dfeocb8qmhtmhnwgqxQb~vptSdZb<monwmob#bm;vptS|retjqsxfb lzfbvpvwbxstjqsc-vpb;a ± bckm%xfbdseogQl<xQbe~xQnw¡[ehbqmhcÄsvwmoetj|b8ct6rblxQn¡[ehbqmcmob~dsc%xQb;|rufnxftr|rbr

¬&

Page 28: Í Î4Ï>Ð Ñ - polytechniquedubos/Men/Omrani09.pdfÍ Î4Ï>Ð Ñ ÒÓÑÔ Õ Î.ÖX× ØÙÑ5Ô Ú>Û Ü[Ý Þ ß à á!ârÝ8ã©ßZÜ ä åçæ èfà!Þ é à áÌÝ Þ èrêfèsã©ë

Z ZE`V"ÅM V"\GS V©V\/¢¨MNVuÅVìVSâ¡aR]`O_¡@VsMNPOQRTSUWV ZEá^@`ÅV©UW¢¡@RT^@\POPÌOPR]UWV©ÅR]RTS ¢BV

ab(xQntj|qfgcmonufb;eobckmhb;vwb;~§8bqsgrusc-ehbdseocb8qmhgrqscvpb9l<gbT$#lnwb8qmxQb(lgreheo8vptjmonpgrqrbm%vpt~dZbqmhb xQb;vt,xQehgrnwmob9x¤ tikuck¥

mhb~bqmab8q¨grqsl<monpgrqxQumob~dscxQb(|ufnxftj|b;qfgreh#tjvpnpco

τndstre%vpb9mhb~dsc%xfb(dfehncnpfnpvwnwmo(t6b8l;xQnw¡[ehbqmetjdfdZgreom%xQb

eh8cogrvpuQmhnwgqα

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

τn

a

α=1/2α=1/4α=1/6α=1/8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

τn

r

α=1/2α=1/4α=1/6α=1/8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

τn

a

α=1/2α=1/4α=1/6α=1/8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

τn

r

α=1/2α=1/4α=1/6α=1/8

(a)

(c) (d)

(b)

º@»¬¼"½ ü o¾ at*dZbqmobabm vpbSl<gbT$#lnwb8qm xQbClgreheo8vptjmonpgrq

rdgufe(vtdZb<mhnmhb ts9b<mvpt*|ehtrqsxQb,lzfbvpvpb lrxs9dgufe

xfn¡[ehbqmobcα

°0gufe%ufqα = 1/2

qsgrusct6rgrqcufqfb;mheoc-gqfqfb(lgreheo8vptjmonpgrqxQb9vt#|retjqxQb8lzfb8vwvpb9dZgrufe%xfb8c-mhb~dscxQb9|rusnpxftr|rb9trvwvtjqmikucou

τ = 0, 7TpQ°0grufe-vt~dZb<monwmob;lzfbvpvpbrQvE gdQmonp(ufcb8(fvpb§<mheob9truQmogufe-xQb

τ = 0, 4Tp

± dstremhnwexQbα = 1/4

vwb~~gQxQvpbq treoehnprb9dfvpusc;CdfeogQxQusnwehb usqfb Zgrqsqfb~dbmonwmob~8lzfb8vwvpbrZdstrelgrqmoehb(vtS|retjqsxQb 8lzfb8vwvpbehb8cmob;lgreheobl1mob

¬ í

Page 29: Í Î4Ï>Ð Ñ - polytechniquedubos/Men/Omrani09.pdfÍ Î4Ï>Ð Ñ ÒÓÑÔ Õ Î.ÖX× ØÙÑ5Ô Ú>Û Ü[Ý Þ ß à á!ârÝ8ã©ßZÜ ä åçæ èfà!Þ é à áÌÝ Þ èrêfèsã©ë

