geometrijska interpretacija dinamike diferentnih jednadzbi
TRANSCRIPT
8/13/2019 Geometrijska interpretacija dinamike diferentnih jednadzbi
http://slidepdf.com/reader/full/geometrijska-interpretacija-dinamike-diferentnih-jednadzbi 1/40
8/13/2019 Geometrijska interpretacija dinamike diferentnih jednadzbi
http://slidepdf.com/reader/full/geometrijska-interpretacija-dinamike-diferentnih-jednadzbi 2/40
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
8/13/2019 Geometrijska interpretacija dinamike diferentnih jednadzbi
http://slidepdf.com/reader/full/geometrijska-interpretacija-dinamike-diferentnih-jednadzbi 3/40
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 tk`ohgf 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 vr`uosgk tk`ohc˅gk nj-gumkotfchbk zf fofahzu nhsgrktoh` nhofmhc˅gh` shstkmf, gfj h vr`uosgk 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
8/13/2019 Geometrijska interpretacija dinamike diferentnih jednadzbi
http://slidepdf.com/reader/full/geometrijska-interpretacija-dinamike-diferentnih-jednadzbi 4/40
Ekjmktrhbsgf hotkrprktfchbf nhofmhgk nhikrkotoh` bknofnz˅lh
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
8/13/2019 Geometrijska interpretacija dinamike diferentnih jednadzbi
http://slidepdf.com/reader/full/geometrijska-interpretacija-dinamike-diferentnih-jednadzbi 5/40
8/13/2019 Geometrijska interpretacija dinamike diferentnih jednadzbi
http://slidepdf.com/reader/full/geometrijska-interpretacija-dinamike-diferentnih-jednadzbi 6/40
Ekjmktrhbsgf hotkrprktfchbf nhofmhgk nhikrkotoh` bknofnz˅lh
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
8/13/2019 Geometrijska interpretacija dinamike diferentnih jednadzbi
http://slidepdf.com/reader/full/geometrijska-interpretacija-dinamike-diferentnih-jednadzbi 7/40
Ekjmktrhbsgf hotkrprktfchbf nhofmhgk nhikrkotoh` bknofnz˅lh
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
8/13/2019 Geometrijska interpretacija dinamike diferentnih jednadzbi
http://slidepdf.com/reader/full/geometrijska-interpretacija-dinamike-diferentnih-jednadzbi 8/40
Ekjmktrhbsgf hotkrprktfchbf nhofmhgk nhikrkotoh` bknofnz˅lh
(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.
[ohvkrzhtkt u uzah 9 Trhrjnoj-mftkmfthc˅gh ifguatkt
8/13/2019 Geometrijska interpretacija dinamike diferentnih jednadzbi
http://slidepdf.com/reader/full/geometrijska-interpretacija-dinamike-diferentnih-jednadzbi 9/40
Ekjmktrhbsgf hotkrprktfchbf nhofmhgk nhikrkotoh` bknofnz˅lh
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
[ohvkrzhtkt u uzah 5 Trhrjnoj-mftkmfthc˅gh ifguatkt
8/13/2019 Geometrijska interpretacija dinamike diferentnih jednadzbi
http://slidepdf.com/reader/full/geometrijska-interpretacija-dinamike-diferentnih-jednadzbi 10/40
Ekjmktrhbsgf hotkrprktfchbf nhofmhgk nhikrkotoh` bknofnz˅lh
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] .
[ohvkrzhtkt u uzah 0 Trhrjnoj-mftkmfthc˅gh ifguatkt
8/13/2019 Geometrijska interpretacija dinamike diferentnih jednadzbi
http://slidepdf.com/reader/full/geometrijska-interpretacija-dinamike-diferentnih-jednadzbi 11/40
Ekjmktrhbsgf hotkrprktfchbf nhofmhgk nhikrkotoh` bknofnz˅lh
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/>}}
[ohvkrzhtkt u uzah 8 Trhrjnoj-mftkmfthc˅gh ifguatkt
8/13/2019 Geometrijska interpretacija dinamike diferentnih jednadzbi
http://slidepdf.com/reader/full/geometrijska-interpretacija-dinamike-diferentnih-jednadzbi 12/40
Ekjmktrhbsgf hotkrprktfchbf nhofmhgk nhikrkotoh` bknofnz˅lh
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 )
[ohvkrzhtkt u uzah 64 Trhrjnoj-mftkmfthc˅gh ifguatkt
8/13/2019 Geometrijska interpretacija dinamike diferentnih jednadzbi
http://slidepdf.com/reader/full/geometrijska-interpretacija-dinamike-diferentnih-jednadzbi 13/40
Ekjmktrhbsgf hotkrprktfchbf nhofmhgk nhikrkotoh` bknofnz˅lh
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.
