geometrijska interpretacija dinamike diferentnih jednadzbi

8/13/2019 Geometrijska interpretacija dinamike diferentnih jednadzbi

http://slidepdf.com/reader/full/geometrijska-interpretacija-dinamike-diferentnih-jednadzbi 1/40



Sfnrz˅fb6 Nhofmhgf bknojnhmkozhjofaoh` nhsgrktoh` nhofmhc˅gh` shstkmf <

6.6 Tjbfm stflhaojsth . . . . . . . . . . . . . . . . . . . . . . . . . ;6.< Yhzufahzfchbf jrlhtf h vrkmkosgh` ohzjvf . . . . . . . . . . . . . 56.3 Jsbktabhvjst pjof˅sfobf rbks˅kobf nhikrkotok bknofn˅zlk . . . . . . 6<6.> Lrzhof ftrfgchbk jrlhtf . . . . . . . . . . . . . . . . . . . . . . 6;6.; Lhiurgfchbk. Lhiurgfchjoh nhbferfmh . . . . . . . . . . . . . . . 60

6.;.6 Lhiurgfchbk . . . . . . . . . . . . . . . . . . . . . . . . . 606.;.< Lhiurgfchjoh nhbferfmh . . . . . . . . . . . . . . . . . . <6

6.9 Ayfpuojvabkvh lrjbkvh . . . . . . . . . . . . . . . . . . . . . . . <3

< Ayokss-jvf nhikrkotof bknofnz˅lf <9<.6 Jsojvoh pjbmjvh . . . . . . . . . . . . . . . . . . . . . . . . . <9

Ahtkrfturf 30



Ekjmktrhbsgf hotkrprktfchbf nhofmhgk nhikrkotoh` bknofnz˅lh

[vjnkjrhbf nhsgrktoh` nhofmhc˅gh` shstkmf h nhikrkotoh` bknofnz˅lh sk hotkozhvoj

prjuc˅fvf u pjsabknobh` trhnksktpkt ejnhof, gfgj zlje zofc˅fboh` mftkmfthc˅gh`rkzuatftf h tkjrhbf, tfgj h zlje vkahgh` prhmbkof u lhjajehbh, stzhch, kgjojmhbh,`kmhbh, nrus˅tvkohm ofugfmf htn.

Gfnf sk prjuc˅fvf nhofmhgf nhikrkotoh` bknofc˅hof, prhbk svkef, rfnh sk jsabknk¾ckm2 jnrknhvfobk tfc˅fgf kgvhahlrhbumf h pkrjhnhc˅oh` tfc˅fgf, fofahzf obh-`jvk stflhaojsth h fshmptjtsgf stflhaojst tk jnrknhvfobk `fjthc˅oje pjofs˅fobf.

Nyofmhcf bk pfgkt rfzvhbko of lfzh pjzoftje rfc˅uofrsgje prjerfmfUjairfm Mft`kmfthcf, gfj gjakgchbf ejtjvh` faftf h iuogchbf zf prjuc˅fvfobk

nhsgrktoh` nhofmhc˅gh` shstkmf h nhikrkotoh` bknofnz˅lh. Tfgkt Nyofmhcf surfzvhah prjiksjrh M. \. S. Guakojvhc h Jrafonj Mkrhoj, sf [ohvkrzhtktf \`jnkHsafon ([SF).

Nyofmhcf sfnrz˅h ohz faftf h tkòhgf faeklfrsgk, oumkrhc˅gk h erfstc˅gk prh-rjnk2 ofafz˅kobk tfc˅fgf kgvhahlrhbumf h pkrhjnhc˅oh` tfc˅fgf, gafshstgfchbf stflha-ojsth kgvhahlrhbumf h pkrhjnhc˅oh` tfc˅fgf, fofahzf pjauchgausf, hzrfc˅uofvfobk hvhzufahzfchbf hovfrhbfoth, hzrfc˅uofvfobk h vhzufahzfchbf Ayfpuojvabkvh` iuogchbfh Ayfpuojvabkvh` lrjbkvf, ekokrhsfobk lhiurgfchjoh` nhbferfmf, vhzufahzfchbfstflhaok h okstflhaok mojejstrugjsth, tk hzrfc˅uofvfobk ljx nhmkozhbk.

Jshm tjef, hzvjroh gjn uoutfr Nyofmhcf sk mjz˅k prhafejnhth h prjs˅hrhthjn strfok gjrhsohgf s prjerfmkrsghm hsgustvjm.

Gjof˅coj, Nyofmhcf sk mjz˅k gjrhsthth zf grkhrfobk vrùosgk tkòhc˅gk nj-gumkotfchbk zf fofahzu nhsgrktoh` nhofmhc˅gh` shstkmf, gfj h vrùosgk erfstgkgjbk sk mjeu jonf hsgjrhsthth u okghm nruehm fpahgfchbfmf.

Nyofmhcf pfgkt, v. 3.4 (buah, <448) zf Ujairfm Mft`kmfthcf v. 9, 5 h0, mjz˅k sk lkspaftoj prkuzkth sf fnrksk2

`ttp2//www.mft`.urh.knu/Nyofmhcf

[ohvkrzhtkt u uzah 6 Trhrjnoj-mftkmfthc˅gh ifguatkt




6 Nhofmhgf bknojnhmkozhjofaoh` nhsgrktoh`nhofmhc˅gh` shstkmf

Nkstohchbf 6.6 Bknofnz˅lf jlahgf

xo +6 = i (xo ), o = 4, 6, . . . (6)

enbk bk i 2 H ↜ H (H hotkrvfa rkfaoh` lrjbkvf), sk ofzhvf nhikrkotojm bkn-ofnz˅ljm prvje rknf , f ubknoj ¾ckmj bk zvfth h bknojnhmkozhjofaoh nh-ofmhc˅gh shstkm .

\bks˅kobk bknofnz˅lk (6) bk svfgh ohz (xo )o∀O 4 gjbh zfnjvjabfvf bknofnz˅lu

(6) zf svk o = 4, 6, . . . .

Nkstohchbf 6.< Bknofnz˅lf jlahgf

xo +6 = f o xo + lo , o = 4, 6, . . . , (<)

enbk su (f o )o∀

O 4 h (l o )o∀

O 4 pjzofth ohzjvh rkfaoh` lrjbkvf, ofzhvf sk ahokfr-ojm nhikrkotojm bknofnz˅ljm prvje rknf .

[ sauc˅fbu gfnf bk lo = 4 (o = 4, 6, . . .), bknofnz˅lf (<) sk ofzhvf `jmj-ekojm , njg sk hof˅ck, tj bkst gfnf bk lo = 4 zf lfr bknoj o ∀ {4, 6, . . .}, bknofnz˅lf (<) ofzhvf ok`jmjekojm .

Jlhc˅oj sk bknofnz˅lh (<) njnfbk tfgjzvfoh uvbkt pjc˅ktoh` vrhbknojsth

x4 = ζ. (3)

Nhikrkotof bknofnz˅lf (<), zfbknoj s pjc˅ktohm uvbktjm (3), c˅hoh tzv. prjlakm pjc˅ktoh` vrhbknojsth (sgrf¾ckoj TTY).

Rf okgk gafsk nhikrkotoh` bknofnz˅lh, prhbk svkef zf okgk ahokfrok, mjeu¾ck bk nj¾ch nj jp¾cke rbks˅kobf. Mknuthm, u jp¾ckohtjm sauc˅fbu tj bk vraj tks˅gjpjsth¾ch. Rlje tjef sk, umbkstj rbks˅fvfobf bknofnz˅lk, pfz˅obf pjsvk¾cubk hsph-thvfobu pjofs˅fobf obkoje rbks˅kobf u jvhsojsth j pjc˅ktojm uvbktu x4.

[ohvkrzhtkt u uzah < Trhrjnoj-mftkmfthc˅gh ifguatkt




Sabknk¾chm prhmbkrjm haustrhrft ¾ckmj ofafz˅kobk tfc˅gk kgvhahlrhbumf h pkrh-jnhc˅oje rbks˅kobf nhikrkotok bknofnz˅lk.

Trhmbkr 6.6 Ajehsthc˅gf nhikrkotof bknofnz˅lf

xo +6 = i (xo ) = >xo (6∜xo ) (>)

hmf nvhbk tfc˅gk kgvhahlrhbumf h c˅kthrh kvkotufaok tfc˅gk kgvhahlrhbumf.

\bks˅kobk. ^fc˅gk kgvhahlrhbumf ofafzhmj rbks˅fvfbu¾ch bknofnz˅lu

x = i (x) = >x(6∜x)

jnfgak bkx(3∜>x) = 4

pf njlhbfmjx = 4 hah ¯x =

3>

.

Nfgak, tfc˅gk kgvhahlrhbumf su 4 h 3> .

Gfgj bk

i (6>

) = 3

>,

i (( i ( 6<

+ 6>− 3)) = 3

>,

i (i (6<

)) = 4 ,

i (i (i (6<

+ 6>− <))) = 4 ,

tj su kvkotufaok tfc˅gk kgvhahlrhbumf2 6>, 6

< + 6>− 3, 6

<, 6< + 6

>− <.

Bknofnz˅lf (>) tfgjnkr hmf pkrhjnhc˅oj rbks˅kobk pkrhjnf nvf

;

0 ∜ 6

0− ; ,

;

0 +

6

0− ;

h nvhbk kvkotufaoj pkrhjnhc˅ok tfc˅gk pkrhjnf nvf

6< ∜

60 64∜<− ; h

6<

+ 60 64∜<− ;

bkr bk

i (i (6< ¶

60 64∜<− ;)) = ;

0 +

60− ; .

[ohvkrzhtkt u uzah > Trhrjnoj-mftkmfthc˅gh ifguatkt




6.6 Tjbfm stflhaojsthNkstohchbf 6.> (Stflhaojst tfc˅gk kgvhahlrhbumf)

6. ^fc˅gf kgvhahlrhbumf x bknofnz˅lk (6) sk ofzhvf stflhaojm , hah ajgfaojstflhaojm , fgj zf svfgj λ ? 4 pjstjbh ξ ? 4 tfgj nf

|x4∜x| 1 ξ ⇜ |xo ∜x| 1 λ, zf svk o ≫4.

<. fc˅gf kgvhahlrhbumf ofzhvf sk okstflhaojm fgj ohbk stflhaof.