* +-,/.10325476

d < : ?ÉÐWÔ ×@< : Ô Ñ´Ö ÑÙKÔ Ñf?ôÖX×eÊÑ5Ô

lÓî ¼ p _y^t lyî ^absuQmxQbl<b$moet6tjnpvr<mtjnwm0x mousxQnpbevtehbdfeh8cobqmhtjmonpgrq;xQbcdfehgQl<bcocouscxfbdZb<mhnmhblzfbvpvwbxftjqcvt~gQxQvpncotjmonpgrq;eh|rnpgrqtjvpbxfuylvwnp#tmb<m-vV npqQþsufb8qsl<b;xQuS|rufnxftj|bcoufevpt tjehnptrfnpvwnwmonwqmhbehqfbxQuy~gQxQvpbbmcotltrdstrlnmh dfeogQxQusnwehbxQbc-lzstj~dscxQbdZb<mhnmhb98lzsbvpvwb;l<geoehb8l<mobdstjeomonpexQb8c-xfgrqfqf8b8c- ©tjnpfvpb;eocgvwufmonpgrq°0gufe-l<b©tjnpeobqsgrusct6rgqsc3ufmonpvwncufqS~gxfvpbustcnw¥À|rgckmheogdfzfnufb xfbuQClgruslzfbcufqy~gxfvpbehbvtmhnwb~bqmconpdsvwb#trnpcSeotjvpnpcmobusndZbeh~b<mSusqfbZgrqfqsbehbdfeh8cobqmhtjmonpgrq xQbvptxf¯qtj~npufb:tmhgcdszfehnpufb*dsnwvpgjmhbdtjeCvE nwqscmhtrfnpvwnwmotjehglvwnpqfb~Jqfb;mousxQb9coufevtdfehnpconpfnwvpnwmo;xQu*c¯Qcmo8b9lvwnp#tmhnpufb;t~<mh;b<¡[b8l<mouf8b9t6rb8lufqbqcb8(fvpb;xQb9cnp(usvptjmonpgrqscquf~8eonufb8c;coufe;vtSbecnpgrq|rvpgrstrvwb~xQuÅ~gQxQvpb~b<m9vtSbecnpgrq:tjnpeobvpnw~nwmoba3 b<¡[b<m(xfu:|ufnxftj|bty<mh#b<ftj~npqf#t6rblvt~<mhzfgQxQb,xQu/~gQxQvpb#dstjeo©tjnwm9ufn$l<gqsconpcmob#|r8qfehbeufqlztj~dztjuQmhb#eocgvwuQmhnwgq¤bqcufnwmob,xfb~Äsvmheob8e9vpb8c;dZb<mhnmhb8clzfbvpvpb8cb<mbqQÄquQmhnwvpnpcobe;lb8cxQgrqfqsb8c,©tjnpfvwbeh8cogrvpuQmonpgrqdZgruse|ufnxQbevwb~gQxQvpb eh|nwgqstjvE bdfvwucusqfb~<mouxQbcoufevE nwqQþufbqslb;xQbvt(mhtjnpvpvwbxQuxQgr#tjnpqfbxQuC~gQxQ8vwbcusevt dfeo8npconwsnwvpnmht <mh~b8qfbcusevt rbeconwgqCtjnpeobvwnp~nmhb;xQbqfgjmheob~gQxQ8vwbab8c;eh8coufvwmhtjmhc~grqmoehbqm9ufb~vptytjehntjfnpvwnwmo~npqmobehqfb,xQuÅ~gQxQvpb#xQdZbqx:¨greomob~b8qm9xQuÅ|ufnpxstj|rbÅJqfb~mhb8lzfqsnpufb#ufncob fvpbtj~vpnwgeob8e0vpt;l8tjdstl<nwmo-xQu~~gxfvpbehbdseogQxQufnpehbvpb8c$lzstj~dsc&xQb-eh<¨8eob8qsl<b9Çvteotjvpnmh;Çjra3 b<¡[b<m$xfu|ufnxftj|bcoufevt;ehbdfeh8cobqmtmonpgrq~xQbc&dbmonwmobc$lzfbvpvwbc&b8cm&moeh8c$nwqfþsufbql<dstre3vpbmob~dscxQbdfehncnpfnpvwnwmoagrecoufbvpb-mob8dc$xQb|ufnxftj|bbckmmoeh8c dZb<monwm~dstjeehtrdfdZgreom tju/mhb~dsc~xQbydfehnpconpfnwvpnwmorvpbC~gQxQvpbSehbdseogQxQufnwm vpb8clzstrdc9|retjqxQbClzfbvpvwbC¨greDCtrqmc tjqcvpb~ltrcgruvwb mob8dcxfb |ufnpxstj|rb(b8cmxQbvE gehxQehb(xQumob8dcxfb dfehncnpfnpvwnwmo[vwb~gxfvpb coblgr~dgemhb lgr~~b(bqtrscobqslbyxQby|rusnpxftr|rb,guHvt|ehtrqsxQbClzfbvpvpbCb8cm~~trveob8dfeogQxQusnmhbCb<m#xQnprb8eo|bSxQbc lzstr~dsc~xQbSeh<¨8eob8qsl<b3abC~gQxQ8vwb¨gufehqfn$vpb8c;~b8nwvpvwb8ufec;eocusvmtmhc9dgufe9ufqÅmhb~dsc(xfb,|rufnxftr|rb#xQb,vE gehxfeob#xQuÅmhb~dsc(xfb,dfehnpconpfnwvpnwmo

