Bifurcation of the state of equilibrium in the system of nonlinear parabolic equations with transformed argument
We consider a system of nonlinear parabolic equations with transformed argument and prove the existence of integral manifolds. We investigate the bifurcation of an invariant torus from the state of equilibrium.
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| Date: | 1999 |
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| Main Authors: | , |
| Format: | Article |
| Language: | Russian English |
| Published: |
Institute of Mathematics, NAS of Ukraine
1999
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/4733 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510896624762880 |
|---|---|
| author | Klevchuk, I. I. Клевчук, И. И. Клевчук, И. И. |
| author_facet | Klevchuk, I. I. Клевчук, И. И. Клевчук, И. И. |
| author_sort | Klevchuk, I. I. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T21:12:54Z |
| description | We consider a system of nonlinear parabolic equations with transformed argument and prove the existence of integral manifolds. We investigate the bifurcation of an invariant torus from the state of equilibrium. |
| first_indexed | 2026-03-24T03:04:17Z |
| format | Article |
| fulltext |
Y,/1K 517.9
H. H. K~eB'~yK (qepuonalt. yn-T)
B H | I IO, I IO~.~EHI , ISI P A B H O B E C H . q
B C H C T E M E I - I E J I H H E H H b I X I I A P A B O d l H t I E C K H X
Y P A B H E H I / I ~ C I I P E O B P A 3 O B A H H b I M A P F Y M E H T O M
We consider a system of nonlinear parabolic equations with a transformed argument. We prove the
existence of integral manifolds. We investigate the bifurcation of an invariant torus from equilibrium.
Po31"Jl,,q/i.a~rl,ol CHtYI'CMa IICJlilliI~IIIHX napa6oJliqllHX pinlDIin, 3 nepe'lBOpellHM apl'yMell'l'OM. ,~[OUCll[ellO
icnylmnna iwrevpaJmlmx t, moronnltin. ~2ocJtilL~ena 6iqbypKaltia immpiawrnoro Topa ia c'rany pin-
lIOBal'14.
1. IIpeo6paaonaune HCXO~HOfi 3a~a~rx. Pacc~oTpnM CHCTeMy ne~nae~max napa-
6o~nqecKHx ypaBneaH~ C npCo6pa3oaaHHhIM apryMcHTOM
bu D(t, + A(t, ~)u + B(t, ~)u a + f(t, x, u, ~%, e.) (1)
H c rlepHo~HqeCKHM yc,rloBHeM
u (t, x + 2re) = u (t, x). (2)
3z~ec~, ~ ~ p-Mepa~a napaMeTp c ManhXMrt nosxo~H-renba~M~x KOMrlOHeHTaMH, U A =
= u ( t , x - A ) , A . ~ c ~ m w apryMCHTa, r~aTpntua D(t, ~), A (t, ~), B(t , e.) n dpynK-
tlklJt f : R 2n+p+2 -") R" IDITb paa nenpep~aano Z~Hqbtl~epcHmlpyeMU no BCeM apry-
MenTaM n 2rc-nepHoz~naecmIe OrnOCHTe~bnO t, X, f(t , X, U, V, e) = O ([ U [2 + IV [2)
npn I u [i+ [ v'[ ---) 0. I'IoaToMy ~yHKtma f ( t , X, U, V. e) yno~zeTsopaeT ycs~osaaM
f ( t , x ,O ,O ,~)= O, [ f ( t , x ,u ,v , 8 ) - f ( t , x , u ' v ' , e ) l <- v ( l u - u ' l 2 + l v - v ' 1 2 ) 1/2,
(3)
l u l < p , l u ' l < p , I v l < p , I v ' l < p ,
rae I "12 : +- . . + nOCTOSSmas n .n .mua v oxer cae ana
KaK yr0~nO Haa'~O}t npri y~crmtucmm p. OyHKH, HIO f ( t , x. u, v, e) MOJgKHO :loonpe-
/Ie~HTb BHe o6nacTn I u I < P, [ v [ < p TaZ, qTO6bl ycdlOBrlC (3) BblrlO.rlHJLrlOCb SO
BCeM npocrpaHca~e, rlycT~ Marpmta D(t, ~) nonox~nTe~n,no onpeaesleHa.
CHCTeMa (1) llprlMenaeTCJi ,RJDI M0]~eJIHp0BaHHa HeJIHSeltHrMX 3CIJ(~DeKTOB a 01ITH-
Ke [I ]. PIrlTerpanbable SHOFOo6paarl~t rl 6a~ypKat~na pcnleH~I~ napa6onn,aecKHx aa-
~taq nayaa.nncb ~ pa6oTax [2 - 6], nprt~eM ~ [5, 6] nccneao~aHa 6ndpypKatma po~ae-
HH.q I.[HK~JIa aBTOHOMHOFO rlapa6o~ll, lqeCKOFO ypaBHeHHa C npeo6pa3oBaHHbiM ap ry -
~errroM. B Haeroatue~4 pa6oTe paccMorpeHa napa6onn,~ecxaa cr~creMa c npeo6paao-
BaHmaM apry~eHTora 6ozee o6mero nn~xa,/~nJ~ K6TOpOfi C noMotub~O MeTo/xa nHTer-
pan~nnx nHoroo6paan~l/xoKaaano cymeca~onalme mmapnammoro Topa.
Hapa~ty c (I) paccMoTpn~t nnHe~lHyao cHereMy
q
~u = D(t,~)~x~ +A(t ,e)u+B(t ,~)uA" (4)
Ot
Petuemm za~a,m (4), (2) 6yae~ nCKaT~ n ~aae papa ~ypbe a Ko~nneKCHO~ ~opMe
U(t,X) = ~ yk(t)exp(-ikx), y_k(t) = y~(t). (5)
rlo~teramaaa (5) ~ (4) n epanmmaa K o ~ b ~ a m ~ e a ~ npa exp (- ikx), nony~aeM
CqeTHyIO CHCTeMy ~ r l~epen t l r l a s lbHl , lX ypaBHCHHI:I 0TH0CHTCJIbHO KO~K~)~HILtleHTOB
papa Oypbc
�9 H. H. KJ'IEBqYK. 1999
1342 ISSN 0041-6053. Yrp. slam. ~'vpn.o 1999, m. 51, N'-' 10
BHOYPKAI.J[H,q FIOJ'IO)KEHH,,q PABHOBECH,q B CHCTEME HEJ'IHHEI::IHBIX ... 1343
dYk(t)
= [-k2D(t ,e ) + A(t,e) + B(t ,e)exp(ikA)]yk(t) , k =0,_+1 . . . . . (6)
dt
CrIcrer~a (6) ~IB.rl.qeTCJI crlcTeMofl JIHHe~HHX/Irlf.~qbepeHl.IHa.rlbHblX ypaneeHrI~t c ne-
prxo~rlqecKI4MU KoaqbqbnuneaTaMn. Cor~acHo TeopeMe O.aoKe cyIaec'rayeT neBu-
po;~t~eHaaa ~ta'rpriua H k( i, e ), H ~( t + 2~, e) " H k( t, e ), TaKa.q, qTO aa~eaa y~=
= Hi( t , e)z k UpnBOanT crlcTeMy (6) K anay
dZk = Ck(E)Z k, C_k(e ) = Ck(E), k = 0,-4- | . . . . ,.