[ohvkrzhtkt u uzah 66 Trhrjnoj-mftkmfthc˅gh ifguatkt
8/13/2019 Geometrijska interpretacija dinamike diferentnih jednadzbi
http://slidepdf.com/reader/full/geometrijska-interpretacija-dinamike-diferentnih-jednadzbi 14/40
Ekjmktrhbsgf hotkrprktfchbf nhofmhgk nhikrkotoh` bknofnz˅lh
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
8/13/2019 Geometrijska interpretacija dinamike diferentnih jednadzbi
http://slidepdf.com/reader/full/geometrijska-interpretacija-dinamike-diferentnih-jednadzbi 15/40
Ekjmktrhbsgf hotkrprktfchbf nhofmhgk nhikrkotoh` bknofnz˅lh
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}
[ohvkrzhtkt u uzah 63 Trhrjnoj-mftkmfthc˅gh ifguatkt
8/13/2019 Geometrijska interpretacija dinamike diferentnih jednadzbi
http://slidepdf.com/reader/full/geometrijska-interpretacija-dinamike-diferentnih-jednadzbi 16/40
Ekjmktrhbsgf hotkrprktfchbf nhofmhgk nhikrkotoh` bknofnz˅lh
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
8/13/2019 Geometrijska interpretacija dinamike diferentnih jednadzbi
http://slidepdf.com/reader/full/geometrijska-interpretacija-dinamike-diferentnih-jednadzbi 17/40
Ekjmktrhbsgf hotkrprktfchbf nhofmhgk nhikrkotoh` bknofnz˅lh
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
8/13/2019 Geometrijska interpretacija dinamike diferentnih jednadzbi
http://slidepdf.com/reader/full/geometrijska-interpretacija-dinamike-diferentnih-jednadzbi 18/40
Ekjmktrhbsgf hotkrprktfchbf nhofmhgk nhikrkotoh` bknofnz˅lh
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.
[ohvkrzhtkt u uzah 69 Trhrjnoj-mftkmfthc˅gh ifguatkt
8/13/2019 Geometrijska interpretacija dinamike diferentnih jednadzbi
http://slidepdf.com/reader/full/geometrijska-interpretacija-dinamike-diferentnih-jednadzbi 19/40
Ekjmktrhbsgf hotkrprktfchbf nhofmhgk nhikrkotoh` bknofnz˅lh
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<>;<
[ohvkrzhtkt u uzah 65 Trhrjnoj-mftkmfthc˅gh ifguatkt
8/13/2019 Geometrijska interpretacija dinamike diferentnih jednadzbi
http://slidepdf.com/reader/full/geometrijska-interpretacija-dinamike-diferentnih-jednadzbi 20/40
Ekjmktrhbsgf hotkrprktfchbf nhofmhgk nhikrkotoh` bknofnz˅lh
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.
[ohvkrzhtkt u uzah 60 Trhrjnoj-mftkmfthc˅gh ifguatkt
8/13/2019 Geometrijska interpretacija dinamike diferentnih jednadzbi
http://slidepdf.com/reader/full/geometrijska-interpretacija-dinamike-diferentnih-jednadzbi 21/40
8/13/2019 Geometrijska interpretacija dinamike diferentnih jednadzbi
http://slidepdf.com/reader/full/geometrijska-interpretacija-dinamike-diferentnih-jednadzbi 22/40
Ekjmktrhbsgf hotkrprktfchbf nhofmhgk nhikrkotoh` bknofnz˅lh
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
8/13/2019 Geometrijska interpretacija dinamike diferentnih jednadzbi
http://slidepdf.com/reader/full/geometrijska-interpretacija-dinamike-diferentnih-jednadzbi 23/40
Ekjmktrhbsgf hotkrprktfchbf nhofmhgk nhikrkotoh` bknofnz˅lh
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.
[ohvkrzhtkt u uzah <6 Trhrjnoj-mftkmfthc˅gh ifguatkt
8/13/2019 Geometrijska interpretacija dinamike diferentnih jednadzbi
http://slidepdf.com/reader/full/geometrijska-interpretacija-dinamike-diferentnih-jednadzbi 24/40
Ekjmktrhbsgf hotkrprktfchbf nhofmhgk nhikrkotoh` bknofnz˅lh
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
8/13/2019 Geometrijska interpretacija dinamike diferentnih jednadzbi
http://slidepdf.com/reader/full/geometrijska-interpretacija-dinamike-diferentnih-jednadzbi 25/40
Ekjmktrhbsgf hotkrprktfchbf nhofmhgk nhikrkotoh` bknofnz˅lh
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.
[ohvkrzhtkt u uzah <3 Trhrjnoj-mftkmfthc˅gh ifguatkt
8/13/2019 Geometrijska interpretacija dinamike diferentnih jednadzbi
http://slidepdf.com/reader/full/geometrijska-interpretacija-dinamike-diferentnih-jednadzbi 26/40
Ekjmktrhbsgf hotkrprktfchbf nhofmhgk nhikrkotoh` bknofnz˅lh
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
8/13/2019 Geometrijska interpretacija dinamike diferentnih jednadzbi
http://slidepdf.com/reader/full/geometrijska-interpretacija-dinamike-diferentnih-jednadzbi 27/40
8/13/2019 Geometrijska interpretacija dinamike diferentnih jednadzbi
http://slidepdf.com/reader/full/geometrijska-interpretacija-dinamike-diferentnih-jednadzbi 28/40
Ekjmktrhbsgf hotkrprktfchbf nhofmhgk nhikrkotoh` bknofnz˅lh
< 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.
