Shadowing for linear systems of differential equations

Publicacions Ma emá iques, Vol 37 (1993), 245-253 . A bs ac SHADOWING FOR LINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS JERZY OMBACH * Fo a sys em o linea o dina y di e en ial equa ions wi h cons an coe icien s a simple p oo is gi en ha hype bolici y is equi alen o shadowing . 1 . The no ion o shadowing o he pseudo o bi acing p ope y (abb .POTP) usually appea s i one conside s a dynamical sys em on a compac mani old . The amous Shadowing Lemma says, ough1y speak- ing, ha hype bolici y implies he POTP . The e a e a numbe o p oo s o his esul : all o hem a he complica ed and edious . In e e y case he compac ness is essen ial . Mo imo o, howe e , in [4] conside ed his p ope y in R' o disc e e dynamical sys ems gene a ed by linea homeomo phisms . He and Kakuba i in [3] p o ed ha hype bolici y is equi alen o he POTP o such sys ems . In [5] we gi e a di e - en p oo which co e s also in ini e dimensional case . In his no e we show he analogous s a emen o sys ems o linea o dina y di e en ial equa ions wi h cons an coe icien s . A p oo ha hype bolici y implies shadowing es ablished o disc e e case in [4] may be ans o med o con- inuous case, [7], ye we gi e he e a di e en p oo which is simple and wo ks also in disc e e case . A p oo o he con e se s a emen mimics he disc e e e sion om [5] . The concep o he POTP comes om Anoso and Bowen . Fo dynamical sys ems wi h con inuous ime i was examined by F anke and Selg ade in [1] and by Thomas in pape s [8] [9] and o he s, see Thomas' pape s o mo e de ails . Fo such sys ems he common de ini ion o he POTP is as ollows . E e y S-pseudo-o bi wi h su icien ly small S > 0 can be a bi a ily close uni o mly app oxima ed by a ue o bi a e some epa ame iza ion o ime on he ue o bi . Wha we a e going o show is ha o a sys em o di e en ial equa ions *Suppo ed by Polish scien i ic g an RP . 1 . 1 0 . 246  J . OMBACH o he o m x' =A - x, A is n x n ma ix, he POTP is equi alen o hype bolici y o he sys em, see below o a de ini ion . Besides, we show ha he POTP can be eplaced by condi ions which a e sligh ly di e en om he o iginal de ini ion o he POTP, ye he abo e equi alen e will s ill hold ue . 2 . Le (X, d) be a me ic space and 0 : X x R --> X be a low Le . 0 is con inuous, O(x, 0) = x, O(O(x, ), s) = O(x, + s) o e e y x E X, , SE R . An o bi o a poin x E X is a se {O(x, ) : E R} . Le T > 0 and 6 > 0 . A pai o sequences ({x°°-__oo},_oo}, { °°__ .}), x c E X, n E R, n > T, o all n E Z, is said e be a (6, T)-pseudo-o bi i o allnEZ ( 1 )  d(O(xn, n), xn+l) < 6 . Fo a gi en (6, T)-pseudo-o bi we deno e by xo * he poin which is uni s om xo along he pseudo-o bi . Mo e p ecisely, xo *  O(xn, - Sn),  o S n -< < Sn+l,  whe e > 0, =  - O(x ., + Sn),  o - S n < <- Sn+l,  whe e < 0, whe e s o = 0, S n = E?