3. ^fc˅gf kgvhahlrhbumf x bknofnz˅lk (6) sk ofzhvf ajgfaohm ftrfgtjrjm

fgj pjstjbh γ ? 4 tfgvj nf

x4 ∀H h |x4∜x| 1 γ ⇜ ahmo↜∟

xo = x.

>. fc˅gf kgvhahlrhbumf x bknofnz˅lk (6) sk ofzhvf ajgfaoj fshmptjtsgh stflhaojm , hah shogjm, hah ftrfgthvojm stgsojm tfc˅gjm prksahgfvfobf i fgj bk jof stflhaof h fgj bk ajgfaoh ftrfgtjr.

;. ^fc˅gf kgvhahlrhbumf x bknofnz˅lk (6) sk ofzhvf eajlfaohm ftrfgtjrjm fgj

x4 ∀H ⇜ ahmo↜∟xo = x.

9. ^fc˅gf kgvhahlrhbumf x bknofnz˅lk (6) sk ofzhvf eajlfaoj fshmptjtsgh stflhaojm fgj bk jof stflhaof h fgj bk eajlfaoh ftrfgtjr.

5. ^fc˅gf kgvhahlrhbumf x bknofnz˅lk (6) sk ofzhvf jnlhbfbu¾cjm tf˅cgjm ,hah rkpkakrjm , fgj pjstjbh r ? 4 tfgvj nf, zf svfgj x4 ∀H , zf gjbk bk 4 1 |x4∜x| 1 r , pjstjbh O ≫6 tfgj nf bk

|xO ∜x|≫r.

^kjrkm 6.6 (^kjrkm ahokfrhzhrfok stflhaojsth) Okgf bk H hotkrvfa rkfa-oh` lrjbkvf h i 2 H ↜H okprkghnoj nhikrkochbflhaof iuogchbf. Jshm tjef, okgf bk x tfc˅gf kgvhahlrhbumf bknofnz˅lk (6). ^fnf vrhbknh2

(h) fgj bk

|i ‼(x)| 1 6,

kgvhahlrhbum x bk ajgfaoj fshmptjtsgh stflhafo7

[ohvkrzhtkt u uzah ; Trhrjnoj-mftkmfthc˅gh ifguatkt




(hh) fgj bk

|i ‼(x)| ? 6,

kgvhahlrhbum x bk jnlhbfbu¾cf stgsof tfc˅gf (rkpkakr), tb. x bk okstflhafo kgvhahlrhbum.

Rf njgfz ^kjrkmf ahokfrhzhrfok stflhaojsth vhnbkth WO].

[ sauc˅fbu gfnf bk |i ‼(x)| = 6, ^kjrkm ahokfrhzhrfok stflhaojsth ok nfbkohgfgvk honhchbk u vkzh sf stflhaojs˅¾cu kgvhahlrhbumf ¯x. fnf su pjtrklofnjnftof hsphthvfobf.

Nkstohchbf 6.; Okgf bk H hotkrvfa rkfaoh` lrjbkvf h i 2 H ↜ H okprkghnojnhikrkochbflhaof iuogchbf h okgf bk x tfc˅gf kgvhahlrhbumf bknofnz˅lk (6).

6. x sk ofzhvf `hpkrljahc˅ojm stgsojm tfc˅gjm fgj bk

|i ‼(x)| = 6 ,

<. x sk ofzhvf ok`hpkrljahc˅ojm stgsojm tfc˅gjm fgj bk

|i ‼(x)| = 6 .

Sauc˅fb ok`hpkrljahc˅oje kgvhahlrhbumf, ok sfmj gjn nhikrkotoh` bknofnz˅lhprvje rknf, okej h gjn nhikrkotoh` bknofnz˅lh lhaj gjbke rknf hah shstkmf nh-ikrkotoh` bknofn˅zlh, spkchstc˅fo bk h mjrf sk pjskloj rfzmftrfth u svfgjmsauc˅fbu pjbknhofc˅oj. Tjgfzubk sk nf bk tj hsphthvfobk vraj gjmpahchrfoj h nfvraj c˅kstj, nfofs, ok mjz˅kmj sf sheurojs˅¾cu nj¾ch nj pjtpuoh` rkzuatftf.

Nkstohchbf 6.9 (Jschafchbk)

(f) Rf ohz {xo}∟o =4 gfz˅kmj nf hmf kvkotufaoj okgj svjbstvj T fgj pjstjbh chbkah lrjb O ≫4 tfgfv nf svfgh c˅afo ohzf {xo}∟o = O hmf tj svjbstvj.

(l) Rf ohz {xo}∟o =4 gf˅zkmj nf jschahrf jgj ouak hah bknojstfvoj jschahrf fgj c˅afojvh xo ohsu kvkotufaoj svh pjzhthvoh ohth kvkotufaoj svh okef-thvoh. Fgj bk nruefc˅hbk, zf ohz gfz˅kmj nf ok jschahrf . Rf ohz {xo}∟o =4gfz˅kmj nf strjej jschahrf fgj zf svfgh o4 ≫ 4 pjstjbk o6, o < ≫ o4

tfgvh nf vrhbknh xo 6 xo < 1 4.

(c) Rf ohz {xo}∟o =4 gf˅zkmj nf jschahrf jgj x fgj ohz {xo ∜x}∟o =4 jschahrf.Rf ohz {xo}∟o =4 gf˅zkmj nf strjej jschahrf jgj x fgj ohz {xo∜x}∟o =4strjej jschahrf.





Nkstohchbf 6.5 (Stflhaojst pkrhjnhc˅ok tfc˅gk) Rf pkrhjnhc˅ou tf˅cgu p bkn-ofnz˅lk (6) mhohmfaoje pkrhjnf g gfz˅kmj nf bk stflhaof , fshmptjtsgh stf-lhaof , okstflhaof , hah eajlfaoh ftrfgtjr fgj bk p rkspkgthvoj stflhaof ,fshmptjtsgh stflhaof , okstflhaof tfc˅gf kgvhahlrhbumf hah eajlfaoh ftrfg-tjr jn i g .

Tkrhjnhc˅ou jrlhtu mjz˅kmj prknstfvhth gfj gjofc˅fo sgup {x4, x6, . . . , x g}.[ tjm sauc˅fbu nkstohs˅kmj mojz˅htkab (muathpahgftjr) jrlhtk gjbke jzofc˅fvfmjsf η2

η = i ‼(x4)i ‼(x6) µ. . . µi ‼(xg ).

Spkchbfaoj, fgj bk

{ p, i ( p)

}pkrhjnhc˅oj rbks˅kobk pkrhjnf nvf bknofnz˅lk (6),

of jsojvu tkjrkmk (6.6) joj bk stflhaoj fgj bk

|η| = |i ‼( p)i ‼(i ( p))| 1 6

h okstflhaoj fgj bk

|η| = |i ‼( p)i ‼(i ( p))| ? 6.

Nkstohchbf 6.0 (Lfzko prhvafc˅kobf) Okgf bk x ajgfaoj fshmptjtsgh stflhaof stgsof tfc˅gf prksahgfvfobf i . Lfzko prhvafc˅kobf jn x, u jzofch L (x), sk nkstohs˅k gfj mfgshmfafo sgup B gjbh sfnrz˅h x, h bjs˅ vrhbknh

ahmo↜∟

i o (x) = x ∄(x∀B ).

Ofafz˅kobk lfzkof prhvafc˅kobf stgsok hah pkrhjnhc˅ok tfc˅gk bk jp¾ckohtj tkz˅fgzfnftfg. Jvnbk ¾ckmj prknstfvhth bknoj jn jsojvoh` tjpjajs˅gh` svjbstfvflfzkof prhvafc˅kobf. Trhbk tjef, mjrfmj nkstohsfth hovfrhbfotoh sgup.

Nkstohchbf 6.8 Sgup M ofzhvfmj hovfrhbfotohm pjn prksahgfvfobkm i fgj bk i (M ) ⊌ M . Nruehm rhbkc˅hmf, M bk hovfrhbfotfo fgj zf svfgh x ∀ M kakmkoth jrlhtk J (x) prhpfnfbu M .

^kjrkm 6.< Okgf bk x ftrfgthvof stgsof tfc˅gf prksahgfvfobf i . fnf bk lfzko prhvafc˅kobf L (x) hovfrhbfotfo jtvjrko hotkrvfa.

6.< Yhzufahzfchbf jrlhtf h vrkmkosgh` ohzjvf^hp erfstgf gjbh sk ofbc˅ks˅¾ck gjrhsth zf vhzufahzfchbu rbks˅kobf bknojnhmkozh-jofaok nhikrkotok bknofnz˅lk (6) ofzhvf sk stkpkofsth nhbferfm . Stkpk-ofsth nhbferfm bk erfstg u prfvjueajm gjjrnhoftojm shstkmu, h sfstjbh sk jnsabknk¾cke2





f) erfstgf iuogchbk y = i (x),

l) erfstgf prfvk y = x,

c) pjahejofaok ahohbk gjbf sk njlhbf spfbfobkm tfc˅fgf

(x4, x6), (x6, x6), (x6, x<), (x<, x<), (x<, x3), (x3, x3), . . . .

Ahohbsgh skemkoth pjahejofaok ahohbk stvfrfbu njbfm stkpkohcf. fgjnkr, vfz˅oj bk prhmhbkthth nf su tfc˅gk kgvhahlrhbumf zfbknohc˅gk tfc˅gk erfstgf y = i (x) sfprfvjm y = x.

Trhmbkr 6.< Yhzufahzhrfth jrlhtu ajehsthc˅gk nhikrkotok bknofnz˅lk (>) stkpk-ofsthm nhbferfmjm, sf pjc˅ktohm uvbktjm x4 = 4.>.

\bks˅kobk. [ jvjm prvjm prhmbkru gjrhs˅tkobf pfgktf Nyofmhcf , rbks˅kobk ¾cklhth oks˅tj nktfabohbk.

Ofbprhbk ¾ckmj uc˅htfth pfgkt Nyofmhcf u Ujairfm Mft`kmfthcf, ofrkn-ljm2

11 Nyofmhcf‐

Ofgjo tjef, nkstohs˅kmj ajehsthc˅gu bknofnz˅lu, prh c˅kmu jnejvfrfbu¾cu iuogchbui bknofnz˅lk (6) njlhbfmj iuogchbjm NK^jMfpW].

Ajehsthckq = xWo + 6] == > xWo] (6 - xWo])7AjehsthcMfp = NK^jMfpWAjehsthckq]7

Trjvbkrhmj sfnf nf bk AjehsthcMfpWx] zfprfvj ofs˅f iuogchbf i (x).