dt
Petueane aa~aqrt (1), (2) 6y~eM riCKaTb a aH~e p ~ a (5). FloJIcTaa~J~ (5) a (1) rt
cpaBnnaa.q Ko3dp~HttrieHTl,I ripn exp ( - ikx) , k E Z, riozyuaeM c~r cricTe~ty
~;adpqbepeHr4ria~bmax ypaBHeltH~i OTHOCrlTe.rlbnO KO~(.13~HUrleHTOB pztta (Dypbe
= M(t , e)y + F(t, y, e), (7)
dt
r~e y = (Y0,YI,Y-I . . . . ) r M(t, e ) ~ 6ecKoaenHaz 6~oqao-~rlaroumubnaz MaTprma c
6JtoKaMH
M k ( t , e ) = - k 2 D ( t , e ) + A ( t , e ) + B ( t , e ) e x p ( i k A ) , k = 0 , _ + l . . . . ;
F(t, y, e)= (f0,3~,f-i .... )T ~ ne~lnnenaaz ~yaKarla, npriueM f~ .qB,'I.qXOTC.q Ko3qb-
d#)HaHeHTaMH rlpn exp ( - i kx ) paa~o~eHria qbynI~mm f ( t , x, u, u a, e) a pzz~ q)yp~,e.
l'loKa~eM, qrO qbyaKtala F(t, y, e) yJlonaeTaopaeT ycaomuo J'lnntumta. BBe~eM
B npoc-rpaHc'rBe nocJ'IenoBaTe.usnocTet,~ aopMy I Y l = (~,'___..ly, I 2 )1/2. PaccMoTpma
~pyror'l aeKTop Z = (Zo,Z~,Z_~ .... )7" ~oa~qbHuI~et~TOa Oypse petuenaz v (t, x) ypaa-
aeHnz (1) rt cooTaeTcTay~omrff~ eMy aeKTop F(t, Z, e)= (go,g~,g-~ .... )7". Hcuom, aya
paaeHcTao FlapceBaaJL rI~eer4
1 2 ~
( i:,-,,r-)"'- =
k----** 0
, ,112 ( 1 2 ! o "~ 1/2
- < - , y , - -
C,ue~o~aTe~,ho, dpyHKtmZ F y~oa.rleraopze'r yc:toamo Tlririmmta c uourOaHHO~
-~:2V.
Hcno.ar~aya Hepa.aeHCT~O Ba~eacKoro, otterm~ pemeHHe Yk(t) CHCTe~a~ (6):
1t
lYk(t)l <- lYk(to)lexp f Ak(t~)dh, (8)
to
rae A ~(t) ~ Har~6ozr, mata xapa~repncau~'cecKHlt KopeHb ~a'rpnttu [M~(t, e) +
+ M~(t ,~)] /2 .
HyeTb ~J~ acex x, npni~a~examrix e~HHHqH01~ Ct.~epe [X [ = 1, amno~a2Kn'c2
Hepaaeac'raa (D(t , ~)x, x) _> ~t > 0, (A x +A rx, x) ~ 2a ; (B( t , ~ )exp( ikA)x +
+ Br ( t , e ) e x p ( - i k A ) x , x ) <_ 2b. Tor/ta
ISSN 004J-6053. Yrp. ~lam. ~hTpu.. 1999. m. 51, N e I0
1344 H, H. KJ'IEBqYK
Ak(t ) = l max(Mk(t, 8)x + M~:(t, 8)x, x) < - k 2 p + a + b
2 Ixtffil �9
Orcm~ta cne~ayer, wro limto** Ak(t ) = - 0-. l'Ipe~nozox~rIra, ,~ro xapaKrepHcTHqe-
CKOC ypaBHeHrl0 det ( C ~ ( e ) - L E ) = 0, k e Z , rlI~ICCT npocT~e KopHI4 " a ,n(8 ) +_
+ i ~m (8), am (0) = 0, [3m (0) > 0, m = 1, .. ~, p, a ocram,H~e KopHn ya0maeTBopa-
m r y c z o B . ~ IRe~.I > ~'+ ~i, T > ~ > 0. YCSmBHe a m (0) = 0 He HMeeT xapaKTep
Bhlpo;K.~eHH~I, Tag. KaX 8 JtBJISIeTCJt p-MepmaM napaMeTpo~,t. ~ n z BCeX Ueylbtx k ,
KpoMe Koae,moro qncna, B~anonnzeTCa Hepa~eHerUo k 2 > (? + 5 + a + b ) / I t .