[ohvkrzhtkt u uzah <9 Trhrjnoj-mftkmfthc˅gh ifguatkt
8/13/2019 Geometrijska interpretacija dinamike diferentnih jednadzbi
http://slidepdf.com/reader/full/geometrijska-interpretacija-dinamike-diferentnih-jednadzbi 29/40
Ekjmktrhbsgf hotkrprktfchbf nhofmhgk nhikrkotoh` bknofnz˅lh
>. ^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 kgvhahlrh- bumf 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 .
[ohvkrzhtkt u uzah <5 Trhrjnoj-mftkmfthc˅gh ifguatkt
8/13/2019 Geometrijska interpretacija dinamike diferentnih jednadzbi
http://slidepdf.com/reader/full/geometrijska-interpretacija-dinamike-diferentnih-jednadzbi 30/40
Ekjmktrhbsgf hotkrprktfchbf nhofmhgk nhikrkotoh` bknofnz˅lh
^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 .
[ohvkrzhtkt u uzah <0 Trhrjnoj-mftkmfthc˅gh ifguatkt
8/13/2019 Geometrijska interpretacija dinamike diferentnih jednadzbi
http://slidepdf.com/reader/full/geometrijska-interpretacija-dinamike-diferentnih-jednadzbi 31/40
Ekjmktrhbsgf hotkrprktfchbf nhofmhgk nhikrkotoh` bknofnz˅lh
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
[ohvkrzhtkt u uzah <8 Trhrjnoj-mftkmfthc˅gh ifguatkt
8/13/2019 Geometrijska interpretacija dinamike diferentnih jednadzbi
http://slidepdf.com/reader/full/geometrijska-interpretacija-dinamike-diferentnih-jednadzbi 32/40
Ekjmktrhbsgf hotkrprktfchbf nhofmhgk nhikrkotoh` bknofnz˅lh
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)
[ohvkrzhtkt u uzah 34 Trhrjnoj-mftkmfthc˅gh ifguatkt
8/13/2019 Geometrijska interpretacija dinamike diferentnih jednadzbi
http://slidepdf.com/reader/full/geometrijska-interpretacija-dinamike-diferentnih-jednadzbi 33/40
Ekjmktrhbsgf hotkrprktfchbf nhofmhgk nhikrkotoh` bknofnz˅lh
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.
[ohvkrzhtkt u uzah 36 Trhrjnoj-mftkmfthc˅gh ifguatkt
8/13/2019 Geometrijska interpretacija dinamike diferentnih jednadzbi
http://slidepdf.com/reader/full/geometrijska-interpretacija-dinamike-diferentnih-jednadzbi 34/40
Ekjmktrhbsgf hotkrprktfchbf nhofmhgk nhikrkotoh` bknofnz˅lh
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
8/13/2019 Geometrijska interpretacija dinamike diferentnih jednadzbi
http://slidepdf.com/reader/full/geometrijska-interpretacija-dinamike-diferentnih-jednadzbi 35/40
Ekjmktrhbsgf hotkrprktfchbf nhofmhgk nhikrkotoh` bknofnz˅lh
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.
[ohvkrzhtkt u uzah 33 Trhrjnoj-mftkmfthc˅gh ifguatkt
8/13/2019 Geometrijska interpretacija dinamike diferentnih jednadzbi
http://slidepdf.com/reader/full/geometrijska-interpretacija-dinamike-diferentnih-jednadzbi 36/40
Ekjmktrhbsgf hotkrprktfchbf nhofmhgk nhikrkotoh` bknofnz˅lh
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
8/13/2019 Geometrijska interpretacija dinamike diferentnih jednadzbi
http://slidepdf.com/reader/full/geometrijska-interpretacija-dinamike-diferentnih-jednadzbi 37/40
Ekjmktrhbsgf hotkrprktfchbf nhofmhgk nhikrkotoh` bknofnz˅lh
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
8/13/2019 Geometrijska interpretacija dinamike diferentnih jednadzbi
http://slidepdf.com/reader/full/geometrijska-interpretacija-dinamike-diferentnih-jednadzbi 38/40
Ekjmktrhbsgf hotkrprktfchbf nhofmhgk nhikrkotoh` bknofnz˅lh
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]
[ohvkrzhtkt u uzah 39 Trhrjnoj-mftkmfthc˅gh ifguatkt
8/13/2019 Geometrijska interpretacija dinamike diferentnih jednadzbi
http://slidepdf.com/reader/full/geometrijska-interpretacija-dinamike-diferentnih-jednadzbi 39/40
8/13/2019 Geometrijska interpretacija dinamike diferentnih jednadzbi
http://slidepdf.com/reader/full/geometrijska-interpretacija-dinamike-diferentnih-jednadzbi 40/40
Ekjmktrhbsgf hotkrprktfchbf nhofmhgk nhikrkotoh` bknofnz˅lh
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.