HoW6]2= AjehsthcMfpWx]JutW6]= > (6 - x) x

Nf lhsmj ofs˅ah okgu htkrfchbu (opr. >4-tu), sf okghm pjc˅ktohm uvbktjm (opr.

x4 = 4.>) pjtrkloj bk hsgjrhsthth iuogchbu HtkrftkWAjehsthcMfp, 4.>, >4] .HoW<]2= HtkrftkWAjehsthcMfp, 4.>, >4]JutW<]= 4.5684;8

[gjahgj z˅kahmj jrlhtu sf prvh` o c˅afojvf (uzkt ¾ckmj >4), gjrhsthmj iuogchbuJrlhtWAjehsthcMfp, 4.>, >4] .





HoW3]2= JrlhtWAjehsthcMfp, 4.>, >4]JutW3]= {4.>, 4.89, 4.6;39, 4.;<44<0, 4.88038;, 4.449>455>, 4.4<;>995,4.488<5<9, 4.3;595, 4.860898, 4.<8509, 4.039;;5, 4.;>9865, 4.88668;,4.43>848, 4.63>596, 4.>99>43, 4.88;>0;, 4.465850;, 4.4549<66,4.<9<;3;, 4.55>>>6, 4.9805<5, 4.0><43, 4.;3<493, 4.88;000, 4.469306<,4.49>>;6<, 4.<>6608, 4.53<490, 4.50>;50, 4.959496, 4.05946, 4.>3>>9;,4.80<0<6, 4.495;39, 4.<;6088, 4.5;350>, 4.5><35>, 4.59;4<, 4.5684;8}

Gjofc˅oj, iuogchbf StfhrcfskNhferfmW] ekokrhs˅k stkpkofsth nhbferfm.

HoW>]2= jrl = JrlhtWAjehsthcMfp, 4.>, >4]7StfhrcfskNhferfmWAjehsthcMfp, jrl, Tajt\foek -? Faa]

h njlhbfmj trfz˅koh erfstg (sahgf 6).

4.< 4.> 4.9 4.0 6.4

4.<

4.>

4.9

4.0

6.4

Sahgf 62 Stkpkofsth nhbferfm jrlhtk J (4.>) zf xo +6 = >xo (6∜xo ).

Nrueh thp erfstgf gjbh sk gjrhsth zf vhzufahzfchbu rbks˅kobf bknojnhmkozh-jofaok nhikrkotok bknofnz˅lk (6) ofzhvf sk vrkmkosgh ohz . Sfstjbh sk jnrkprkzkotfchbk vfrhbflak xo gfj iuogchbk jn o. Jlhc˅oj, `jrhzjotfaof jsf prkn-stfvabf o h vkrthgfaof jsf prknstfvabf xo . Jlhc˅oj ef ekokrhs˅kmj s jlzhrjm ofjnejvfrfbu¾ck tfc˅gk rfvojtkz˅k.

Trvj ofafzhmj tfc˅gk kgvhahlrhbumf ajehsthc˅gk nhikrkotok bknofnz˅lk (>).

HoW;]2= SjavkWAjehsthcMfpWx] == x, x]JutW;]= {{x -? 4}, {x -? 3/>}}

Njlhah smj nvhbk stgsok tfc˅gk, tk ¾ckmj h` hmkojvfth gfj itW6] h itW<] 2

HoW9]2= {itW6], itW<]} = {x} /. %JutW9]= {{4}, {3/>}}





Of jvfb ofc˅ho smj njlhah vrhbknojsth vfrhbflah itW6]WW6]] = 4 h itW<]WW6]]= 3/> .

Rf crtfobk erfstgf vrmkosgje ohzf gjrhsth sk Nyofmhcf iuogchbu ^hmkSkrhksTajtW] .Hsgjrhstht ¾ckmj prkt`jnoj ekokrhsfou jrlhtu jrl sf pjc˅ktohm uvbktjm x4 =4.>2

HoW5]2= ^hmkSkrhksTajtWjrl, FxksJrheho -? {4, itW<]WW6]]}]

h njlhbfmj trfz˅koh erfstg (sahgf <).

4 64 <4 34 >4

4.4

4.<

4.>

4.9

4.0

6.4

Sahgf <2 Erfstg vrkmkosgje ohzf zf jrlhtu J (4.>) zf xo +6 = >xo (6∜xo ).

Sahc˅fo erfstg sk njlhbf fgj sk hzjstfvk ahohbk gjbk spfbfbu tfc˅gk of erfstgu<. Jvfb erfstg sk hzvjnh pjmj¾cu Nyofmhcf iuogchbk JrlhtTajtW] 2

HoW0]2= JrlhtTajtWAjehsthcMfp, 4.>, >4]

h njlhbfmj ofvknkoh erfstg (sahgf 3).

4 64 <4 34 >4

4.4

4.<

4.>

4.9

4.0

6.4

Sahgf 32 Erfstg tfc˅fgf zf jrlhtu J (4.>) zf xo +6 = >xo (6∜xo )





Trhmbkr 6.3 Tjsmftrfbmj nhofmhc˅gh shstkm u zfvhsojsth jn pfrfmktrf F2

xo +6 = x<o + F. (;)

\bks˅kobk.

HoW6]2= 11 Nyofmhcf‐

Ofgjo tjef, nkstohs˅kmj nftu bknofnz˅lu, prh c˅kmu jnejvfrfbu¾cu iuogchbu q bknofnz˅lk (;) njlhbfmj iuogchbjm NK^jMfpW].

HoW<]2= kqo = xWo+6]==xWo]_< + F7

HoW3]2= q = NK jMfpWkqo]7Tfrfmktru F ¾ckmj njnbkabhvfth pjsklok h gjogrktok vrhbknojsth zf jlfvabfobkoumkrhc˅gk shmuachbk.

HoW>]2= F = -4.;7

Sfnf ¾ckmj ekokrhsfth erfstg vrkmkosgje ohzf2

HoW;]2= ^hmkSkrhksTajtWq, 4.6, <4]

4 ; 64 6; <44.;

4.>

4.3

4.<

4.6

4.4

Sahgf >2 Erfstg vrkmkosgje ohzf zf jrlhtu J (4.6) zf xo +6 = x<o + F

fc˅gk kgvhahlrhbumf nhikrkotok bknofnz˅lk (;) su2

HoW9]2= SjavkWqWx] == x, x]JutW9]2= {{x -? -4.3994<;}, {x -? 6.39943}}

Erfstg vrkmkosgje ohzf (sahgf >.) pjgfzubk nf jrlhtf gjovkrehrf gf tfc˅ghkgvhahlrhbumf.

Ekokrhsft ¾ckmj h erfstg tfc˅fgf zf hstu jrlhtu gfj h zf vrkmkosgh ohz.





4 ; 64 6; <44.;

4.>

4.3

4.<

4.6

4.4

Sahgf ;2 Erfstg tfc˅fgf zf jrlhtu J (4.6) zf xo +6 = x<o + F

6.3 Jsbktabhvjst pjofs˅fobf rbks˅kobf nhikrkotok bknofnz˅lkOf prhmbkru prksahgfvfobf i (x) = x< + F , ¾ckmj ujc˅hth nf ‑rkafthvoj mfak”prjmbkok pfrfmktrf F mjeu nfth rfzahc˅htj pjofs˅fobk rbks˅kobf.

Tjsmftrfbu¾ch jrlhtu J (4.6) zf F = ∜6.<< mjz˅kmj vhnbkth nf pjstjbhfshmptjtsgj pkrhjnhc˅oj rbks˅kobk pkrhjnf <.

HoW6]2= F = -6.<<7HoW<]2= jrl = Jrlht Wq, 4.6, 94]7HoW3]2= ^fgk Wjrl, {;4, 94}]JutW3]={-6.60995, 4.600605, -6.60>;8, 4.603<>3, -6.609><, 4.605;85,-6.60>06, 4.603590, -6.609<3, 4.6056>, -6.60>80}

Erfstg vrkmkosgje ohzf (sahgf 9.) pjgfzubk fshmptjtsgh gfrfgtkr pkrh-jnhc˅oje rbks˅kobf pkrhjnf <.

4 64 <4 34 >4 ;4 94

6.<

6.4

4.0

4.9

4.>

4.<

4.4

4.<

Sahgf 92 Erfstg vrkmkosgje ohzf2 ^hmkSkrhksTajtWjrl]

Erfstg tfc˅fgf (sahgf 5.) zf hstu jrlhtu, jnojsoj Nhofmycf ofrknlf JrlhtTajtW]nfbk hstu hoijrmfchbu u bfsohbkm jlahgu.

[ohvkrzhtkt u uzah 6< Trhrjnoj-mftkmfthc˅gh ifguatkt




4 64 <4 34 >4 ;4 94

6.<

6.4

4.0

4.9

4.>

4.<

4.4

4.<

Sahgf 52 Erfstg tfc˅fgf2 JrlhtTajtWjrl]

Nrueh tkst, sf pfrfmktrjm F = ∜6.3 ugfzubk of pjstjbfobk fshmptjtsgkpkrhjnhc˅ok tfc˅gk pkrhjnf >. Trhgfzf¾ckmj htkrfchbk u rfspjou jn x;4 , . . . , x 94 .

HoW>]2= F = -6.37HoW;]2= jrl6 = JrlhtWq, 4.6, 94]7HoW9]2= ^fgkWjrl6, {;4, 94}]JutW9]= {-6.<889<, 4.308468, -6.6>099, 4.468>343, -6.<889<, 4.308468,-6.6>099, 4.468>343, -6.<889<, 4.308468, -6.6>099}

Fshmptjtsgj pjofs˅fobk mjz˅k lhth tks˅gj zf vhzfuahzfchbu pjmj¾cu erfstgf vr-

mkosgje ohzf (sahgf (0)).

4 64 <4 34 >4 ;4 94

6.4

4.;

4.4

Sahgf 02 Erfstg vrkmkosgje ohzf2 ^hmkSkrhksTajtWjrl6]

Of erfstgu tfc˅fgf ljabk ujc˅fvfmj fshmptjtsgf pjofs˅fobf.Tjmj¾cu ofrknlk IhonMhohmfaTkrhjnW] mjz˅kmj jnrknhth mhohmfaoh pk-

rhjn pkrhjnhc˅oh` jrlhtf.