T0rz~a - k 2 g + a + b _< - ( T + 5), At( t ) < - ( 7 + 5) H Ha Hepa~encTBa (8) c:~eayeT
oueaKa
[yk(t)[ < lye(t0)[ e x p [ - ( T + 5 ) ( t - t 0 ) ] , t > t o. (9)
l'lepermmeM CHCTeMy (7) B ~a~ae
dY~ = N ~ ( t , 8 ) y ~ + F l ( t , y , e ) , d ~ = N2(t, 8 ) y 2 + F 2 ( t , y , e ) , (10)
dt dt
r a e y = (y~,~)r , Yt = (Yo, Y~,Y-~ . . . . . Yko,Y-ko) r, Y2 = (Yk,+t,Y-to-I . . . . ) r , N~ (t,
e) = diag (M o, Mr, M_~ . . . . . Mto, M ko ), N 2 (t, 8) = diag (Mt~,+~, M_~,_t . . . . ) . F ~ (t, y,
8) = (f0,J~,f-l,: .- . ,fko,f-ko)r F 2 ( t , y , 8 ) = (ftq,+l,f-ko-I . . . . )T, k o =
= [ 4 ( ? + 8 + a + b ) / g ]. Marpauy C ( 8 ) = diag (Co, Ct,C_ ~ . . . . . Cko,C_ko) npa~ene~
K BHny C ( e ) = T ( g ) F ( 8 ) T - t ( 8 ) , rae F (8) = diag(A~(8), F ~(8)), A ~ (8) =
= diag (A3(8), A4(8)),' cO6CT~eHmae 3na'~egr~a MaTpHU~a A 3(8) yaounerr~opa~o-r yc-
nOBmO Re~. > ? + 5, A 4 ( 8 ) ~ aHaronan~,Haa MaTprlua c qHCYla~H (:~m(E)q"
+ i 13m(8) no armroHa.nH, a co6cT~enH~e zHa'~emta ~taTpaur~ F ~ (8) y~on~eT~opa~o'r
ycJ]OnHm Re X < - ? - 5. Tahoe rrpeo6pa~oBaHrm MO~HO no.ny~HT~ nyTeM npH~eRe-
rata Maxpm~ C(8) K ~op/IaHonolt uopr~azbgo~.t qbopMe. B CHCTeMe (I0) c~tenaeM
aaMeHy Yt (t) = H( t, 8) T(8) Y3 (t), l'ae H ( t, 8) = diag (H 0, Hi, H_~ . . . . . Hk0, H_ko ).
B pc3y.qbTaTC nony,~nM CnCTC~y
d ~ = F(8) Y3 + T'I(8) H-I( t, 8) F~(t, y, 8),
dt
01)
dY~
= N 2 ( t , e ) Y 2 + F 2 ( t , y , 8 ).
dt
Hoczo.rlbKy MaTprltta F ( s ~a~'I~C'I'C~l 6.rloqao-/~Harona.rrbHofl, TO CHCTeMy (1 1) MO~-
HO IICpCHHCaTb B BH~[C
dw I
dt
dt
= A l ( 8 ) w l + G l ( t , w , 8 ) ,
= A2( t , 8 ) t u 2 + G 2 ( t , w , 8 ) ,
(12)
r1~e A2( t , 8)= diag (1" I (~), N2(t , 8)), I" 3 = (to t, Y4 )7", w2 = (Y4, Y2)T to I ~ RI+2P,
to 2 nptma~ne~Kwr 6aBaxoaoMy npocTpaHc'r~y M.
B cxay npeano.no~eaaa OTHOCHTCJII,HO CO~'I'I~HHUX 3HatleaHPl MaTpHII~ l" 1 (8)
cnpaeen~HBaoUerma l e x p [ F t ( 8 ) t ] l - < N e • t > 0 , N >_ 1. Torlla
~Jut qby~aMerrram,HOll MaTpmtu L( t, s) CHerebr, a dw2/dt = A2(t , 8 )w 2 H3 Hepa-
eeHcrea (9) cne~yeT OUCHKa
[L(t,s) I ~ Nexp[-(T+8)(t-s)] , r>_s. (13)
I$SN 004 i-6053. YKp. ~tam, :~. pn., ! 999. In. 5 !. Iq~ ! 0
BHOYPKAI.[H~[ FIOJIO)KEHHfl PABHOBECH~[ B CHCTEME HE.FIHHEI~HblX ... 1345
AHa.rIOrH,mo MO)KHO rlos~y,~a~ oueaKy
]exp[A~(e) t ] l < N e x p [ - ( y + 8 ) t ] , t < 0 . (14)
HOCgO~bgy BeKXOp-~yaKttna F y~oa~eTaopaeT yc~omi~o J-lrmmrma a F (t, 0,
e ) m 0 , T O
G~(t, 0 , ~ ) = G2(t, 0, e ) = 0, (IGt(t, to, e)-G~(t,u,e)[a+
+ [Gz(t, to, e)-Ge(t,v,e)[2) u2 <- v~lto-vl, (15)
r~e
v , = 4-2v max {1, IT-'(a) H- ' ( t ,e) l} max {I, [H(t,e.)T(E)[}.
2. CymecTnonaHHe n cnoitcTna HHTerpaJ1bH~qX l~HOrOOfpasHfi.
Teope~aa 1. /7ycmb o~moAn~7omca ot~enKu (13) - (15). Tozaa npu
8
v t < (16)
N(1 + 2N)
cytqecmsyem dpymr w 2 = h ( t, w l, e ), onpeOe,~enna~ tta R z+3p+l, yc)o6.aemoo-
p~otqa:t ycAoaua~t
h ( / , 0 , g ) = 0, ]h(t, to l ,g ) -h ( t , Wl,~.) t <_ l l w l- t011 (17)
u mmca:~, umo ~mo~ecmeo S- = { (t, w l, w 2) [ t ~ R , w I ~ Rt+2P, w ~. = h (t, to r,
e), w 2 ~ M } . ~ a e m c n tmmezpaAbnbt~t ~tnozoogpa3ue~t cucme~tb~ (12). ~2n.~
ato6ozo peutemt~ w (t) = (Wl( t ) , h (t, w l(t ), e)) cucme~tbt (12), npunaO.~e~Ka~ezo
S-, cnpaoeomtoa o~en~a
Iw(t) l < 2N]w~(cy ) l exp[y (cr - t ) ] , t<_cr. (18)
]l[otcaaame~R,cmoo. Hapally c CnCTeMoii (12) pacct, mTpm, t CHCTeMy naTerpasm-
m,~x ypaBHemff~
13
to~(t) = exp [a~(e)(t - o ) ] q - I exp [A~(e) (t - s)] G~ (s, w(s), r
l
(19)
t
w2(t) = f L(t's) G2(s'w(s)'~')ds"
OTKyIla
(I t
w(t) = H+(t- fr)c - f H+(t-s)G(s ,w(s) ,E)ds + f H_(t,s) G(s,w(s),~.)ds,
l - - o a
(20)
l'Lle
H+(t )=diag[exp[a t (E) t ] ,O] , c = [ c t , 0 ] r,
H _ ( t , s ) = d i a g [ O , L ( t , s ) ] , G = [ G I , G 2 ] r
Cytuec'rBoBaHHe pemeHH~ ypaBnenn~ (20) /IOKa~<.eM C nOMO~hIO MeTO~a nocs~e-
~OBaTe.abHtaX npn6mDgeHHfl
r
w(0)(t) = 0, wr = H + ( t - o ) c - I H+(t-s) G(s'w(n)(s)'l~)ds +
I
t
+ I H( t , s ) G(s, wO~ r n = O, I, 2 . . . . . (21)
1SSN 004t-6053..YKp. ~tam. ~xypn,. 1999. m. 51. IV'-' I0
1346 H . H . KJIEBqYK
I'[o HHRyKRHH/~OKa~KeM, t r r o cupaBe/~JqHBO HepaBeHCTBO
] w (m) it) - w (m-l) (t) I <- N I cl (vl K) m-l exp [ y ( ~ - t)] , (22)
r~te m = 1,2, , t '~cr, K = 2 N / 5 .