ó i, o a = 1, 2, 3 . . . , Sn = L i ln i, o a = -l, -2, -3,  . . . A (S, - )-pseudo-o bi is c- aced, E > 0 is gi en, by he o bi o a poin x i he e is a epa ame iza ion o ime Le . a a inc easing homeonlo phism h : R --> R, h(0) = 0, such ha (2)  d(O(x, h( )), xo * ) < E, o all E R . We say ha he low 0 has he pseudo-o bi s acing p ope y wi h e- spec o (POTP(T)) i o e e y E > 0 he e exis s S > 0 such ha any (S, T)-pseudo-o bi is E- aced by some o bi . The low has he POTP i i has he POTP(T) o all T > 0 . The abo e de ini ion was es ablished by Ranke and Selg ade and hen used by many au ho s, see o example [6], [8] and e e en es he e in . Ye , o lows o a Rn we will see ha his de ini ion may be weakened o s eng hen in a ious ways and he new condi ions such ob ained a e s ill equi alen o he o iginal de ini ion o he POTP . We say ha he low has he s ong POTP(- ) (SPOTP(T)) i in he abo e de ini ion o he POTP(T) we ake h( ) = o all E R . We say ha he low has he no mal POTP(T) (NPOTP(T)) i in he abo e de ini ion o he POTP(T) we may es ic ou sel es o (6, T)-pseudo- o bi s ha ing all n = T . We say ha he low has he NSPOTP(T) i SHADOWING FOR LINEAR SYSTEMS  247 i has bo h SPOTP(T) and NPOTP( , ) . A las , he low has SPOTP, NPOTP o NSPOTP i i has he co esponding p ope y wi h any T > 0 . A semi-o bi o a poin x E X is a se {O(x, ) : >_ 0} . A semi-(S, T)- pseudo-o bi is a pai o sequences ({x,ñ o }, { ñ °_ o }), xn E X, n>_ T, such ha (1) holds o all nE N . Now, any o he abo e de ini ion may be e o mula ed in e ms o semi-o bi s and semi-pseudo-o bi s . Co - esponding concep s hus ob ained will be deno ed by semi-POTP(T), semi-POTP, e c . Le us no e ha any semi-(S, T)-pseudo-o bi may be ex ended o a (S, T)-pseudo-o bi by pu ing xn = O(xo, nT) and n = T o all n = -1, -2, -3, . . . . Hence any mu a ion o he POTP de ined abo e implies he co esponding semi-p ope y . Le us no e ha in he de ini ion o he POTP(- ) (and in all o he de ini ions) we may assume ha o all n E Z we ha e n <_ 2- . In ac , i we ha e n > 2- o some n hen he e is k > 2 such ha kT _< n < (k + 1)T . We modi y he (S, T)-pseudo-o bi by inse ing be ween poin s xnand xn+1 poin s xni = W(xn, iT), whe e i = 0, . . . . k-1 and by pu ing numbe s ni = T o i = 0, . .', k-2, n (k_ 1 ) = n-(k-1)T in place o n .  I is clea ha a e such modi ica ions he new (S, T)- pseudo-o bi shows he same xo * o all E R bu now all n < 2T . We also ema k he e ha he abo e de ini ions do no depend on a pa icula me ic used bu a he en he uni o m s uc ü e on he space X . We conside a sys em o o dina y linea di e en ial equa ions wi h cons an coe icien s x'=A-x and i s low O(x, ) = exp( A) - x, whe e A is ce ain n xn ma ix . The sys em (o i s low) is said o be hype bolic i all eigen alues o he ma ix A ha e non-ze o eal pa s . Ou main esul s a e es ablished in he ollowing wo p oposi ions and, in mo e comple e o m, as he heo em . P oposi ion l . I sys em (3) is hype bolic, hen i s low has he SPOTP . P oposi ion 2 . I he ow o sys em (3) has he semi-NPOTP(T) o some T > 0, hen he sys em is hype bolic . Theo em . Fo sys em (3) all de ini ions o he a ious ypes o he pseudo-o bi s acing p ope y s a ed abo e a e equi alen o each o he andany o hem is equi alen o hype bolici y . 24 8  J . OMBACH The p oo s o he p oposi ions will be p esen ed in he nex sec ion . The p oposi ions easily imply he heo em, and a p oo o he heo em is shown a he ollowing igu e . Le T > 0 be ixed . Then, all implica ions poin ed ou a he diag am a e ob ious . 3 . In o de o p o e P oposi ions 1 and 2 we will need he h ee ol- lowing lemmas . The i s wo Na e s aigh o wa d p oo s . SHADOWING FOR LINEAR SYSTEMS  24 9 Lemma 1 . Le (Xi, di), i = 1, 2 be me ic spaces and Wi lows on X i . Le X = Xl x X2 be equipped wi h a me ic compa ible wi h he uni o m p oduc s uc u e . Le <P be he p oduc low on X i .e . 