HoW5]2= IhonMhohmfaTkrhjnWJrlhtWq, 4, <44]]Mhohmfa pkrhjn = >Tkrhjnhc jrlht = {-6.<889<,4.308468,-6.6>099,4.468>343}





4 64 <4 34 >4 ;4 94

6.4

4.;

4.4

Sahgf 82 Erfstg tfc˅fgf2 JrlhtTajtWjrl6]

Ihofaoh tkst ¾ckmj urfnhth sf pfrfmktrjm F = ∜6.30. Trhgfzf¾ckmj htkrf-chbk u rfspjou jn x;4 , . . . , x 94 .

HoW0]2= F = -6.307HoW8]2= jrl< = JrlhtWq, 4.6, 94]7HoW64]2= ^fgkWjrl<, {;4, 94}]JutW9]= {-6.3995, 4.>0505<, -6.6>680, -4.45;0509, -6.35><>, 4.;40;><,-6.6<630, -4.6<<>89, -6.39>88, 4.>03<6, -6.6>9;6}

Crtfobk erfstgf tfc˅fgf (94 tfc˅fgf) suekrhs˅k of mjeu¾ck pjstjbfobk pkrhjnc˅oktfc˅gk pkrhjnf >, gjbf bk ajgfaoj fshmptjtsgh stflhaof.

4 64 <4 34 >4 ;4 94

6.4

4.;

4.4

4.;

Sahgf 642 Erfstg tfc˅fgf2 JrlhtTajtWjrl<]

Mknuthm, crtfobk hstje erfstgf sf 944 tfc˅fgf nfbk nruefc˅hbu prknstfvu.\bks˅kobk gjovkrehrf pkrhjnhc˅ojb tfc˅gh pkrhjnf 0.

Of grfbu, mjz˅kmj gjrhshth h `hstjerfm zf ujc˅fvfobk pkrhjnhc˅ojsth rbks˅kobf.@hstjerfm prvh` 3444 c˅afojvf jrlhtk J (4.6) ugfzubk nf rbks˅kobk gjovkrehrfpkrhjnhc˅ojm rbks˅kobu pkrhjnf 0.

[ohvkrzhtkt u uzah 6> Trhrjnoj-mftkmfthc˅gh ifguatkt




4 644 <44 344 >44 ;44 944

6.4

4.;

4.4

4.;

Sahgf 662 Erfstg tfc˅fgf2 JrlhtTajtWq, 4.6, 944]

6.4 4.; 4.4 4.;

644

<44

344

>44

Sahgf 6<2 @hstjerfm2 @hstjerfmWJrlhtWq, 4.6, 3444], 644]

6.> Lrzhof ftrfgchbk jrlhtfSfnf ¾ckmj pjgfzfth gfgj mojz˅htkab pkrhjnhc˅ok jrlhtk jnrknubk lrzhou ftrfg-chbk jrlhtk. [nfabkojst jn stgsok tfc˅gk trklf h¾ch gf 4 kgspjokochbfaoj lrzj.

Tjsmftrft ¾ckmj jrlhtu zf F = ∜4.>.

HoW66]2= F = -4.>7HoW6<]2= jrl3 = JrlhtWq, 4, >4]7

HoW63]2= ^hmkSkrhksTajtWjrl3, Tajt\foek -? Faa]

Ekokrhs˅hmj h erfstg tfc˅fgf jrlhtk.Sfnf ¾ckmj vhnbkth gfgj bk lrzj htkrfthvoj prhlahz˅fvfobk tfc˅gh kgvhahlrhbumf

pjmj¾cu rfc˅uofaok mktjnk.Jnrknhmj stgsok tfc˅gk.

HoW6>]2= SjavkWqWx] == x, x]JutW6>]= {{x -? -4.349<<9}, {x -? 6.349<3}}

[ohvkrzhtkt u uzah 6; Trhrjnoj-mftkmfthc˅gh ifguatkt




4 64 <4 34 >4

4.>

4.3

4.<

4.6

4.4

Sahgf 632 Erfstg vrkmkosgje ohzf pjgfzubk lrzu gjovkrekochbu ohzf tfc˅fgf

4 64 <4 34 >4

4.>

4.3

4.<

4.6

4.4

Sahgf 6>2 Erfstg tfc˅fgf jrlhtk

Mjeah smj njlhth rfzumou fprjgshmfchbu tfc˅gk kgvhahlrhbumf h vkahghmlrjbkm htkrfchbf. [zmhmj opr. 6444 htkrfchbu.

HoW6;]2= ihxpjhot = HtkrftkWq, 4, 6444]JutW6;]= -4.349<<9

Sfnf jnuzhmfmj jvu vrhbknojst ( ihxpjhot ) jn svfgje c˅afof jrlhtk ( jrl3 ),h njlhbfmj ahstu rfzahgf gjbu mjz˅kmj hsgjrhsthth gfj hnkbu zf lrzhou gjovkr-ekochbk.

HoW69]2= nhii = jrl3 - ihxpjhot7

Ajefrhtfmsgh erfstg jvk rfzahgk (ahstf nhii ) mjz˅kmj hsgjrhsthth zf prhgfzprhlahz˅fvfobf ouah gfj kgspjokochbfaof iuogchbf jn o.

HoW65]2= AhstAjeTajtWFlsWnhii]]

Ahstf jmbkrf pjgfzubk nf bk svfgf rfzahgf jnojsoj svfgh c˅afo ahstk nhii ,fprjgshmfthvoj 4.96< putf prkt`jnoh c˅afo.





64 <4 34 >4

64 5

64 ;

4.446

4.6

Sahgf 6;2 Ajefrhtfmsgh erfstg

HoW60]2= rfthjs = ^flakWnhiiWWh + 6]]/nhiiWWh]], {h, 6, 34}]jutW60]= {-4.349<<9, -4.549<<9, -4.;>9<<9, -4.9>09<9, -4.;00800, -4.9<9<56,-4.943585, -4.965955, -4.948<<>, -4.96>>60, -4.966<>3, -4.96368,-4.966888, -4.96<5<8, -4.96<<0<, -4.96<;;9, -4.96<300, -4.96<>86,-4.96<><0, -4.96<>99, -4.96<>>3, -4.96<>;5, -4.96<>>0, -4.96<>;>,-4.96<>;, -4.96<>;<, -4.96<>;6, -4.96<>;<, -4.96<>;6, -4.96<>;<}

Ofrknoh erfstg (sahgf 69.) pjgfzubk lrzu gjovkrekochbu ahstk rfthj gfobkojb erfohch.

HoW68]2= AhstTajtWrfthjs, Irfmk -? ^ruk]

4 ; 64 6; <4 <; 34

4.969

4.96>

4.96<

4.964

Sahgf 692 Erfstg ahstk rfthj

Iuogchbf MuathpahkrW] mjz˅k sk hsgjrhsthth zf hzrfc˅uofvfobk gjostfotojejmbkrf. [stvfrh, tj bk mojz˅htkab pkrhjnhc˅ok jrlhtk, s thm nf zf pkrhjnhc˅outfc˅gu uzmkmj stgsou tfc˅gu.

HoW<4]2= MuathpahkrWq, {ihxpjhot}]JutW<4]= -4.96<>;<





6.; Lhiurgfchbk. Lhiurgfchjoh nhbferfmh6.;.6 Lhiurgfchbk

Ofbpjzofthbf okahokfrof nhsgrktof bknofnz˅lf gjbf sk gjrhsth zf mjnkahrfobk bknok vrstk bk ajehsthc˅gf nhikrkotof bknofnz˅lf 2

xo +6 = i p(xo ) = pxo (6∜xo ), o = 4, 6, . . . , p ? 4, 4≮x≮6. (9)

Nf lhsmj ofs˅ah tfc˅gk kgvhahlrhbumf njvjaboj bk rhbks˅hth bknofnz˅lu2

i p(x) = px(6∜x) = x.

HoW<6]2= Ajehsthckq = xWo + 6] == p xWo] (6 - xWo])7HoW<<]2= AjehsthcMfp = NK^jMfpWAjehsthckq]7HoW<3]2= SjavkWAjehsthcMfpWx] == x, x]JutW<3]= {{x -? 4}, {x -? (-6 + p)/p}}

Stjef su tfc˅gk kgvhahlrhbumf 4 h p∜

6 p . Hsphtfbmj stflhaojst svfgk jn ofvknkoh`

tfc˅fgf. Hzrfc˅uofbkmj prvh hzvjn prksahgfvfobf i p.

i ‼ p(x) = ( px(6∜x))‼ = ( px∜ px<)‼ = p∜< px.

(f) fc˅gf kgvhahlrhbumf bk 4. Gfgj bk i ‼ p(4) = p, of jsojvu tkjrkmf 6.6

hmfmj2

(h) 4 bk ajgfaoj fshmptjtsgh stflhaof stgsof tf c˅gf zf 4 1 p 1 6,

(hh) 4 bk okstflhaof stgsof tfc˅gf zf p ? 6.

(l) fc˅gf kgvhahlrhbumf bk p∜

6 p . Rlje

i ‼ p p∜6

p= p∜< pµ

p∜6 p

= p∜< p + < = <∜ p,

of jsojvu tkjrkmf 6.6 hmfmj2(h) p

∜6

p bk ajgfaoj fshmptjtsgh stflhaof stgsof tfc˅gf zf 6 1 p 1 3,

(hh) p∜

6 p bk okstflhaof stgsof tfc˅gf zf p ? 3.

Nf lhsmj jnrknhah pkrhjnhc˅oj rbks˅kobk pkrhjnf nvf, njvjaboj bk rhbks˅hth bkn-ofnz˅lu2

i p(i p(x)) = p( px(6∜x))(6∜ px(6∜x)) = p<x(6∜x)(6∜ px(6∜x)) = x.





Ohbk tks˅gj prhmhbkthth gfgj sk rfzmftrfobk gfrfgtkrhsthgf ajehsthc˅gk bkn-ofnz˅lk svjnh of prjuc˅fvfobk okgh` spkchstc˅oh` tfc˅fgf sfmk iuogchbk h obkzhofpfrfmktrf p.

Nfgak, gfnf pfrfmktfr p prjmhbkoh svjbu vrhbknojst, pjofs˅fobk rbks˅kobfsk tfgjnkr mhbkobf. Jzofc˅hmj sf lg vrhbknojst pfrfmftrf p u gjbjb sk bfvabfprjmbkof pjofs˅fobf u g-tjb pjsmftrfojb tfc˅gh. fnf hmfmj sabknk¾cu tflahcu.