FIpH m = 1 HepaseHcrBo (22) c~e/lyeT n3 (14). l'Iyc'r~ HepaBeUCTBO (22) cnpa-
Be~smno npn m = n. TorRa, yqr r r~saz (13) - (15), nonyqaeM
o
Iw'"+"(o - ~'"'(ol ~ f Noxpt(a- v)C,-s)lv, ~'"'(~) - ~'"-'~('1~ +
t
l
+ f N exp[(5 + V)(t-~)]v,l~")(~) - ~c"-'~(s)Jds _<
<_ NJcJ(vlK)nexp[y(r
Csle~osavesu, no, HepaBeHCTBO (22) cnpaBe~nBo npn m = n + 1, nO3TOMy OnO enpa-
Be~aBO npn Bcex HaTypanbn~X m. I-I0esm~t0Bave~,mae npn6nrg~eHnJ~ CXO~J~TC~ K
pemeHmo ypaaHennJ~ (20) npn yesIoBrm v I K < 1.
Bta6Hpaa B paBeHCTBe (20) BMeeTO ~ C ~tpyvyxo nOCTO~mly~o c ' , noslyuaeM
t
w'(t) = H+Ct-~)c' - ~ H+Ct-s) GCs, w'(s),E)ds + f H_(t,s) G(s,w" (s),~)ds.
f - - o o
Hcrio~ar~3ya (13) rt (14), o t lermz paarloc'n,
r
I~o)- ,,,'(t)l- N~xp[(a-~')(t-o)]l~-~'l + j" N~xp[(~-,t)(,-~)] v, lw(~)-
t
- w'(~)la~ + j Nexp[ (a+v) (~-0]v , lw(~)- w'(s)la~.
I l o s i o x n ~ x(t) = exp [y( t - o ) ] [w(t) - w ' ( t ) [ , ror t la
13
~ o ) -~ Nl~-~'loxpt~('-'~)l + v,N ~ exp[-~lt-sl]x(s)ds,
oaxyaa corsmcHo [7, c. 156] naxo~n~
2aNl~-c'l [462-2v,6N(t o)]. x(t) <- ~ exp
~ + ~ / 8 2 - 2 V l S N
Yqm'~Ba~ o 6 o 3 h a q e a n ~ / ~ x(t), no~yuaeM
]w(t ) _ w,(t)l <_ . ~ e x P t c y + _ ~ S 2 _ 2 ~ N I ~ - ~ ' I [ ( 4 2v,~N)(r (23)
~ + ~ / 8 2 - 2 v t f N
H o n a r a z B (19) t = ~ , naxo~r~M npel~cvaBnenae nHTcrpaYlbHOrO Mnoroo6pa3riz
WI(CY ) = c 1, h (O, Cl,~:) = iL(o , s )G2(s ,w(s ) ,e )ds .
- - e e
~OKa~eM cnpaee~tsm~iocT~ ottemc.H (17):
[h(~'c1'e)-h(~r'c~'e)l <- I N ~xpt(n+r)(~-o)]~,lw(~)- w'(~)lds ~-
~; 2vtaN21c-dl
(a+4a2- 2viaN) 2"
ISSN 0041,6053. YKp. ~lam. ~.'ypu.. 1999, m. 5 1 . 1 ~ / 0
BHOYPKAI3.Hfi i'IO.rIO;KEHH~I PABHOBECH,q B CHCTEME HF.3IHHEI;IHblX ... 1347
BEa6npaeM v I rta yCJIOBH,,q
2 v I ~ N 2 < 1
2 - 2
,~.FI~I BblIIOJIHeHH$1 3TOFO Hepar~eHcTBa /~OCTaTOqHO, qTO6hl Bblr10.r1115.tIOCb yc.rloB11e
(16). Ho Tor~a crlpaBe~.n11Ba OUeHKa (17). OuertKa (18) cJle/~yeT 113 (23), e c m I noao-
~Kr~Tb C '= 0. TeopeMa lloKa3aHa.
TeopeMa 2. l l ycmb 8bznoanmomca ycaooua (13) - (16). Toz3a cyt~ecmsyem
dpynKtCua W 1 = g ( t , w 2, r onpe3eaennaa na R p+l x M , y3oeaemeopatoucaa ycao-
eua~, g ( t , 0, e ) = 0 , I g ( t , w , e ) - g ( t , w ' , e ) I < u marax, ~mo zmoJKe-
cmeo S + = { ( t , Wl, W 2 ) [ t ~ R, w 2 ~ M , w l = g ( t , w 2 , e ) , w 1~ R/+2t '} a e a a -
emcx unmezpant, nbl~t ~moeoo6pa3ue~t cucme~tbt (12). f lax mo6oeo pemenua W ( t ) =
= (g ( t , w 2(t), e ) , w 2 (t)) cucmeztbt (12), npuna3ne~aur S +, cnpaeec3nusa
o~r I w(t)l = 2NIw2(~)l exp [ 7 ( c - t)], t_> ~.
,[~or, a3ame ,~ t , cmao aHa.norrlqHo/~oKa3aTeJibc'rBy TeopeH~ 1.
YlycTb t = r ~ HeKOTOpOe qHCaO (Haqa~mm~}i MOMeHT). rloKa~eM, qTO Hrrrer-
paYlbHOe MH0~eCTBO S- yCTO~qrIBO B TOM CMt4CJIe, qTO OHO rlp11T~ir11BaeT K ce6e Bce
6.rtrlaKHe petueHH.a w (t), t > ~, no ~KCrIOHeHUHa.nI, HOMy 3aKony.