τ = 0, 5Tp<!at

mtjnpvwvpb(xQuxQgr#tjnpqfb;bckmufqfb9tjehntjfvpb9l<vpb<3xftrqscvt#vE mousxQb xQb(vt~dfehnpconpfnwvpnwmo(xQua ± sª3qb¡Zbm8Zdfvpuscb8vwvpb(tjus|r~bqmobdsvwuscvt dfeo8npconwsnwvpnmhxfuyco¯Qckmh~bxfnw~npqusbrb8lnc b<Qdfvpnusbdstrevpb©tjnwmusbvpgrecoufbqfgjmheobxQg~trnwqsbbckmcou$,cotr~b8qm|ehtrqsx¤vwbc$npqmob8ehtl1monpgrqc$b8qmheobc$vpb8cxQnw¡Z8eob8qmhcl<gdZgchtjqmhb8c$xfuSco¯cmo8~btmh~gcodfzfehnusb%cgqmmheoc$np~dgemtjqmobclbufntruf|r~b8qmhb;cgqyl8tjetrl1mhehblzstrgjmhnpufbb<m-dstrel<grqcusbqslbvpnw~nwmob9cht(dseo8nconwfnpvpnmhrQª3qehbtjqslzsbustrqsxCvpb9xQg~trnwqsbbckmdZb<mhnmsvE txQrbl1mhnwgqSxQb;vE npqQ¨geo#tmhnwgqC|ehtrqsxQb8lzfb8vwvpb¨grufehqfnpbtruQCZgrexfc-cobdseogdstj|bt6b8lufqCmhb~dsc-dsvwusc-lgrufeomufbvwbmhb~dscxfb;dfeo8npconwsnwvpnmhb<m|rusnpxQb;truscocon¤vpb9a ±

¿=q º^ xÂq ¼$¹utñÀ q ^ab8c&b8qikbuf bq~~gQxQvpncotjmonpgrqeo8|rnpgrqstrvwbxQu#l<vpnw#tjm$cgqm&8qfgreh~b8c&xQu©tjnwm$xQb-vt¨greomobxQb8~trqsxQb-coglnwmhtrvwbbq~dstjeomonl<usvwnpbecoufe0vtustrqmhnÄZltmhnwgq xQbcnwqslbeomonwmousxfb8c0xQb8cdfehgjikbl1mhnwgqsc!lvwnp#tmhnpufbc!eh|nwgqstjvpb8c!b<m0vpbufec0l<grqcusbqslb8ccufe0vE 8tjvpustmhnwgqxfb vt,ufvpqfetjsnwvpnmh xQb qfgccoglnwmocxftrqscufql<gqmob<mhb xQbeolzstjuf¡Zb8b8qml<vpnw#tjmonusbr«n0l<b8c%moet6tjuQehdZgrqsxQb8qmxQb©tC<gqHtrchcb8¾#dstjeomonpbvpvpbSlbeomhtjnpqsc xQbSlb8c(ufbckmhnwgqsc(lvwc0xQbSqfgr feob8uscob8c;ehb8cmob8qmbqslgrehb#grufbeomobc!°0treo~nl<bvpvpb8c¥ÀlnVqsgrusc-dZgrusrgrqcnxQb8qmhnÄbe-mheognpcufndstrehtrnpchcobqmvpb8cdsvwuscdb8emhnwqfb8qmob8c-~lgrufeomob8lzs8tjql<b~²³ ikuschu #trnwqmob8qstjqm3qfgusc$t6grqscuQmhnwvpncusq#|rufnxftj|b-qfgrq#xQnchl<ehnw~npqstmhgrnpeobjlj b8cm¥£¥£xQnwehbqfgruc3t6rgqsc0|rufnxQmhgruQmhb8cvpb8clzfbvpvpb8c%coufe-mhgruQmvV b8codstl<b9xQuxfgr#tjnpqfbrZat,ufbckmhnwgqufn!cob9dZgcob9b8cml<b u ufq|rufnxftj|b;dstje%ufqfb(~mozfgQxQb(usvmhn¥À8lzfb8vwvpb gqsxQbvpb<momobch<fufqb<¡[b<m%lgr~dstjetjsvwbtjuy|rufnxftr|rb;cdZb8l<moetjv&fcobetjnwm-dfvwuctjdfdseogdfeonp!È