Hotkrvfa pfrfmktrf p Yrstf pjof˅sfobf rbks˅kobf Grhthc˅of vrhbknojstpfrfmtkrf p

4 1 p 1 6 fc˅gf kgvhahlrhbumf 4 bk ajgfaoj fshmp-tjtsgh stflhaof

l 4 = 6

6 1 p 1 3 fc˅gf kgvhahlrhbumf p ∜ 6

p bk ajgfaoj

fshmptjtsgh stflhaofl 6 = 3

3 1 p 1 6 + − 9 Tkrhjnh˅cof rbks˅kobf p6 h p< su ajgfaojfsphmptjtsgh stflhaof

l < = 6 + − 9

Yrhbknojsth lg sk ofzhvfbu lhiurgfchjok vrhbknojsth pfrfmktrf.Ofstfvabfbu¾ch jvfb prjcks, mjz˅kmj of¾ch sabknk¾cu lhiurgfchjou vrhbknojst

l 3 gjbf ok jnejvfrf pkrhjnhc˅ojm rbks˅kobu prjstje pkrhjnf trh, fah jnejvfrfpkrhjnhc˅ojm rbks˅kobu prjstje pkrhjnf c˅kthrh.

Sabknk¾cf lhiurgfchjof vrhbknojst l > jnejvfrf pkrhjnhc˅ojm rbks˅kobu pkrhjnfjsfm. Jshm tjef, mjz˅k sk njgfzfth nf bk zf p∀(l <, l 3), pkrhjnhc˅oj rbks˅kobk

prjstje pkrhjnf c˅kthrh stflhaoj, f pkrhjnhc˅oj rbks˅kobk pkrhjnf nvf pjstfbkokstflhaoj.Hstj tfgj, mjz˅k sk njgfzfth nf bk zf p∀(l 3, l >), pkrhjnhc˅oj rbks˅kobk prjs-

tje pkrhjnf jsfm stflhaoj, f pkrhjnhc˅oj rbks˅kobk pkrhjnf c˅kthrh pjstfbk oks-tflhaoj.

Ofstfvabfbu¾ch jvfb prjcks, mjz˅kmj vhnbkth nf pjstjbh ohz{l g}∟o =4 lhiurgf-chjoh` vrhbknojsth pfrfmktrf sf sabknk¾chm svjbstvjm2 zf p ∀ (l g , l g+6 ) pkrh-jnhc˅oj rbks˅kobk prjstje pkrhjnf < g bk stflhaoj, f pkrhjnhc˅of rbks˅kobf svfgjejn pkrhjnf < , . . . , <g

∜6 pjstfbu okstflhaof.

Jvfb ikojmko sk ofzhvf put u `fjs unvjstruc˅fvfobkm pkrhjnf lh-iurgfchbk .

Jvj zofc˅h, fgj sk pfrfmktfr p pjvk¾cf hzofn l6, tfc˅gf kgvhahlrhbumf skerfof of pkrhjnhc˅oj rbks˅kobk pkrhjnf nvf7 f hzofn vrhbknojsth l< pkrhjnhc˅ojrbks˅kobk pkrhjnf nvf sk erfof of pkrhjnhc˅oj rbks˅kobk pkrhjnf c˅kthrh, htn.

Ohz lhiurgfchjoh` vrhbknojsth pfrfmktrf tkz˅h lrjbu l∟ = 3.;988> . . . enbk bknofnz˅lf (9) hmf pkrhjnhc˅of rbks˅kobf svh` pkrhjnf gfj h okgf okpkrhjnhc˅ofrbks˅kobf. Jvfgvf shtufchbf sk c˅kstj jphsubk gfj fjthc˅oj pjof˅sfobk hah `fjs .

[ohvkrzhtkt u uzah <4 Trhrjnoj-mftkmfthc˅gh ifguatkt




Ohz

{l g

}∟o =4 lhiurgfchjoh` vrhbknojsth pfrfmktrf hmf hzuzktoj svjbstvj2

ahmg↜∟

l g∜l g∜

6

l g+6 ∜l g= ξ ≀ >.998<4. . . .

Lrjb ξ = >.998<4. . . sk zjvk Ikhekolfum-jv lrjb , f ofzhv bk njlhj pjobkejvjm jtgrhvfc˅u, stzhc˅fru Mhtc`kaa Ikhekolfum.

[stvfrh, Ikhekolfum bk ofprfvhj mojej vk¾ck jtgrh¾ck. Lrjb ξ bk uohvkrzf-afo h okzfvhsh jn jlahgf ifmhahbk prksahgfvfobf i p. Mknuthm, lrjb l ∟ zfvhsh jnifmhahbk prksahgfvfobf gjbf sk rfzmftrfbu.

6.;.< Lhiurgfchjoh nhbferfmh

Yk¾c smj vhnbkah, lfrkm s˅tj sk thc˅k ajehsthc˅gk nhikrkotok bknofnz˅lk, nf rfzahc˅htkvrhbknojsth pfrfmktrf mjeu nfth gvfahtfthvoj rfzahc˅htf rbks˅kobf. Tjskloj ofszfohmf prjmbkof prhrjnk rbks˅kobf gfnf sk pfrfmktfr mhbkobf, f zf hsphthvfobktfgvh` prjmbkof gjrhsthth ¾ckmj lhiurgfchjoh nhbferfm .

Lhiurgfchjoh nhbferfm pjgfzubk mojek hzokofnok gvfahtfthvok prjmbkoku ftrfgthvojb stgsojb tfc˅gh gfj h u pkrhjnhc˅ojb jrlhth.

Bknof jn prvh` jtgrhvkoh` lhiurgfchbf bk lhiurgfchbf unvjstruc˅fvfobf pk-rhjnf gjbf sk pjbfvabubk u sauc˅fbu bknojnhmkozhjofaok ajehsthc˅gk bknofnz˅lk(9).

Nhofmycf iuogchbf zf ekokrhsfobk lhiurgfchjoje nhbferfmf bkLhiurcfthjoTajtON .

LhiurcfthjoTajtONWiuogchbf,{p, pmho, pmfx},{skkn},Stkps -? lrjb6, Htkrftks -? lrjb<, IhrstHt -? lrjb3]7

Lhiurgfchjoh nhbferfm ekokrhs˅kmj pjofvabfbu¾ch sabknk¾ch pjstupfg, enbk suI hrstHt, Htkrftks, Stkps, pmfx, pmho uofprhbkn nfth lrjbkvh.

(f) Hzflkrkmj vrhbknojst pfrfmktrf p, pjc˅kvs˅h sf hochbfaojm vrhbknojsth pmho.

(l) Hzflkrkmj pj c˅ktou tfc˅gu skkn = x4 u prjmftrfojm sgupu.(c) Hzrfc˅uofmj jrlhtu u x4.

(n) Jnlfchmj prvh` I hrstHt ∜6 htkrfchbf h crtfmj jrlhtu pjc˅kvs˅h sf htkrf-chbjm I hrstHt h zfvrs˅fvfbu¾ch ofgjo Htkrftks htkrfchbf.

(k) Tjvk¾cfmj p zf vrhbknojst ( pmfx ∜ pmho)/Stkps h pjojvhmj pjstupfg.

Ofbprhbk ¾ckmj nkstohsfth ajehsthc˅gu nhikrkotou bknofnz˅lu sf pfrfmktrjm, fzfthm ekokrhsfth lhiurgfchjoh nhbferfm grjz pfgkt Nyofmhcf.





HoW6]2=ajehsp = xWo + 6] == p xWo] (6 - xWo])7HoW<]2=ajehspmfp = NK^jMfpWajehsp]7HoW3]2=LhiurcfthjoTajtONWajehspmfp, {p, <.;, >.4}, {4.;},IhrstHt -? >;4, Htkrftks -? <44, Stkps -? 3;4]

<.9 <.0 3.4 3.< 3.> 3.9 3.0 >.4

4.4

4.<

4.>

4.9

4.0

6.4

Sahgf 652 Lhiurgfchjoh nhbferfm zf <.; 1 p 1 >.4

Gjofc˅oj njlhbfmj lhiurgfchjoh nhbferfm ajehsthc˅gk nhikrkotok bknofnz˅lk zfpfrfmktfr < .; 1 p 1 >.4. Of jvjm nhbferfmu, `jrhzjotfaof jsf prknstfvabfpfrfmktfr p f vkrthgaof jsf rbks˅kobk xo .

Lrjb prksbkgf sf vkrthgfaojm ahohbjm u lhaj gjbjb tfc˅gh nfbk pkrhjn ftrfg-thvoje rbks˅kobf zf jnejvfrfbu¾cu vrhbknojst pfrfmktrf p.

Mjz˅k sk vhnbkth rbks˅kobk pkrhjnf nvf, pkrhjnf c˅kthrh h pkrhjnf jsfm.Jvj bk pjc˅ktfg putf u `fjs unvjstruc˅fvfobkm pkrhjnf.Of nhbferfmu sk tfgjnkr vhnk ‑lhbkah prjzjrh” zf vrhbknojst pfrfmktrf

oks˅tj vk¾cu jn 3.0, s˅tj suekrhs˅k of pjstjbfobk pkrhjnhc˅oje rbks˅kobf pkrhjnf 3lahzu 3.0.

Lhiurgfchjok vrhbknojsth zf pkrhjnhc˅of rbks˅kobf pkrhjnf <, > h 0 mjeu skprkchzohbk jnrknhth of nhbferfmu zf 3.> 1 p 1 3.9.

Okgh lhbkah vkrthgfaoh prjzjrh smbks˅tkoh hzmknu p = 3.;0 h 3.;8 su bfsohbkvhnabhvh.

HoW>]2=LhiurcfthjoTajtONWajehspmfp, {p, 3.>, 3.9}, {4.;},IhrstHt -? >;4, Htkrftks -? <44, Stkps -? 3;4]

[ohvkrzhtkt u uzah << Trhrjnoj-mftkmfthc˅gh ifguatkt




3.>4 3.>; 3.;4 3.;; 3.94

4.>

4.;

4.9

4.5

4.0

4.8

Sahgf 602 Lhiurgfchjoh nhbferfm zf 3.> 1 p 1 3.9

6.9 Ayfpuojvabkvh lrjbkvhNrueh ofc˅ho zf mbkrkobk sajz˅kojsth pjofs˅fobf rbks˅kobf nkstrkotok bknofnz˅lksk rfc˅uof sf Ayfpuojvabkvhm lrjbkm. [ sauc˅fbu bknojnhmkozhjofaoje nh-ofmhc˅gje shstkmf

xo +6 = i (xo ), o = 4, 6, . . . , i 2 (f, l )↜(f, l ) (5)

‑Ayfpuojvabkv kgspjokot” bk mbkrf jnstupfobf nvhbu jrlhtf gjbk pjc˅hobu sfokzoftoj rfzahc˅hthm pjc˅ktohm uvbkthmf x4 h x4

¶ξ 4, ξ 4 ? 4.