3abieTrtM, qTO:nonez~emm petUenHt-I CrlcTeMr~t (12) ~a rtrlTerpa.ribHOM MHoroo6pa-
arm S- onacbmaeTcz ypaBHermeM
d..vv = A l (E)v + G 1 (t, v, h'(t, v, e), e). (24)
dt
TeopeMa 3. Hycmt, w ( t ) = (w l( t) , w 2 ( t ) ) ~ npouzeoatmoe petuenue cucme~u,t
(12) c tla~taabnbl~t zna~entte~t w(~) npu t = t~. Ilpu ycaoauu (16) cyucecmeyem
peutenue ~ ( t ) = ( v ( t ), h ( t, v ( t ), e)) , ae~aucee na S - u ma~:oe, qmo cnpa~eOauoa
ot(enKa
[ w ( t ) - ~ ( t ) [ <_- 2 N I w 2 ( ~ ) - h ( ( r , u ( o ) , e ) l e x p [ 7 ( ~ - t ) ] , t>>.~. (25)
]JoKazame.abcm~o. O6oaHaqm, t qepea v (t) pememte ypaBnellaJ~ (24) C aaqa.m,-
m,u,t ycaoaaeM v(t:r) = a . T o r a a ~( t ) 6y/leT aaanceTr. OT a H nMeT~, naqa.ru, Hoe
aHaneHHe ~ ( ~ ) = (a , h (r a , e)) . Btanom~aa a CHcTeHe (12) aaMeny r~epeMeHm,~x
x( t ) = w i ( t ) - v (t), y ( t )= w 2 ( / ) - h( t , v ( t ) , e), noayqaeM
dx = A ~ ( e ) x + G l ( t , rt + ~ , a ) - G l ( t , ~ , ~ ) ,
dt
(26)
d_..yy = A2(t , l~)y + G2( t , T I + ~ , 8 ) _ G 2 ( t , ~ , 8 ) ,
dt
r ae rl(t) = (x( t ) , y ( t ) ) . Oya~ur ta G(t , "q +~, e ) - G( t , ~, e.) yaOB.rleTBopJter no ne-
peMermofl rl yC~OBmO J-lanturma c nOCTOammlt v ~.
B crmy TeOpeMbl 2 crlcreMa (26) H~eeT a r r re rpa~ ,noe MHoroo6paane S +, npezt-
CTaBtlMOe B Brl~e x =g( t , y , a, e), rlIe ~y11KU~a g y~IOKneraopzeT yc:qomtaM
g ( t , o , a , e ) = o , I g ( t , y , a , e ) - g ( t , j " , a , e ) l <- ~ l y - y l . (27)
~ a z zao6oro pemeuna rl ( t) = (x ( t ) , y ( t ) ) CHcTema (26) c naqa.m,11~11 Z~aHma-
url y(r = ~, x ( ~ ) = g ( ~ , ~, a , e) , ~ a M , c n p a a e a m m o aepaBenCTBO Ir l ( t ) l <
- 2 N l y ( c ~ ) l e x p D ( ~ - t ) ] , t_> ~ .
H o ~ a ~ e H Tenep~, cymecTaoaarme TaXnx ~ a a , qTO ~.na p e m e u a a w (t) =
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1348 H, H. KJIEBqYK
= ( to l ( t ) , to2(t)) C14CTeMr~ (12) H pemenna rl(t ) = (x(t), y( t )) CnCTeMm (26) rip14
ncex t > ~ BIdlIO.rlH$1IOTCJI paBeHcTna
x( t ) = W l ( t ) - v ( t ) , y(t) = w 2 ( t ) - h ( t , v ( t ) , ~ ), (28)
oTKy/xa a 6y~eT cne~toBarb ouenxa (25).
Ecna paBeHcTBa (28) BI-drHOJIHJIIOTC.~I npn t = if, TO B CHJIy TcopeM~a e/:I14HCTBeH-
HOCTrt OHH ~tanonnJUOTC~ n npI4 Bcex t --> r Hpn t = a (28) nMeIOT Bad
g ( O , 4 , a , e ) = w l ( o ' ) - a , 4 = w2(c~)-h(c~,a,c:). (29)
SytleM paccMaTpnBaTb (29) KaK CnCTeMy ypasnen14~t OTHOCHTea-lbHO 4 14 a .
HMCeM
a = w t (o ) - g ( o , t o 2 ( O ' ) - - hCa, a , e), a , e) . (30)
I'loKa~c.eM, q r o aTO ypaaaeHrie 14Meet petuen14e npn ~I06bIX w i (O) 14 to 2(~) .
PaCCMOTp14M oxo6paz~eane H (a) = to l (~) - g ( ~ , w 2(6) - h ( a , a , e) , a , e ) . H c-
nosu, aya CaO~CTBa (27) d,bynKu1414 g, Haxo~HM ottemcy I H ( a ) - to l ( o ) [ < I w 2 ( c ) -
- h ( o , a , e ) [ / 2 , OTKyZta
i n ( a ) _ w l ( o ) l < 1 _ ~ I w2(~) - h(~, to l(c~), e)l + ~ [h(~, to ~(~), e ) - h (~, a, e)l.
fl)yHKnna h y~qBneTnopaex yc~on14m d'Ianm14tta (17), no3ror~y 143 noc~e~aHero Hepa-
~e14cT~a o n e , yeT
| I ,"-to (~') I- InCa):.tot(o)l <- ~lto, , (o)-h(o,to~(o),~:) l + ~
Pacc~0TpaM ~ (l + 2p)-MepnoM npocTpaHCT~e map H , onpez~esxaeMb~ Hepa-
BerlCT~O~ (OTHOC14Te.m, HO a ) [ a - to I (O') I < I to 2 (o ) - h (or, to I (~ e) l- H:~ 14epa-
BeH~TBa
3 IH(a) - to~(~)l <- ~ Ito~(,::r)- h(~, to ~(o). e.)l .01)
cne~ycT, ,aWO OTO6p.ay, c.en14e H (a') rlepeBo,l~14T map H B ce6a, rlO~TOMy, cornacno
TeOpeMc 13pay~pa, ~TO OTo6paz~en14e nMeeT 14enomn14a~.HylO TO'~Ky a*.
HTaX, ypaBHcnne (30) 14MeeT pemeHae a = a* , KOTOpOe m crony (31) y/IoBneT~O-
p a e r OUeHKe
3
la* - t o l ( o ) l <- ~ I to2(c0 - h ( o , t o t (o ) , e) l . (32)
H o ~ c T a ~ a pemenae a* BO BTOpOe ypaBHeH14e (29), HaXO/IrlM, HTO 14apa a*, 4 *,
r~xe 4" = to2(~) - h (~ , a*, ~), y/~OBJIeTBOpaeT c14c'reMe (29). Teope~a ~o~aaaHa.