0((xl, x2), ) _ (01(x1, ), 02(x2, )) . (i) I 01 and 02 ha e he SPOTP(- ) hen <P does . (ii) I 0 has he semi-NPOTP(T) hen bo h 01 and 02 do . Lemma 2 . Le 0 be a low on a me ic space X sa is ying he ollow- ing condi ion : Fo e e y T > 0 and E > 0 he e exis s 6 > 0 such ha : d(x, y) < S and ¡ i < T imply d(O(x, ), O(y, )) < e . (This condi ion is sa is ied by he ow o sys em (3)) . Then, i 0 has he SPOTP(T) hen he e e se ow 0, O(x, ) = O(x, - ), has he same p ope y . Lemma 3 . Le 0 be a ow on Rn . I 0 has he semi-SPOTP(- ) hen i has he SPOTP(T) . P oo . Le 6 > 0 be chosen o a gi en e > 0 by he semi-SPOTP(T) . Le ({x°° { °°__ .}) be a (6, T)-pseudo-o bi . Then o each k E N ({x°° _k}, { n°__k} is a semi-(6, T)-pseudo-o bi s a ing om he poin x_ k . So, he e exis~`s 15oin s y_k E Rn such ha d(O(y_k, ), x_k * ) < c, o all > 0 . I ollows ha he poin s zk = O(y-k, s_k) belong o he ball B(xo, e) . By he compac ness o he closed ball we ge a poin x E Rn and a sequence k i -j oo such ha zk i --> x . This poin e- aces he abo e (6, T)-pseudo- o bi . Fo i , ix E R and conside such kis ha -S-k¡ <_ (i is so o almos all kis because i > 7') . We ha e : d(O(zk i, ), xo * ) = d(O(O(y-ki, s-kj, ), xo * ) = Le ing ki ---> oo we ha e d(O(x, ), xo * ) < e . = d(O(y-ki , s-k ¡ + ), x-ki * (s-k¡ + )) _< e . P oo o P oposi ion 1 : Fix T > 0 . Fi s we show ha any sys em o he o m (3) wi h a ma ix A which all eigen alues ha e nega i e eal pa s sha es he semi-SPOTP(- ) . Then, by Lemma 3 such a sys em has he SPOTP( , ) . Hence, any sys em o he o m (3) wi h a ma ix A which all eigen alues ha e posi i e eal pa s has, by Lemma 2, he SPOTP (T) . Now, any hype bolic sys em is, by he Jo dan Decomposi ion, a p oduc o wo sys ems ; one ha ing all eigen alues wi h posi i e and he o he wi h nega i e eal pa s . Lemma 1(i) will comple e he p oo . 250  J . OMBACH So we assume ha all eigen alues o he ma ix A ha e nega i e eal pa s . I is known, see o example [2], ha he e a e a no m on Rn and a cons an c > 0 such ha Fix e > 0 and le 5 > 0 be small enough o be de e mined la e . Le ({x'_ o }, { ,~ o }) be a semi-(b, T)-pseudo-o bi . We show ha his semi- o bi is e'- aced by he poin xo . Recall ha so = 0, s n = Ez ~ 2 o n=1,2,3,  . Fi s we ha e : II0(x0, Sn+1) - Hence by induc ion 11 exp( A) - xii < e - ` - llxil, o all > 0 and x E R n . xn+lll :~ II4'(x0,sn+1)-O(xn, n)II+II0(xny n)-xn+1ll :~ < 110(x0, sn), n) - W(xn, n) II + b <- 11 exp( nA) - (4'(x0, Sn) - xn)11 + b < < e-CT 110(xo, Sn) - xnll + S . 110(X0, sn) - xn l l o any e', i b is su lcien ly small . Fix > 0 . The e exis s n E N wi h s n < < Sn+l . By he ema k ha we made a e he de ini ions o he POTP we may assume : 8 14 _1 - s n <- 27 - . We ha e : II0(x0, ) - xo * i¡ = 110(x0, Sn), - Sn) - `V(xn, - Sn)11 = _ 11 exp( - sn)A - (exp(s nA) - xo - xn)11 < e ( -sn)IIAII - II0(x0, Sn) - xnll < e2-IIAII . E ' < e o small e' > 0 . The p oo is comple e . B= (000 . . . 0 010 . . . 0 001 . . . 0 ,B=C,B= b e -C% , < -  < É - 1 __ e-C7- P oo o P oposi ion 2 : Assume ha he low o sys em (3) has he semi-NPOTP(- ) wi h some posi i e . Assume ha he ma ix A is no hype bolic and le A = i~3, ~3 E R, be an eigen alue o A wi h ze o eal pa . Le B be he eal Jo dan block o A co esponding o A . By Lemma 1( i) he low o he sys em x' = B - x has he NPOTP(- ) . Ma ix B may ha e one o he ollowing o ms : B = 0, (CO . . . 