Fgj bk x4 pkrhjnhc˅of tfc˅gf pkrhjnf g nhofmhc˅gje shstkmf (5), h fgjzfpjc˅okmj jrlhtu sf lahsgjm tfc˅gjm x4 ¶ ξ 4, jonf ofgjo prvk htkrfchbk,enbk bk x6 = i (x4) h x‼6 = i (x4 ¶ ξ 4) rfzahgf mknu obhmf bk prhlahz˅oj

x6∜x‼6 = i (x4)∜i (x4 ¶ ξ 4) ≀ i ‼(x4)ξ 4.

Jzofc˅hmj sfξ h = |xh∜x‼h|, M h = |i ‼(xh)|.

Hzξ 6 = |x6∜x‼6| ≀ |i ‼(x4)ξ 4| = |i ‼(x4)|ξ 4 = M 4ξ 4

vhnhmj nf bk M 4 ifgtjr uvk¾cfvfobf zf prvu htkrfchbu.Ofgjo nruek htkrfchbk, njlhbfmj

ξ < ≀ |i ‼(x6)|ξ 6 = M 6ξ 6 ≀ M 4M 6ξ 4,

enbk bk M 6 ifgtjr uvk¾cfvfobf zf nrueu htkrfchbu.Ofstfvabfbu¾ch nfabk of jvfgfv ofc˅ho, mjz˅kmj zfgabuc˅hth nf bk ugupoh ifg-

tjr uvk¾cfobf zf pkrhjnhc˅ou jrlhtu pkrhjnf g (tjgjm bknoje chgausf) bknofgprjhzvjnu2

M 4M 6 . . . M g∜

6.





Lunu¾ch nf bk jvfb prjhzvjn fgumuafchbf ifgtjrf uvk¾cfobf, hmf smhsaf uzkthu jlzhr okgu prjsbkc˅ou vrhbknojst. Ofbpjvjabohbh bk ekjmktrhbsgf srknhof2

(M 4M 6 . . . M g∜

6)6/g .

Fgj uzmkmj ajefrhtfm prkt`jnok ekjmktrhbsgk srknhok, h jonf tj jzofc˅hmjsf η njlhbfmj2

η = ao (M 4M 6 . . . M g∜

6)6/g

η = 6

g (aoM 4 + aoM 6 . . . aoM g

∜6)

η = 6

g (ao|i ‼(x4)|+ ao |i ‼(x6)| . . . ao|i ‼(xg∜

6)|)Hotuhthvoj, uvbkt zf stflhaojst pkrhjnhc˅ok jrlhtk bk nf prjsbk˅coh ifgtjr

uvk¾cfobf lunk mfobh jn 6, s˅tj bk kgvhvfakotoj sf

η 1 6 stflhaofη ? 6 okstflhaof.

Sfnf ¾ckmj jvfb prhstup (sf pkrhjnhc˅ojm jrlhtjm) hsgjrhsthth gfgj lh ef prjs˅hrhahof prjhzvjabou jrlhtu.

Nkstohchbf 6.64 (Ayfpuojvabkv kgspjokot) Okgf bk i okprkghnoj nhik-rkochbflhaoj prksahgfvfobk of \ h okgf bk x4 nftf pj˅cktof tfc˅gf. Ayfpuojv-abkv kgspjokot η(x4) prksahgfvfobf i sk nkstohs˅k gfj

η(x4) = ahmg↜∟

6g

(ao|i ‼(x4)|+ ao |i ‼(x6)| . . . ao|i ‼(xg∜

6)|), (0)

ugjahgj ahmks pjstjbh. [ sauc˅fbu gfnf bk lhaj gjbh jn hzvjnf bknofg ouah, tfnf uzhmfmj nf bk η(x4) = ∜∟. Ayfpuojvabkv lrjb A(x4) sk nkstohs˅k gfjkgspjokot Ayfpuojvabkvje kgspjokotf, ugjahgj pjstjbh2

A(x4) = kη(x 4 ) .

Nkstohchbf 6.66 (Fshmptjtsgh pkrhjnhc˅of jrlhtf) Rf jrlhtu {x6, x<, x3, . . .}gfz˅kmj nf bk fshmptjtsgh pkrhjnhc˅of fgj pjstjbh pkrhjnhc˅of jrlhtf {y6, y<, y3, . . .}tfgvf nf vrhbknh ahm

o↜∟|xo ∜yo | = 4 . (8)

Sfnf mjz˅kmj nkstohsfth h `fjthc˅ou jrlhtu.

[ohvkrzhtkt u uzah <> Trhrjnoj-mftkmfthc˅gh ifguatkt




< Ayokss-jvf nhikrkotof bknofnz˅lf

<.6 Jsojvoh pjbmjvhOkgf bk H hotkrvfa rkfaoh` lrjbkvf, h okgf bk

i 2 H ÕH ↜H

okprkghnoj nhikrkochbflhaof iuogchbf. fnf svfgh sgup pjc˅ktoh` uvbktf x4, x∜

6 ∀H nhikrkotok bknofnz˅lk

xo +6 = i (xo , xo∜

6), o = 4, 6, . . . (64)

hmf bknhostvkoj rbks˅kobk {xo}∟o =∜

6.

fc˅gf kgvhahlrhbumf bknofnz˅lk (64) bk tfc˅gf x∀H , tfgvf nf bk

i (x, x) = x

jnojsojxo = x zf svfgj o≫4

bk rbks˅kobk bknofnz˅lk (64), hah kgvhvfakotoj, ¯x bk stgsof tfc˅gf prksahgfvfobf i .

Ayokss-jvf nhikrkotof bknofnz˅lf bk rfchjofaof nhikrkotof bknofnz˅lf nrueje rknf jlahgf

xo +6 = i (xo , xo∜

6) = F + xo

xo∜

6(66)

enbk bk pfrkmktfr F okokefthvfo rkfafo lrjb h pjc˅ktoh uvbkth x∜

6 h x4 prj-hzvjaboh pjzhthvoh rkfaoh lrjbkvh tfgvh nf bk

xo ? 4, zf svfgj o ? 4.

Nkstohchbf <.6 (Stflhaojst tfc˅gk kgvhahlrhbumf)

6. ^fc˅gf kgvhahlrhbumf x bknofnz˅lk (64) sk ofzhvf stflhaojm , hah ajgfaoj

stflhaojm , fgj zf svfgj λ ? 4 pjstjbh ξ ? 4 tfgj nf x4, x

∜6 ∀H h |x4∜x|+ |x∜6∜x| 1 ξ ⇜ |xo∜x| 1 λ, zf svfgj o ≫∜6.

<. fc˅gf kgvhahlrhbumf ofzhvf sk okstflhaojm fgj ohbk stflhaof.

3. ^fc˅gf kgvhahlrhbumf x bknofnz˅lk (64) sk ofzhvf ajgfaohm ftrfgtjrjm fgj pjstjbh γ ? 4 tfgvj nf

x4, x∜

6 ∀H h |x4∜x|+ |x∜6∜x| 1 γ ⇜ ahmo↜∟

xo = x.





>. ^fc˅gf kgvhahlrhbumf x bknofnz˅lk (64) sk ofzhvf ajgfaoj fshmptjtsgh stflhaojm , hah shogjm, hah ftrfgthvojm stgsojm tfc˅gjm prksahgfvfobf i fgj bk jof stflhaof h fgj bk ajgfaoh ftrfgtjr.

;. fc˅gf kgvhahlrhbumf x bknofnz˅lk (64) sk ofzhvf eajlfaohm ftrfgtj-rjm fgj

x4, x∜

6 ∀H ⇜ ahmo↜∟

xo = x.

9. ^fc˅gf kgvhahlrhbumf x bknofnz˅lk (64) sk ofzhvf eajlfaoj fshmptjtsgh stflhaojm fgj bk jof stflhaof h fgj bk eajlfaoh ftrfgtjr.

5. ^fc˅gf kgvhahlrhbumf x bknofnz˅lk (64) sk ofzhvf jnlhbfbu¾cjm tf˅cgjm ,hah rkpkakrjm , fgj pjstjbh r ? 4 tfgvj nf, zf svfgj x4, x

∜6 ∀H , zf gjbk

bk 4 1 |x4∜x|+ |x∜6∜x| 1 r , pjstjbh O ≫6 tfgj nf bk

|xO ∜x|≫r.

Bfsoj, rkpkakr bk okstflhaof tfc˅gf kgvhahlrhbumf.

Okgf p =

∌i ∌u

(x, x) h q = ∌i ∌v

(x, x)

jzofc˅fvfbu pfrchbfaok hzvjnk jn i (u, v) hzrfc˅uoftk u kgvhahlrhbumu ¯x bkn-ofnz˅lk (64).fnf sk bknofnz˅lf

yo +6 = pyo + qyo∜

6, o = 4, 6, . . . (6<)

zjvk ahokfrhzhrfof bknofnz˅lf prhnruz˅kof bknofnz˅lh (64) u tfc˅gh kgvhahlrhbumf x.

Trktpjstfvhmj nf rbks˅kobf bknofnz˅lk (6<) hmfbu jlahg

yo = ηo , o = 4, 6, . . .

enbk vrhbknojst η = 4 trklf jnrknhth. Rfmbkojm jvk vrhbknojsth u bknofnz˅lu(6<) njlhbfmj2

ηo +6 = pηo + q η∜6

jnfgak nhbkabkobkm sf ηo∜

6 = 4, sahbknh

η< = pη + q (63)

Jvf bknofnz˅lf sk ofzhvf gfrfgtkrhsthc˅ojm bknofnz˅ljm nhikrkotok bkn-ofnz˅lk (64), f obkoh gjrhbkoh η sk ofzhvfbu svjbstvkohm vrhbknojsthmf .





^kjrkm <.6 (^kjrkm ahokfrhzhrfok stflhaojsth)

(f) Fgj jlf gjrhbkof gfrfgtkrhsthc˅ok bknofnz˅lk (63) akz˅k u jtvjrkojm bk-nhohc˅ojm nhsgu |η| 1 6 jonf bk kgvhahlrhbum x bknofnz˅lk (64) ajgfaojfshmptjtsgh stflhafo.

