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SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

6.4 A C ~ center

6.4 A C ~ center manifold 134 So far we have obtained a Lipschltz smooth center manifold. In this section we will prove that this manifold is actually smooth. Recall that the center manifold is obtained by first solving the fixed point equation (6.3) Theorem 6.11 Corollary 6.12 u = ~'(u, ¢) with ~(u, ¢) = T(.)¢ + lC?r~o~(U). The mapping ¢ ~ u*(¢) obtained in Theorem 6.6 is C ~. The center manifold is C k. Idea of the proof. Our basic ingredients are the smoothness of the substitution operator and contractions on scales of Banach spaces. We have freedom in choosing the exponent by which we allow solutions on the center manifold to grow exponentially. This fact we exploit carefully. Proof. Choose ~, ry, 6 and $ positive such that 0 < k~ < ~ and IIK:IIL~,,~(~ < ½ for all r/E [@,@]. (Note that []K;II depends on r/and e.) To avoid too much notation we write out the details for k = 1, 2. The proof for general k is a straightforward generalization of the case k = 2, but involves a lot of (trivial) notation, which we will save the reader. k = 1. Choose t¢ such that 0 < ~: < @. View ~;,noa as a mapping from BC-~(R;X) into BC~+"(lt; X®*). We noticed in Remark 6.7 that if in Xo, I1¢11 < ~ then )l(I-Po)u*(¢)ll < ~. Then Lemma 6.8 implies that rn, og(~) is C a in u*(¢). /C is a bounded linear operator from BC-~+~(R; X ®*) into BC~+~(R; X). We are now in the position to apply Lemma 6.10 with Yo = r = BC~(~; Z), h = Xo and Ya = BC~+~(R; X) In g(X°; BC~(R; X)) we solve Lemma 6.10 tells us that if we view its solution u0)*(¢), and u*(¢), as mappings from X ° into £:(X°; BC'~+~(R; X)), and BC ~ (R; X), respectively, then the mapping ~b --} u*(¢) is C 1 with derivative ¢ --* u0)*(¢). k = 2. We consider in £(Xo2; BC:~(R; X) the equation (6.7) uO) = T(.) + K.D?,~od(6)(u*(¢))uO) (6.8) = y~(u0), ¢). u(2) = ]CDF'n°d(~)(u*(¢))u(~) + ICD~Fm°'t($)(u*(¢))(u(~)*(¢))~ (6.9) = y2(~(~), ¢). We would like to apply directly Lemma 6.10. There are two problems. ~'~(u(1), ¢) is not continuously differentiable with respect to both u0) and ¢. This forces us to apply Lemma 6.10 to its full strength, that is using three different spaces Y0, Y, Y~. Differentiation with respect to u0) becomes continuous if we embed BC2~(R; X)in BC2~+~(R; X). Now to see that differentiation with respect to ¢ is actually continuous we observe that u*(~b) and u(X)*(¢) have arbitrarily small exponential growth rate; if we would divide everywhere ~/ by 2 we would not find different solutions to the various fixed point equations. We apply Lemma 6.10 with Yo = BC 2, y = BCa~ and Y1 = BG -a'~+'. We then meet the conditions of Lemma 6.10. O

135 Theorem 6.13 (Center Manifold) Assume that g E C k, k > 1, g(O) = O, Dg(O) = 0 and let A0 ~ 0. There exist a Ck-mapping ¢ --* C(¢) of a nelghbourhood of the origin in Xo into X and a positive constant ~ such that (i). Im(C) is locally invariant in the sense that u*(f)(t) satisfies the equation C(Po(u*(¢)(t))) = u*(¢)(t) and u*(¢) is a solution of (6.1) on the interval I = [S,T], S < o < T, provided~or t in this interval Ilu'(¢)(t)ll < ,s, dC /O~.t. (il}. Im(C) is tangent to Xo at zero: C(O) = O and ~ )~, = ¢, (iii). :Trn(C) contains all solutions of (6.1) which are defined on R and bounded above by ~ in the supremum norm. We conclude this section by stating the attraction property of the center manifold. For the proof we refer to [Bal73]. Theorem 6.14 (Attraetlon of the center manifold) For every positive constant v there exist positive constants C and ~ such that, (i). if u and v are solutions of (6.1) on the interval / = [T,0], T < O, satisfying (a) (pC+, + Po%~(0) -- (p+O. + Po%v(0), (b} for all t E I, Ilu(t)ll < 5 and Ilv(t)ll _-< 5, then IIP_°*(u(0)- v(0))ll _< ClIP_e*(~,(T) - v(T))lt e-('-+~)T. (ii). if u and v are solutions o/'(6.1) on the interval I = [0,T], T > 0, satisfying (a) (V_O* + poe*)u(O) = (p_e. + Voe,)v(O), (b) for all t E I, Ilu(t)ll _< 5 and IIv(t)ll _< 5, then liPS'(u(0) - v(O))ll < CiIPe+*(u(T) - v(T))lle-('r+-~)T. Finally we remark that if we let y(t) = PoC*(u*(¢)(t)) then y(t) satisfies the ordinary differ- ential equation in Xo fl = Ay + Peo" r®* g(C(y)). (6.10) 6.5 Parameter dependence We need to modify the theory such that we can deal with parameter dependent systems. We have in mind the FDE { :~(t) = yohd¢(#,#)x(t-O)+g(xt,#), t>_0, (6.11) x(a) = ¢(a), -h

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