¬ -l<g~b8qm xQ<ÄqfnwexQb#©tC<grqÅgQikb8l<monprb#vtmhtrnwvpvwbSxQuxQgr#trnwqfb#bm(vt*eocgvwufmonpgrq/xQu/~gQxQvpb,dZgrufe9usqfbC~bnpvpvwb8ufeobehbdfeh8cobqmhtjmonpgrqyxQbvpt#lnwel<ufvtmhnwgqStjmo~gcodfzf8eonufb;xQbÄsqfb;8lzsbvpvwb;xftrqscufqfb;eh|nwgqSxQgqfqfbÈ

ófdZbuQmo¥Egq#nwqmoehgQxQufnpeob-lb<mmhb~<mhzfgQxQb%xftrqsc&ufq#etjnQ~gQxQvpbeh|rnpgrqtjvft6rblufq,xQb|eo-xQbl<gr~dfvpb<Qnwmo-dfvpusc$vpbr-ufbvpb;gQxQ8vwb;ustcnw¥E|gckmheogdfzfnusbmob8vufb ï btmhzfbePb8cob8tjelzStrqsx ï grehb8l8trcmonpqf|~*gQxQbv fÈ

¬ ð

Page 30: Í Î4Ï>Ð Ñ - polytechniquedubos/Men/Omrani09.pdfÍ Î4Ï>Ð Ñ ÒÓÑÔ Õ Î.ÖX× ØÙÑ5Ô Ú>Û Ü[Ý Þ ß à á!ârÝ8ã©ßZÜ ä åçæ èfà!Þ é à áÌÝ Þ èrêfèsã©ë

×Ï>Ð-×@<ûÙKÎ×Ñ

± vwbftjqsxQehu ± p b9ª3vpnpts3;ptrqsx*atrdfehnpcobr~; V¬r­r­ ír1¢£qmobehqstrvtjehnptrfnwvpnwmk¯Cnpq*eob8|rnpgrqstrvl<vpnw#tmhb(xQg½-qschltrvwnpqf|#tm%mozfbcob8trcogrqtjv¤col8tjvpbr! #"%$'&)(+*-,/.10324&)(06587906:<;06=& ³ ó ÿs©® ² ó ¬¬Q³ñ¾ ó ¬ órðf

± qmozfbcB; ± wBÉufg?>B9p ò truf~zfb<¨qsbe0 &°0w&ª$eoehnplgs;&/p3tjqsx ò bmmh|rbr&¿ k³® ð ÿ 1Å°3ehb8xQnl1mtjfnpvpnmk¯/gr~b8cogchltrvwbtmhgcdszfbehnpl~gjmhnwgqsc8A@9B%:%2%$DC30FEG;)$IHJ03"LK(M,NEF;C6E ¬ ððC² ³ÿr®ß¾f¬j­¬