(l) Fgj bk lfrkm bknfo jn gjrhbkof gfrfgtkrhsthc˅ok bknofnz˅lk (63) pj fpsj-autojb vrhbknojsth vk¾ch jn bknfo, jonf bk kgvhahlrhbum x bknofnz˅lk (64)okstflhafo.

(c) Tjtrklfo h njvjabfo usajv nf jlf gjrhbkof gfrfgtkrhsthc˅ok bknofnz˅lk (63)akz˅k u jtvjrkojm bknhohc˅ojm nhsgu

|η

|1 6, bk

| p| 1 6∜q 1 <. (6>)

[ jvjm sauc˅fbu, x bk shog .

(n) Tjtrklfo h njvjabfo usajv nf jlf gjrhbkof gfrfgtkrhsthc˅ok bknofnz˅lk (63)hmfbu fpsjautou vrhbknojst vk¾cu jn bknfo, bk

|q | ? 6 h | p| 1 |6∜q |.[ jvjm sauc˅fbu, x bk rkpkakr .

(k) Tjtrklfo h njvjabfo usajv nf bknfo gjrhbko gfrfgtkrhsthc˅ok bknofn˅zlk (63) hmf fpsjautou vrhbknojst vk¾cu jn bknfo, h nf nrueh hmf fpsjautou vrhbknojst mfobu jn bknfo, bk

p< + >q ? 4 h | p| ? |6∜q |.[ jvjm sauc˅fbu, okstflhaoh kgvhahlrhbum sk zjvk sknafstf tfc˅gf .

(i ) Tjtrklfo h njvjabfo usajv nf gjrhbko gfrfgtkrhsthc˅ok bknofnz˅lk (63) hmf fpsjautou vrhbknojst bknofgu bknfo, bk

| p| = |6∜q |hah

q = ∜6 h | p|≮<.

[ jvjm sauc˅fbu, kgvhahlrhbum x sk zjvk ok`hpkrljahc˅of tfc˅gf .





Nkstohchbf <.<

(f) Rf rbks˅kobk {xo}∟o =∜

6 bknofnz˅lk (64) gfz˅kmj nf bk pkrhjnh˅coj sf pk-rhjnjm p fgj bk

xo + p = xo zf svfgj o≫∜6. (6;)

(l) Rf rbks˅kobk {xo}∟o =∜

6 bknofnz˅lk (64) gfz˅kmj nf bk pkrhjnhc˅oj sf prjsthm pkrhjnjm p, hah p-chgah˅coj fgj bk joj pkrhjnhc˅oj sf pk-rhjnjm p h fgj bk p ofbmfobh prhrjnfo lrjb zf gjbh vrhbknh (6;).

Yfz˅fo faft zf hsphthvfobk stflhaojsth h fshmptjtsgh` pjofs˅fobf okgh` nhik-rkotoh` bknofnz˅lh bk prjuc˅fvfobk obh`jvh` hovfrhbfoth.

Nkstohchbf <.3 Okgjostfotoj okprkghnoj prksahgfvfobk H 2 \ <↜

\ sk ofzhvf hovfrhbfotf zf (64) fgj vrhbknh

H (xo +6 , xo ) = H (xo , xo∜

6) zf svfgj o = 4, 6, . . . .

Spkchbfafo sauc˅fb bknofnz˅lk (66), zf F = 6, Ayokss bk jtgrhj 68><. ej-nhok rfnk¾ch of prjlakmhmf hz kjrhbk lrjbkvf. [ jvjm spkchfbaojm sauc˅fbu, bknofnz˅lf pjstfbk

xo +6 = 6 + xo

xo∜6, o = 4, 6 . . . (69)

f svfgj obkoj rbks˅kobk bk pkrhjnhc˅oj sf pkrhjnjm pkt. Rfhstf bk rbks˅kobk bkn-ofnz˅lk (69) sf pj c˅ktohm uvbkthmf x4 h x

∜6 ;-chgahc˅oj2

x∜

6, x4, 6 + x4

x∜

6, 6 + x

∜6 + x4

x∜

6x4, 6 + x

∜6

x4, . . . .

Bknofnz˅lf (66) pjsbknubk hovfrhbfotu

H (xo , xo∜

6) = 6 + 6xo

6 + 6xo∜

6(F + xo + xo

∜6) = cjostfot

hz c˅kef sahbknh nf bk svfgj rbks˅kobk bknofnz˅lk (66) jerfohc˅koj jnjznj h jnjzejsf pjzhthvojm gjostfotjm.

[ pjsabknobh` okgjahgj ejnhof Ayokssjvf bknofnz˅lf (66) bk prhvugaf vk-ahgu pfz˅obu struc˅obfgf hz jlafsth nhsgrktoh` nhofmhc˅gh` shstkmf h nhikrkotoh` bknofnz˅lh. Bknfo jn rfzajef zf tfb hotkrks bk ljeftf nhofmhgf gjbu pjsbknubkjvf bknofnz˅lf

Sfnf ¾ckmj rfzmjtrhth prjlakm prjofafz˅kobf h fofahzk tfc˅fgf kgvhahlrhbumfAyokssjvk bknofnz˅lk. Ayokssjvj prksahgfvfobk ¾ckmj nkstohrfth u zfvhsojsth jnpfrfmktfrf F , of sabknk¾ch ofc˅ho2





HoW6]2= AWxZ, yZ] 2= (F + x)/y7

Jnrknhmj stgsok tfc˅gk.

HoW<]2= SjavkWAWx, x] == x, x]JutW<]= {{x -? 6/< (6 - SqrtW6 + > F])}, {x -? 6/< (6 + SqrtW6 + > F])}}

HoW3]2= {it6, it<} = {x, x} /. %JutW3]= {{6/< (6 - SqrtW6 + > F]), 6/< (6 - SqrtW6 + > F])},{6/< (6 + SqrtW6 + > F]), 6/< (6 + SqrtW6 + > F])}}

Yhnhmj nf hmfbu nvhbk stgsok tfc˅gk. [ vfrhbflak it6 h it< smj pj`rfohah prvuh nrueu stgsou tfc˅gu.

Jnrknhmj pfrchbfaok hzvjnk pj jlbkmf prjmbkoabhvhm.

HoW>]2= p = NWAWx, y], x]JutW>]= 6/y

HoW;]2= q = NWAWx, y], y]JutW;]= -((F + x)/y_<)

Jnrknhmj vrhbknojst p h q u stgsojb tfc˅gh it6 ,

HoW5]2= {x, y} = it67HoW0]2= p6 = pJutW0]= </(6 - SqrtW6 + > F])

HoW8]2= q6 = q JutW8]= -((> (F + 6/< (6 - SqrtW6 + > F])))/(6 - SqrtW6 + > F])_<)

f zfthm h u stgsojb tfc˅gh it< .

HoW5]2= {x, y} = it<7HoW0]2= p< = pJutW0]= </(6 + SqrtW6 + > F])

HoW8]2= q< = q JutW8]= -((> (F + 6/< (6 + SqrtW6 + > F])))/(6 + SqrtW6 + > F])_<)

Jnrknhmj gfrfgtkrhsthc˅ou bknofnz˅lu.

HoW64]2= gp = QWAfmlnf]_< - p QWAfmlnf] - q == 47HoW66]2= gp = IuaaShmpahiyWgp]JutW66]= (QWAfmlnf] + SqrtW6 + > F] QWAfmlnf] + < F (6 + QWAfmlnf]_<))/(< F)





Hzrfc˅uofbmj gfrfgtkrhsthc˅ou bknofnz˅lu u it6 ,

HoW6<]2= {x, y} = it67HoW63]2= gp6 = gp7

HoW6>]2= gp6 = IuaaShmpahiyWgp6, F ? 4]JutW6>] = (QWAfmlnf] + SqrtW6 + > F] QWAfmlnf] + < F (6 + QWAfmlnf]_<))/(< F)

f zfthm hzrfc˅uofbmj svjbstvkok vrhbknojsth gfrfgtkrhsthc˅ok bknofnz˅lk u it6 .

HoW6;]2= SjavkWgp6 == 4, QWAfmlnf]]

HoW69]2= {sv6, sv<} = QWAfmlnf] /. %JutW69]2={(-6 - SqrtW6 + > F] - SqrtW< + > F - 69 F_< + < SqrtW6 + > F]])/(> F),(-6 - SqrtW6 + > F] + SqrtW< + > F - 69 F_< + < SqrtW6 + > F]])/(> F)}

HoW65]2= IuaaShmpahiyWsv6*sv<, F ? 4]JutW65]= 6

Yhnhmj nf bk prjhzvjn svjbstvkoh` vrhbknojsth u it6 bknofg bknfo.

Nf lhsmj utvrnhah bksu ah svjbstvkok vrhbknojsth rkfaoh hah gjmpakgsoh lrj-

bkvh, pjsmftrfbmj zofg nhsgrhmhofotk gfrfgtkrhsthc˅ok bknofnz˅lk gp6 .HoW60]2= nhsc6 = NhscrhmhofotWgp6, η]7

nhsc6 = IuaaShmpahiyWnhsc6]JutW60]= 6+< F

∜0F < + − 6+> F<F <

Erfstg nhsgrhmhofotk (<4) pjgfzubk nf jof mjz˅k lhth pjzhthvof hah okef-thvof u zfvhsojsth jn pfrfmktrf F .

HoW68]2= TajtWnhsc6, {F, 4, >}, Tajt\foek -? {-;, ;}]

Jnrknhmj ouak nhsgrhmhofotk.

HoW<4]2= SjavkWnhsc6 == 4, F]JutW<4]= {{F -? -(6/>)}, {F -? 3/>}}

Nhsgrhmhofotf bk ouaf u thm vrhbknojsthmf pfrfmktrf F . Sf erfstgf (<4)ujc˅fvfmj nf bk nhsgrhmhofotf okefthvof of hotkrvfau ( 3

>, + ∟), jnojsoj nfsu u tjm hotrvfau gjrhbkoh gfrfgtkrhsthc˅ok bknofnz˅lk gjmpakgsoh lrjbkvh.

\fzmjtrhmj sauc˅fb gfnf bk F = 3> . ^fnf su jlbk svjbstvkok vrhbknojsth u

it6 bknofgk ∜6.





6 < 3 >

>

<

<

>

Sahgf <42 Erfstg nhsgrhmhofotk gfrfgtkrhsthc˅ok bknofnz˅lk gp6

HoW<6]2= sv6 /. F -? 3/>JutW<6]= -6HoW<<]2= sv< /. F -? 3/>JutW<<]= -6

Ihgsof tfc˅gf it6 zf F = 3> bk ∜6

< .