0 ICO . . . 0 OIC . . . 0 ~o . . . 010/  ~o . . . OIC) whe e exp( B) = Hence SHADOWING FOR LINEAR SYSTEMS  251 C=(~  0), andl=(0 1 o) Ou goal is o cons uc a semi-(5, T)-pseudo-o bi , wi h ,, = T, o any small S which is no aced by any poin . I will comple e he p oo . In ac , we cons uc such a semi-(S, T)-pseudo-o bi in all he abo e cases ye a p oo is p esen ed only o he las , mo e in e es ing one . In he case B= 0 he low ac s on some one-dimensional subspace which can be iden i ied wi h he eal axis R . We de ine a semi-(S, T)- pseudo-o bi by xn = n5, n = T o n E N . In he second case he low ac s on ce ain k-dimensional subspace, say Rk . We de ine a semi-(S, T)- pseudo-o bi as xo = 0 E R k and He e, a( ¡ ) deno es he i- h coo dina e o a ec o a om R k . In he hi d case he low ac s on a plane, say R 2 . We de ine a semi-(S, T)-pseudo- o bi by  ' xn _ - (nS - cos(nT0) ) '  n = 7-- n6 - sin(nT0) In he las case he low ac s on some 2k-dimensional subspace, say R2k . A semi- (5, T)-pseudo-o bi is de ined by : xo = 0 E R 2 k, To see i is in ac a semi- (S, T)-pseudo-o bi no e ha : (n + 1)8  cos(n + 1)- o (n + 1)S  sin(n + 1), O O(xn)(3)  ,  n = T . 0 R / (n + 1)8 O( x n)( 2 ) ) cos /O  - sin 43 whe e R = ( sin ~3  cos ~3 (exp( B) . x)(1)  =R .  x(1) ( (exp( B)  x) (2) )  (x(2)) ' 25 2  J . OMBACH Hence d(O(xn, n), x . .+1) = II W(xn, T) - X .+1 11 = - l~R  «x .) (1) x n)( 2 )  -  (xn+1)(2))  b .  ( sm(n + 1), ~3) This semi-(S, T)-pseudo-o bi is no aced by any poin o R2k . In ac , gi en x E R 2k and an inc easing homeomo phism h : R --> R, h(0) = 0, we ha e : d(O(x, h(n- )), xo * nT) = 11 exp(h(nT)B) - x - xn 11 > (exp(h(n- )B) ' x)(1)  (xn+1)(1)  _  x(1)  -, (exp(h(n- )B) ' x)(2)) - ( ( x n+I)(2))  >  x(2))  -nS as n -> oo . One could ask why we did no use he complexi ica ion me hod as i was done in [5] o he disc e e case . The eason is ha he complex- i ica ion me hod would equi e ha he con e se s a emen o ha in Lemma 1(ii) holds ue and his is no so ob ious . Re e en es 1 .  FRANKE, J . AND SELGRADE, J ., Hype bolici y and chain ecu - ence, Jou nal o Di e en ial Equa ions 26 (1977), 27-36 . 2 . HIRSCH, M . AND SMALE, S ., "Di e en ial equa ions, dynamical sys ems and linea algeb a," Academic P ess, New Yo k, 1974 . 3 .  KAKUBARI, S ., A no e on a linea homeomo phism on Rn wi h he pseudo o bi acing p ope y, Sci . Rep . Niiga a Uni . Se . A 23 (1987), 35-37 . 4 .  MORIMOTO, A ., Mani ols and Lie g oups, P og ess in Ma h . 14 (1981),283-299 . 5 .  OMBACH, J ., The Shadowing Lemma in he linea case, Ac a Ma h- ema ica, Uni e si a is Jagellonicae . 6 .  OMBACH, J ., Sinks, sou ces and saddles o expansi e lows wi h he pseudo-o bi s acing p ope y, Annales Polonici Ma hema ici 53 (1991), 238-252 . 7 .  OMBACH, J ., The pseudo-o bi s acing p ope y o linea sys ems o di e en ial eqúa ions, Glasnik Ma ema icki 27(47) (1992), 49-56 . 8 .  THOMAS, R ., S abili y p ope ies o one-pa ame e lows, P oc . London Ma h . Soc . 45 (1982), 479-505 . SHADOWING FOR LINEAR SYSTEMS  253 9 .  THOMAS, R ., Topological s abili y : some undamen al p ope ies, Jou nal o Di e en ial Equa ions 59 (1985), 103-122 . Ins y u Ma ema yki Uniwe sy e Jagiello iski ul Reymon a 4 * 30 059 K aków POLAND Rebu el 17 de Gene de 1992