t6nwbcI9 k³® í 1 ± vtmob8eousv!Zgrufqxftjeh¯C¨geo ufvtmonpgrq¨ge ufvwmonw¥EvpbbvdfeobxQnl1monpgrqgQxQb8vpc8PORQ245F&S0T5F*-,VU+"4Q+5F$W2%*X"ZY&([0\78"%,24*] #0T&S03"%5L"%*^"L_4;C324*?`a"bCT;06&c, ³­¬f©ü ó ¬ -² ü­ÿ ¾üs³ ðs

bª$vwntf;watrdfehnpcobr9wtjqsx bqfnc8 ò E¬j­­¬ < ï grehb8ltckmhnwqs|cnpvwvvwnp~nmc,gj;qfbckmhb8x¤vpnw~nwmob8xQ¥ÀtreobtÅgQxQb8vpc² ±dZbeo¨b8l1mo¥E~gQxQb8vtjdfdseogtrlz #"%$&(M*d,/.1032%&([0T5J780T:b;0T=& ³ ó ­ ð%² ¬j­r­ß¾f¬j­¬ óf

bqsnpc8 ò w¶rmorÅp[trqsx:atjdseoncb39 V¬j­­¬ 1 «dZb8l<moetjv3xQbl<gdZgconwmonpgrqgr3mk½gj¥£xQnw~b8qscnpgrqtjv0tmo~gcdfzsbehnpl(Äsbvxfcgqvpnw~nwmob8xQ¥Àtreobt(xQg~trnwqcucnpqf|9mhzfb;xQncoleobmobl<gconpqfbmhehtrqsck¨geo xfl1m11 e"4$&(M*-,.f032%&([0T5R7906:<;06=& ³ ó ­ íj² ³ ð ³6¬ ¾Z³ ð ¬r®

npgreh|rnE ï tjqsx ò nE?g( V¬r­r­­ 1 ± cmousxf¯:gr-nwqmob8eoqtjv3tjehntjfnpvwnwmk¯:gjteob8|rnpgrqstrv$l<vpnp~tjmob,~gQxQb8vV/U+h\HJ03"LK[(+,NETh790FEThp³8­ÿ ² ¬r®ÿj­ ó ¾f¬j®ÿ¬r¬ tjnxQrg|rb8vV ( ò trqsxìb8vpx¤¢1/ k³8® ð ­ 1Yg~gr|rb8qfbgusc,ustcnw¥E|bgckmheogdfzfnlCmhufehfufvpbqslbxQehnwbq ¯ t/ufqfnw¨grehmob8~db8ehtjmoufehb;|retrxQnpbqm!U+"%Q+5F$D2%*i"jYk&)(0\@9&clm"NEcK[(065F;Cn`DCT;0T$WC30FE1sóí ³¬ %² ¬&jüü¾f¬&[­

grqsb8c8~;Zps*ufehdfz¯r~s/pstrqsx # gr|ufbef/ ³8®®ÿ <$«np(ufvtmhnwgqSgr0lvwnp#tmob9lztjqf|bgbebuseogdbusconwqs|,t qsb8cmob8xehb|rnpgrqtjvw¥Àlvwnp#tmob&~gQxQb8vV8n² ± chcbcoco~bqmgjslgrqmoehgrvl<vpnw#tmhbrnwql<vpusxQnpqf|%cobqsconmhnwnwmk¯%mhg%vpgQltmhnwgqgjfvtmob8ehtrvjZgrufqsxstjehnwbcORQ2%5F&o065F*d,pUM"4Q+5F$W24*i"ZYk&([0\78"4,2%*a #0T&S03"%5L"%*^"L_4;C324*?`a"bCT;06&c, ³6¬Q³ò ² ³ü³ ó ¾Z³üü®

a!tjdfehncb;p tjeh~ts/;pQ %b8qfnpc8 ò pst6¯tsr (pQtjqsxq[t6½txQ¾nEj¢1 V¬j­­r­ 1[°3ehb8xQnl1mtjfnpvwnwmk¯~gj!t9qfb8cmobx,vpnw~nwmob8xQ¥Àtreobt~gQxQbvER #"%$&(M*d,/.1032%&([0T58780T:<;06=& ³6¬ ð k³¬ ² üs³8ü®¾üs³6ÿü