HoW<3]2= it6 /. F -? 3/>JutW<3]= {-(6/<), -(6/<)}

Sfnf ¾ckmj nfth eafvoh zfgabuc˅fg u pjeaknu svjbstvkoh` vrhbknojsth u smhsautkjrkmf (<.6). [ tu svr`u, grkhrf¾ckmj erfstg svjbstvkoh` vrhbknojsth u stgsojbtfc˅gh it6 u zfvhsojsth jn pfrfmktrf F .

HoW<>]2= kqo = gp6 == 47sja = SjavkWkqo, QWAfmlnf]]7{sja6, sja<} = {QWAfmlnf]} /. %7e6 = TajtWsja6, {F, 4, 3/>}, Tajt\foek -? {-3, 6}, TajtStyak -? \kn]7e< = TajtWsja<, {F, 4, 3/>}, Tajt\foek -? {-3, 6}, TajtStyak -? Erkko]7e3 = AhstTajtW{{3/>, 4}}]7

S`jwWe6, e<, e3]Gjrhstk¾ch svk prkt`jnoj rkc˅koj gfj h erfstg (<6) zfgabuc˅ubkmj nf fgj bk

F 1 3> jonf pjstjbk nvhbk rkfaok svjbstvkok vrhbknojsth, bknof bk pj fpsjautojb

vrhbknojsth mfobf jn bknfo (zkakoh erfstg) f nruef bk vk¾cf jn bknfo (crvkoherfstg). [ tjm sauc˅fbu, stgsof tfc˅gf gjbu smj jzofc˅hah sf ip6 bk okstflhaof , htj sknafstf tfc˅gf .

Ekokrhs˅hmj jrlhtu Ayokssjvje prksahgfvfobf sf 644 tfc˅fgf, h prh tjmkuzmhmj nf bk (6, <) pjc˅ktof tfc˅gf. [zkt ¾ckmj nf bk F = <, h prhgfzfth prvh`nkskt c˅afojvf jrlhtk.

[ohvkrzhtkt u uzah 3< Trhrjnoj-mftkmfthc˅gh ifguatkt




4.6 4.< 4.3 4.> 4.; 4.9 4.5

3

<

6

6

Sahgf <62 Erfstg svjbstvkoh` vrhbknojsth zf it6 u zfvhsojsth jn F

HoW<;]2= F = <7HoW<9]2= jrl = JrlhtWAyokssMfp, {6.4, <.4}, 644]7HoW<5]2= ^fgkWjrl, {6, 64}]JutW<5]= {{6., <.}, {<., >.}, {>., 3.}, {3., 6.<;}, {6.<;, 6.40333},{6.40333, <.>9995}, {<.>9995, >.6<340}, {>.6<340, <.>0<33},{<.>0<33, 6.40563}, {6.40563, 6.<>39>}}

Ofcrtfbmj erfstg tf c˅fgf prkt`jnok jrlhtk sf 644 rbks˅kobf.

4 <4 >4 94 04 644

4

6

<

3

>

Sahgf <<2 Erfstg tfc˅fgf jrlhtk Ayokssjvk bknofnz˅lk

Ekokrhs˅hmj jnejvfrfbu¾ch erfstg vrkmkosgje ohzf zf prkt`jnoh erfstg tfc˅fgf.Ifzoh pjrtrkt pjgfzubk nf bk jrlhtf sfnrz˅fof u zftvjrkojb grhvjb.Ifzoh pjrtrkt bk ekokrhsfo sabknk¾cjm Nyofmhcf ofrknojm2

HoW<0]2=i6 = T`fskTjrtrfhtWjrl, Tajt\foek -? {{4, ;}, {4, ;}},Cjajrs -? {\kn}]

Sfnf ¾ckmj ekokrhsfth bjs˅ nvf ifzof pjrtrktf zf nruek nvhbk jrlhtk Ayoksjvk bknofnz˅ok, h prhgfzfth svf trh ifzof pjrtrktf of bknojm erfstgu.





4 <4 >4 94 04 644

4

6

<

3

>

Sahgf <32 Erfstg vrkmkosgje ohzf jrlhtk Ayokssjvk bknofnz˅lk

4 6 < 3 > ;4

6

<

3

>

;

Sahgf <>2 Ifzoh pjrtkrt jrlhtk Ayokssjvk bknofnz˅lk

HoW<8]2= jrl< = JrlhtWAyokssMfp, {6.4, 6.4}, <44]7HoW34]2=i< = T`fskTjrtrfhtWjrl<, Tajt\foek -? {{4, ;}, {4, ;}}, Cjajrs -? {Lauk}]7HoW36]2= jrl3 = JrlhtWAyokssMfp, {6.<, <.;}, <44]7HoW3<]2=i3 = T`fskTjrtrfhtWjrl3, Tajt\foek -? {{4, ;}, {4, ;}}, Cjajrs -? {Erkko}]7HoW33]2= S`jwWi6, i<, i3]

Ekokrhs˅hmj erfstg hovfrhbfotk Ayokssjvk nhikrkotok bknofnz˅lf. Ayokssjvknhikrkotok bknofnz˅lf bk vk¾c nkstohsfof u pfgktf Nyofmhcf.

HoW3>]2= Ayokss

[ohvkrzhtkt u uzah 3> Trhrjnoj-mftkmfthc˅gh ifguatkt




4 6 < 3 > ;4

6

<

3

>

;

Sahgf <;2 Ifzoh pjrtkrth jrlhtf Ayokssjvk bknofnz˅lk

JutW3>]= xW6 + o] == (F + xWo])/xW-6 + o]

Rf ofafz˅kobk hovfrhbfotk, gjrhsth¾ckmj Nyofmhcf ofrknlu \fthjofaHovfrhfotW] .

HoW3;]2= hovAyokss = \fthjofaHovfrhfotWAyokss]JutW3;]= cW6] + ((6 + F) cW0])/xW-6 + o] + cW0] xW-6 + o] + ((6 + F) cW0])/xWo] + (F cW0])/(xW-6 + o] xWo]) + (cW0] xW-6 + o])/xWo] +cW0] xWo] + (cW0] xWo])/xW-6 + o]cW6] + ((6 + F) cW0])/xW-6 + o] + cW0] xW-6 + o] + ((6 + F) cW0])/xWo] + (F cW0])/(xW-6 + o] xWo]) + (cW0] xW-6 + o])/xWo] +cW0] xWo] + (cW0] xWo])/xW-6 + o]

HoW39]2= hovAyokss = hovAyokss /. {cW0] -? 6, cW6] -? F + <}JutW39]= < + F + (6 + F)/xW-6 + o] + xW-6 + o] + (6 + F)/xWo] + F/(

xW-6 + o] xWo]) + xW-6 + o]/xWo] + xWo] + xWo]/xW-6 + o]

HoW35]2= hovAyokss = hovAyokss /. {xWo] -? x, xWo - 6] -? y}JutW35]= < + F + (6 + F)/x + x + (6 + F)/y + F/(x y) + x/y + y + y/x

HoW30]2= hovAyokss = IfctjrWhovAyokss]JutW30]= ((6 + x) (6 + y) (F + x + y))/(x y)HoW38]2= F = <.47HoW>4]2= Tajt3NWhovAyokss, {x, 4.446, ;}, {y, 4.446, ;}, Tajt\foek -? {4, 644},

TajtTjhots -? 34]

[ohvkrzhtkt u uzah 3; Trhrjnoj-mftkmfthc˅gh ifguatkt




4

<

>

4

<

>

4

;4

644

Sahgf <92 Erfstg hovfrhbfotk Ayokssjvk nhikrkotok bknofnz˅lk zf F = <

Gjmpakgsoj pjofs˅fobk Ayokssjvk bknofnz˅lk zf rfzahc˅htk vrhbknojst hpf-rfmktf F mjz˅k sk vhnbkth hz lhiurgfchjoje nhbferfmf.

HoW>6]2= CakfrWF]HoW><]2= LhiurcfthjoTajtONWAyokssMfp, {F, 4.8;, 6.4;}, {6.<, 6.4},

Stkps -? 344, Htkrftks -? 6;4, IhrstHt -? >44]

4.89 4.80 6.44 6.4< 6.4>

6.4

6.;

<.4

<.;

Sahgf <52 Lhiurgfchjoh nhbferfm Ayokssjvk bknofnz˅lk zf F ∀(4.8; , 6.4;)

\fohbk smj rkgah, nf bk zf F = 6 svfgj rbks˅kobk Ayokssjvk bknofnz˅lkpkrhjnhc˅oj sf pkrhjnjm pkt, f vhzufaou pjtvrnu tjef ofm nfbk lhiurgfstjohnhbferfm, gjbke ¾ckmj uvk¾cfth u lahzhoh tfc˅gk F = 6.

HoW>3]2= LhiurcfthjoTajtONWAyokssMfp, {F, 4.88, 6.46}, {6.<, 6.4},Stkps -? 344, Htkrftks -? 6;4, IhrstHt -? >44]





AhtkrfturfWK] S. Kafynh, Fo Hotrjnucthjo tj Nhfflkrkock Kqufthjos - ^`hrn Knhthjo ,

Sprhoekr, Okw Xjrg, <44;.

WGM] M.\.S. Guakojvh¾c fon J. Mkrhoj, Nhscrktk Nyofmhcfa systkms fon Nhfflkrkock Kqufthjos wht` Mft`kmfthcf , C`fpmfo fo @faa/C\C, Ljcf\ftjo - Okw Xjrg, <44<.

WGA] M.\.S. Guakojvh¾c fon E. Afnfs, Nyofmhcs ji Skcjon Jrnkr \fthjofa Nhfflkrkock Kqufthjos , C`fpmfo/C\C, Ljcf \ftjo, <44<.

WO] M. Ourgfojvh¾c, Nhikrkotok bknofnz˅lk - kjrhbf h prhmbkok , Nkoifs, ^uzaf,<440.

WGA] M.\.S. Guakojvh¾c fon E. Afnfs, Nyofmhcs ji Skcjon Jrnkr \fthjofa Nhfflkrkock Kqufthjos wht` Jpko Trjlakms fon Cjobkcturks , C`fpmfo fon@faa/C\C, Ljcf \ftjo/Ajonjo/Okw Xjrg/Ufs`hoetjo, N.C., <446.

geometrijska interpretacija dinamike diferentnih jednadzbi

Documents