C14cTe~y (24) nepenmuer, t ~ ~14~e
dto3 = A3(e) to 3 + G3(t, to3, to4, h(t , to3, to4, e ) , e ) ,
dt (33)
dto4 = A4(~.)to 4 + G4tt , to3,to4, h( t , w3,to4, e ) , e ) , ~
dt
FaO 1~ ----(1123, I//4), G I = (G3 , G 4 ) . B cnny npeanono~Kenalt OTHOC14Tea'IbHO
CO~:~2TBCHHhlX 3HaqCH141t MaTp14tI A 3(E) 14 A4(e ) cnpaaca;n14n~ OtleHK14
lexp[A~(e)t] l --- N e x p [ ( r - ~ ) t ] , t > 0,
(34)
[exp[A4(~)t]l < N e x p [ ( y + ~ ) t ] , t < 0.
H p a ycno~H14 (16), c o r n a c a o [8, c. 42], cymecT~yCT 14nTerpan~aoc ~taoroo6paa14e
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BHcDYPKAI_[HJ:I I'IOJ'IO)KEHH,q PABHOBECH,q B CHCTEME HE./'IHHEI~HblX ... 1349
S~" CHCTeM~a (33), KOTOpOe Mo>KeT 6UTb npe~tcTa~mno B BH~te W 3 = r ( t , w 4, ~).
Hprl 3TOM txbyHKI~H~l r (t , W, e) yaOBYieTB0p.qeT yCYIOBHZH
l
r i t , O ,E)=O , I r ( t , w , ~ ) - r i t , v , E ) l < ~ l w - v l, w E R 2p, v E R 2p.
OTClO~a c-qeayeT cymecTaoBamle tteHTpa~IbHOrO MHOroo6paarta. 06oanaqrIH r I (t ,
w, e) = h (t, r(t , w, e), w, e).
TeopeHa 4. Flycmb ebtno~n~lomcn o,enKu (13) - 0 6 ) , (34). Toeaa cytqecmsYem
~enmpanbuoe ~mozoo6pa3ue S= { (t, w 3, w 4, w 2) I t e R , w 4 ~ R 2p, w 3 = r(t ,
w4, e), w 3 ~ R t, w g = r l ( t , w4, e ), w2~ M } cucme~aa(12).
YpaBHeHne rla MHOroo6pa3rm S HMeeT BHZ~
dw4. = A4(I~)w 4 + G 4 (t, r(t, w 4, e), w4, r I (t , W4, eL e). (35)
dt
Bo MHOrHX c~yaa:ax ~ a Hccne~oBaHnJ~ 6n~ypKaurm rpnBaam, Horo pemem~a
ypaBHeHaa (35) ~OCTaTOqHO onpeae~HTb dpynKImH r (t , W, e) rI r l(t , w, e) npH-
6an~eHHO. ~ z ~TOrO Haliz~eM cHa~aza npn6~HmeHHOe B~pa~KeHHe ~yHKImn h ( G,
c I , ~) B npeano~o~eHrm, aTO Ma'rpHI~a A2(t , e ) = A2(e) He aaBnCHT OT t. IIycTb,
HanpHMep,
O
h,,(c, c~, e) = exp [ A2 (~) (G ~- s)] G2 (s, w('~ (s), E)ds ,
r~ae w OO (s) naxo~HTCa c nOMOm~IO peKyppeHTa0~ dpopMy~I,i (21). B ~aCTa'OCTa,
Hy~eB0e H nepBoe npa6~H~cenrta r~MelOT ImZ~
O
h 0 ( ~ , c l , ~ : ) = 0, h l ( ~ , c l , e ) = j" exp[A2(e) f~r-s)]G2(s ,H+(s-o)c ,e)ds .
O'rcto~a Haxo~HM Hy~eBoe H nepBoe n p a 6 ~ r ~ e H n a ~ynKtmR r(c~, w , e), r t ( ~ ,
1/3, E):
/ 0 ) (or, w , e ) = 0, ri ~~ (cr, to, ~) = 0,
r (I) (G, to, e) = - f exp [A 3 (E) (G - s)] G 3 is, 0, exp [ A 4 (e) is - G)] w,.O, e)(Is,
O
~i (t) (G, w, e) = f exp [A 2 (e) (o - s)] G 2 is, 0, exp [A~ (e) (s - O')] w, 0, e)ds.
- o o
I-IycTb ~yHKaHH G2 (t, O, w, O, E) n G 3 (t, O, w, O, e), w ~ R 2p,. aHaJmTHqHM
OTHOCHTCJ'IbHO W B HCKOTOpO~ 0KpeCTH0CTH TOqKH 1/; = 0"
G 2 ( t , O , w , O , e ) = E g a ( t ) w a, G3( t ,O,w,O, l~)= fa( t )w a,
lal=q Ictl=q
rae q > 2, a = ( a I . . . . . a2p), aj, j= 1 . . . . 2p , ~ HeoTpHttare:mHr~e Ite~r~e, [ a t =
= ( X l + . . . . . . + ff '2p' Wa= W7 i u"2p'"(X2P
I[pe;xnoJxOmHM, qTO HMr MCCTO paaJXOXCHH~
f(x (t) = t ftxkeiit' gct(t)= t g~ eikt"
k=-o. k=--
Tor~taonpeae~eHrm d~yHKI.U.I~I r O) (O, w, e) H ~0) (O, w, 8) cB0a~rca K BbI~HCJIe=
HHJtM HHTer'paJIOB Bl, I~a
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�9 1350 H.H. KJIEBqYK
;exp [l.t (~) (ff J)] exp [iks +/i(~ m (~) 4- !~,,, (E)) (s - (~)]ds, k
exp [~,(~) (~ - s)] cxp [iks + n(a m (~) +_ ip,,, (~)) (s - ~)]ds,
FAe n e N, m = l . . . . p , ~, (E), ~t ( E ) - - C06CTBeHHble 3HaqeHHa MaTpHu A2(E ) H
A 3 (e) Coo'rBeTCTBeHHO.