agrehbqf¾Qª # k³®[ ó1& b<mob8eo~npqfnckmhnplqfgqfdZbehnwgQxQnlþsg½;rUM"4Q+5F$W24*?"ZYG&([0k@9&slm"NEcK([0T5F;Ct`aC6;0T$WC306E1 ¬j­sE¬ ² ³ ó ­¾Z³8üs³ °$zfnwvpvpnwdsc8 # ± ³8®ÿ[ <!¿zfb|rb8qfbetjv[l<npel<ufvtmhnwgqCgj!mozfb;tjmo~gcodfzfb8eob² ± qusb8eonltrv[b<Qdb8eonp~bqm8\ORQ2%5F&o065F*d,fU+"4Q+5F$W2%*"ZYt&)(0979"%,2%*] e0T&o0L"45u"%*v"u_;C32%*i`D"<C6;0T&s,Zð ¬s ó ÿ¬ ² ³6¬ ó ¾[³rü

«ftjvtj~bz¿w %ehgrfnpqsconV°ptrqsxÌ %usgc¿ V¬r­r­® 1ÌAqÌquxQ|rnpqf|Åmonp~bgrdfmonpnp¾8tjmonpgrqÌnpq eob8|rnpgrqstrvl<vpnw#tmhb~gj¥xQb8vwvpnwqs|:² tjdsdfvwnltjmonpgrqmogSmozfb~~b8xQnwmob8eoetjqsb8tjqstrconpqeORQ2%5F&o065F*d,wU+"4Q+5F$W2%*A"jYx&([0x78"4,2%*y #06&o03"%5u"4*^"L_4;C32%*z`D"<C6;0T&c, bqehnpconpgrq<

trvwvpnc3KÉ~ k³8® ðró1¸Aq/mozfbdfehb8xfnpl<mhtjsnwvpnmk¯Ågrustrcon¥À|rb8gcmoehgrdfzsnplþsg½ ²&¿zfbyb<¡[b8l<mhcgjbmht:tjqsx/tjehglvwnpqfnl<nwmk¯rU+"%Q+5F$D2%*i"jYG&)(09@9&clf"%EcK[([0T5F;Cn`DCT;06$DC30FE1 ü­sk³ ² ³­ß¾Q¬ í

usnl<b8nplr$¿3tjqsx ª3eheonl<gK;3/ k³®r®­ 1Ë¿zfbnwqfþsufbql<bgjtjeomonwÄl<ntjvtrqsxdfz¯Qconpl8tjv©trl<mogehc ufdZgrqÌdfehb8xQnl1mtjfnpvwnwmk¯b8cmonp#tmhb8c-usconpqf|#tlgr~dfvpb<Cvpnpnwmobx¥£tjehb8t gQxQb8vVR #"4$&(M*-,/.f032%&([0T58780T:b;0T=& ³r³ ð íj%² ³8ür­¾Z³8ü ð ¬

usnl<b8nplr!¿!trqsx:°&tjb|vwb ± # ³8® ð ® <S¿zfb#npqQþsusbqslb,gj-grqsb<¥À½t6¯*npqmobetrl<monpqf|*vtmob8ehtrv&gufqsxftreo¯:l<gqsxQnwmonpgrqscufdZgrqdfehb8xQnl1mtjfnpvpnmk¯,gr!þg½npqyZgrusqsxQb8xyquf~behnltjv[~gQxQbvc8R e"4$&(M*-,I.1032%&([0T5J780T:b;0T=& ³r³ í E¬ ² ó ü­¾ ó ÿr­

tjqs|s4>wab8ufqf|sl<|eob8|rge8a3pabbr ( ¥¥É~w trqf|s ¥w %npqf|4> ptrqsxLÉnp(ufetf ï E¬j­­jü <3b|rnpgrqtjvfl<vpnp~tjmob~gQxQbvpnwqs|#²j°3ehgr|rehb8chc86lztjvpvwb8qf|rbcjtrqsxdfeogcdZb8l<mhc8?U+"%Q+5F$D2%*W"jYr&([08 #06&o03"%5u"4*^"u_;C32%*W`D"<CT;06&c,x"ZY8UM23K[24$srð ¬ ² ³6ÿj®®ß¾Z³ ¬ ðf

¬j®