3aMeTaM, HTO ~JI.~ HCCJ'I(~OBaI-IH~ ycJIOBrlil 6~ypKaUHH ]IOCTaTOX.IHO orpaH~4-
,:m'rbca ~J~eHa~H SToporo nops~a s paa.noaceHm~ dpyHZtmf~ r m (~, W, 0), r: ~) (C~,
w, 0) n pa~ Tci~.nopa. Toraa npasa.a ~acT~ CHCTCM~ (35) npa ~ = 0 6y~CT ~3secTaa
c TOqHOC'rb~o ~O a.ncHOS TpcT~ero nop~za.
3. Hcc~esonauHe 6H~ypKa~UH no.ao~zeHH~ panHonecHa. C~cTe~y (35) he-
perlHllleM B BH,Re
du_.~k =
dt
dt
[~k (e) "h i~k (E)]V k + Vk(t, 1~, ~, E),
(36)
V k ~ KOMq.rleKcHaa nepcMe~maz, v = (v t . . . . . vp) T, Vk( t + 2re, v , ~, e) = r~e
=Vk(t,v, ~,e), Vk(t,v, ~,e)=O([u[ 2) npH l u l - ~ O , k = l . . . . p.
I'[yCTb BhlHOJIHaCTC~I yc.nosI~e A :
n t ~ l ( O ) + ... + npI3v(0 ) ~ m npn 0 < [ n t [ + ... + [ n v [ < 6 ,
r~te m, n I, . . . , np ~ tte:nae.
Ilpeo6paayeM CHCTCMy (36) C nOMOmbIO rlO,RCTaHOBKrl
4
V = X + "~ W k ( t , X , ~ , E ) ' (37)
k=2
r s e W 2, W 3, W4 - - qbop~u Coo'rneTCTBeHHo BTOpOrO, TpeTbero a qeTBepToro no-
pARKa C nepao]xaqeCKHMa KO~b(1)HUHeHTaMH. lIpco6pazoBaHne (37) MO~HO HOaO-
6paTh TaxHM o6paaOM, aTO ypaBHCHHZ ~.na x H ~ npHMyT BHa [9, l 0]
dxk
= [(Zk(e)+i~k(g)]X k + X k ~ a k j ( s + X k ( t , x , Z , e ) ,
dt j=l
dXk' "- [0~ k (E) - i~ k (E)] ~k + Xk ~ a--kj (s x j~ + Xk (t, x, ~ , E),
d t j=t
r~ae X t ( t + 2 r c , x , ~ , e ) = X k ( t , x , ~ , s ) , X k ( t , x , ~ , e ) = O(Ixl 5) npr~ Ixl--> 0.
IIepeia]~eM Z noJlap~I~aM zoopnHHarat4, n o ~ a r a z xk = rkcxp( i~k) , Xk =
= rkcx p (-- iq) k) . B pe3yJmTaTe noJ1y~HM cae re~y ypanaeHH~
dt j=i
d,pk = * + r %(t, r, q,, e),
dt jffil
r~Io b.~j(g) fReat j (g) , ck j (~) f lmakj (~) , Rk(t, r, ~p, ~)= O ( I r l S ) , ~ k ( t , r, ~p, ~ )=
=O(irl 5) npH Irl-*0.
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BHOYPKAIAHJ:t HOd[O~KEHH.q PABHOBECH~ B CHCTEME HEdlHHEI~HI:)I'X ... 1351
Paccr, lOTprIM 6rI~ypKattHOHHOe ypa~Henne B (8) r 2 + a ( s = 0, r~;e B (8) ~ MaT-
pmla c 3:IeMeHTaMH b k / ( 8 ) , a(s H r 2 --:- aeKTOpU C a~eMeHTamI ak(8 ) n ~2 COOT-
BeTCTBeHHO. O603naqHM qepea p (8) = ( P l , - '- , P p ) pememie ypaBaenHa B ( 0 ) r 2 +
+ -~-a (0)8 = 0 a paCCMOTpHM MaTp)my Q ( 8 ) = diag [91 . . . . . p p] B (0)
/
diag [p l
. . . . 9p].
da
TeopeMa 5. l ' lycmb d e t B ( 0 ) ~: 0 , det ~--~ (0) ~:. 0 , oce a,~e~tenmta ~eKmopa
B -~(0) ~da(o)e omput~amenbnbt, 8btno~an~emc~ ycnosue A u ~tamputla Q(8)
uerpumu~tecKa~. ToeOa cyucecmoyem unaapuanmnbt~ mop cucmeztbl (1).
YTBepz~enne c.ae~yeT Ha cymecTBoBariHa mmaprmHTHOrO Topa CHCTeMbl (36).
HI-IBaptla/-ITHbIfl T o p 6 y / l e T yC.rIOBHO yCTOflqrlBbIM. H y C T b I = 0 , B/~IIIOJ-/HaIOTCJI
yC.rIOBHJI TeopeMbl 5 rl Bce C06CTBeHHbIe 3HaqerlH~l MaTprlIlbl Q ( 8 ) HMeIOT OTpHl2a o
TeJIbHl:,Ie BeUleCTBeHHble qaCTH. Torz~a rtHBapHaHTHbnrl T o p CHCTeMbl ( 3 6 ) 6y,/IeT y c -
TO~qrlBblM, IIOaTOMy H3 TeopeMbl 3. caleayeT yCTO~'armOCT~ XaHBapHarrrnoro Topa CH-
c'reM~I (1).
"PetaeHHa Ha Tope 6y/IyT K~a3nnepHo~naecKm, m, ec.nrf I (m, 13 ( 0 ) ) + q l >
>~ ' lm[ - p - ' , [3 = (13i,--:,j13p) nprx HIe~oTopoM "~> 0, ae~oM q H aeKTope m =
= ( m I . . . . . m p ) c u e s I o q n c s l e H H ~ M r l ~zeMeHTaMH [1 1 ,C. 4 7 ] .
3a~e ,cauus t . 1. K p o M e T 0 p a M a z c m ~ a s m H O l t paaMepHOCTa MOryT c y m e c a ~ O B a T b
TaKT, Ce TOpbl Merlbt l l r lx paaMepHOCTe~, ~ X KOTOpblX r k = 0 rlpH HeKOTOpblX k .
2. Peay~,Ta-n,t pa6oTU MO~KHO o6o6ma-n, Ha CHCTeMb~ anna
02 u
O--U-U = Dj(t, 8)=-'w + A ( t , 8 ) u + B( t , 8)u a + f ( t , x , u , ua, 8 ),
Ot ~x 7 j=l
x = (x~ . . . . . xm), A = (A ~ . . . . . Am), c nepHormqecznMH yCS~OBH~H U (t , X~ . . . . . Xj +
+ 2 n . . . . ;X, , , )= u ( t , x 1 . . . . . Xm), j = 1 . . . . m.
1. Ax,~tuno8 C. A., Bopottqor M. A.. HtJanor B. I0. rellepaltlta c ' tpyx'ryp n orrrnqec~r~x cHe'rcMax c
/tl)yMeplloft o6pa'lllOll CBJ,13hlO: lla lly'rH K CO3/I.alIHIO 11r alliL/IOI'OB llel~lpOll-
m:ax ce'tefl / / Homae qbHanqecxHe np14nnnm,~ orrrHtleCKOlt o6pa6om14 m~qbopMa~mH. - M.: Hayxa,
1990. - C. 263 - 325.
2. Xenpu ]1. Feo~4e'rpr~qecKa~l "reopna no~ty~mHeltmax napa6o~mqecKr~x ypanue14ult . - M.: Mnp,
1985. - 376 c.
3. XaccupO I;.. Ka3uputtor H., Bau kl. Tcopua n npnJ~O~KCmm 6rlqbypKaRnn po~lteHn14 !IHKJIa. -- M.:
M14p, 1985. - 2 8 0 c.
4. Eenan E. 17, Jl~ur O. E. TeopeMa o tteu'rpa~a,noM MUOroo6pa~14H ne:mHeflnovo napa6oamqr
CKOrO ypam~enn:,t / / YKp. ~,mT. ~ y p m -- 1996. --48, N'-' 8. - C. 1021 - 1036.
5. KattfettKo C.A. ACHMII'ro'rHKa npocTpallCTBelillO-IleOlOlOpOlOIblX c ' rpygTyp IS KoFepeII'I:IlhlX
IICJIHIIe~IIO.-OIITHqeCKt'IX CH~'I'eMaX // )KypII. Bi~IMHCJIH'r. MR'i'C~Ma'r14KH H MaT. ~H3HKH. -- 1991. --
31, N'-' 3. - C. 467 - 473.
Pusey/~tut A. B. 0 6 aBTOKOJIe6alIHZX B II~:JIH[I~I~IIOI;I riapa6oJ~HqecKofl 3a/[aqo c rlpeo6pa3oi~a14~h~M
apvyMeirroM // TaM ;~e. - 1993. - 3 3 , N ~ 1 . - C . 69 - 8 0 .
]1anet{Kuii i0..11., Kpeiitt M.F. YCTOl~lqUltOC'rh pellIellnl~/l.nqb~pr ypaallenHfl a 6ana-
XO~OM npocTpancn~r - M.: Hay~a, 1970. - 534 c.
Ilaucc B.A. H~rrcrpa.m, nme Mno~eerBa ncpnollrltleeKHX crlcTeM /l[14qbtt~Cpr llblX ypamle-
n1411. - M,: Hayxa, t977. - 3 0 4 c.
Ca~toiineuKo A. M., i"lo.aec.~ H. 8. Po~/lenHr aHBapHaH'lllraX MIIO~KeC'rB B OKp~c'rlIOCTH no.tlo~r
m~a pam~onecna // J3,aqbqbepemt. ypaBnenHa. - 1975 . - 11, N~8. - C. 1409 - 1415.
10. 8u6uKon 1(9.'H. MuoroqacToTmae ~e~mneflmac goJle6allaa 14 )Ix 614qbypga~an. - JI.: Ha/~-no J'lr
rosin'p, y~-'~a, 1991. - 144 c.
t 1. ~oeoato6oa H. H., M~unponon~cxuii 1(9. A., Ca,~toiineutr A. M. Me'toil yc~ope)molt CXO/~14~OCTa a
lleJi1411ettllOfl Mexamlge. - Knen: Hayg. lltyMga, 1969. ' 248 c.
HOSlyqeUO 15.10.97,
noc~|r ltOpa6OTK14 ~ 11.05.98
6.
7.
8.
9.
ISSN 0041,6053. Yxp. ,~tam. :~'y. pn., 1999, m. 51, bl e 10
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| id | umjimathkievua-article-4733 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T03:04:17Z |
| publishDate | 1999 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/73/d48357874a453acc72b3cc3a98292d73.pdf |
| spelling | umjimathkievua-article-47332020-03-18T21:12:54Z Bifurcation of the state of equilibrium in the system of nonlinear parabolic equations with transformed argument Бифуркация положения равновесия в системе не линейных параболических уравнений с преобразованным аргументом Klevchuk, I. I. Клевчук, И. И. Клевчук, И. И. We consider a system of nonlinear parabolic equations with transformed argument and prove the existence of integral manifolds. We investigate the bifurcation of an invariant torus from the state of equilibrium. Розглядається система нелінійних параболічних рівнянь з перетвореним аргументом. Доведено існування інтегральних многовидів. Досліджена біфуркація інваріантного тора із стану рівновагі. Institute of Mathematics, NAS of Ukraine 1999-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4733 Ukrains’kyi Matematychnyi Zhurnal; Vol. 51 No. 10 (1999); 1342–1351 Український математичний журнал; Том 51 № 10 (1999); 1342–1351 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/4733/6167 https://umj.imath.kiev.ua/index.php/umj/article/view/4733/6168 Copyright (c) 1999 Klevchuk I. I. |
| spellingShingle | Klevchuk, I. I. Клевчук, И. И. Клевчук, И. И. Bifurcation of the state of equilibrium in the system of nonlinear parabolic equations with transformed argument |
| title | Bifurcation of the state of equilibrium in the system of nonlinear parabolic equations with transformed argument |
| title_alt | Бифуркация положения равновесия в системе не линейных параболических уравнений с преобразованным аргументом |
| title_full | Bifurcation of the state of equilibrium in the system of nonlinear parabolic equations with transformed argument |
| title_fullStr | Bifurcation of the state of equilibrium in the system of nonlinear parabolic equations with transformed argument |
| title_full_unstemmed | Bifurcation of the state of equilibrium in the system of nonlinear parabolic equations with transformed argument |
| title_short | Bifurcation of the state of equilibrium in the system of nonlinear parabolic equations with transformed argument |
| title_sort | bifurcation of the state of equilibrium in the system of nonlinear parabolic equations with transformed argument |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4733 |
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