Journal of Dynamics and Differential Equations, Vol. 14, No. 3, July 2002 (© 2002)

Small Delay Inertial Manifolds Under Numerics: A Numerical Structural Stability Result 1 Gyula Farkas 2 , 3 Received July 16, 2001 In this paper we formulate a numerical structural stability result for delay equations with small delay under Euler discretization. The main ingredients of our approach are the existence and smoothness of small delay inertial manifolds, the C 1-closeness of the small delay inertial manifolds and their numerical approximation and M.-C. Li’s recent result on numerical structural stability of ordinary differential equations under the Euler method. KEY WORDS: Delay equations; small delay inertial manifolds; smoothness; C 1-estimates; Euler method; numerical structural stability.

1. INTRODUCTION In recent years, there has been an increased interest in understanding the behavior of numerical discretizations of differential equations. One key problem is to determine how well the dynamics of the underlying equation is captured by the discretization, see, e.g., [11, 14, 23, 24, 26]. It is well known that conjugacies play a fundamental role in the qualitative theory of ordinary differential equations. Indeed, when a conjugacy exists between two dynamical systems then the dynamical systems have the same orbit structure, they are qualitatively the same. We want to claim that under certain conditions the dynamics of the discretization considered as a discrete dynamical system and of the original system are the same. Thus it is natural to seek for conjugacies between the dynamical system and its 1

Posthumous publication. Department of Mathematic, Istva´n Széchenyi University of Applied Sciences, H-9026 Gyo˝r, Héderva´ri u. 3., Hungary. 3 Correspondence should be addressed to George R. Sell, University of Minnesota, School of Mathematics, Minneapolis, Minnesota 55455. E-mail: [email protected] 2

549 1040-7294/02/0700-0549/0 © 2002 Plenum Publishing Corporation

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numerical approximation. This yields us the concept of numerical structural stability. Numerical structural stability results for ordinary differential equations can be found in [10, 19, 20]. Conjugacies can also be constructed near a fold bifurcation point, see [8]. However, when one wants to prove similar results for infinite dimensional dynamical systems one has to face several difficulties. First of all, in general, structural stability fails even for the continuous problem. (A simple example showing that there is no Hartman–Grobman theorem for delay equations on a neighborhood of a hyperbolic equilibrium point can be found in [7].) This fact shows that structural stability is not the best dynamical concept to seek under perturbation of infinite dimensional systems. Instead, one should look for ‘‘points of continuity’’—in the sense of dynamics—for the perturbations, see [23, 24]. On the other hand, in general (esp. for delay equations), there are no error bounds between the continuous dynamical system and its discretization on a fixed neighborhood of the phase space. This makes standard perturbation results inapplicable. The usual method to overcome these difficulties is to reduce the original problem to a finite dimensional invariant manifold and apply existing results for ordinary differential equations. The aim of the present paper is to show that such a process works for delay equations with sufficiently small delay. The outline of our proof is as follows. First we construct exponentially attractive n-dimensional invariant manifolds (small delay inertial manifolds) of class C 2. (The construction is based merely on the method of [5].) The C 2 norms of these manifolds tend to zero as the time delay goes to zero. (This allows us to obtain structural stability with respect to delay.) Secondly (via the same construction) we prove that the small delay inertial manifolds are well approximated under the Euler method. Finally we apply a recent result of [20], where a numerical structural stability result was proved for C 2 dynamical systems under the Euler discretization. Let us note here that instead of the Euler method we could also treat any one-step numerical scheme with properties (H0)h –(H3)h and (i)–(vi) from Lemma 8, see Section 5. The reasons for dealing with the Euler method are its simplicity and its coarseness, i.e., it is plausible that if numerical structural stability holds for the Euler method then it holds for more accurate (higher order) methods as well. We note that for all sufficiently small fixed delay time the existence of small delay inertial manifolds would follow by applying the abstract result of [3]. The use of our own construction is twofold. First, we need (at least) C 2 smoothness in order to apply the main result of [20]. Secondly, we have

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to control the C 2 norms of the constructed manifolds as well. (It is also true, see Theorem 2 later, that the rate of the exponential attractivity can be chosen independently of the delay time.) Recently in [2] a numerical structural stability result was proved for scalar parabolic partial differential equations under spatial discretization. Roughly speaking their method is to construct a family of inertial manifolds on which the errors tend to zero in the C 1 norm. To the contrary the dimension is fixed in our construction. It is not clear how one can prove numerical structural stability on a fixed inertial manifold since (at least) C 2 smoothness is crucial in deriving error estimates, but inertial manifold may lose smoothness, see [6]. Our result says that when the delay is small the delay equation is close to the ‘‘limiting ODE,’’ i.e., when the delay time is zero. To the best of our knowledge this question was first studied in [17], see also [18]. A similar result was proved in [22] under the condition that the function acting on the delayed argument is small and has small Lipschitz constant. In this paper we do not assume this smallness condition. Small delay inertial manifolds were used in [21] to study the behavior of the attractor of the sunflower equation with small delay. Similarly, in [1] inertial manifolds were constructed for retarded semi-linear parabolic partial differential equations. In order to prove the existence of the inertial manifold the delay time must satisfy some sort of smallness condition, see Theorem 3.1 in [1]. The results can be applied to ordinary functional differential equations as well. They also studied the continuity properties of the inertial manifolds with respect to the delay time. Although smoothness is very important in applications neither [21] nor [1] contain smoothness results. We admit that our result works only for delay equations close to ODE’s. It is known that even scalar delay equations may possess very rich dynamical behavior (when the delay time is sufficiently large). In recent works, see [15, 16], the complete characterization of the attractor of a scalar delay equation was presented. (Unfortunately, we cannot obtain such a complete description of the attractor of the discretization, the results in [9] show lower semi-continuous convergence of the approximating attractors to the true one. Upper semi- continuity of the Morse decomposition of the attractor of a scalar delay equation under a ‘‘spatial-like’’ discretization, i.e., the delay equation was approximated by a family of ODE’s with increasing dimension, was proved in [12]. The results of [9] work for the Euler method, which is a ‘‘full’’ discretization, i.e., the delay equation is approximated by a family of finite dimensional mappings with increasing dimension.) Although our result says something for systems in arbitrary dimension (provided the delay is small enough) for higher dimensional systems very little is known, see, e.g., [4].

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The rest of the paper is as follows. After a preliminary section, in Section 3 we prove the existence and smoothness of small delay inertial manifolds. In Section 4 we study the existence of asymptotic phases. We show that the small delay inertial manifolds are well approximated under the Euler method in Section 5. Finally, we show the desired numerical structural stability in Section 6.

2. PRELIMINARIES Consider the following delay differential equation x˙(t)=Ax(t)+f(x(t))+g(x(t − e)),

(1)

where e > 0, A ¥ R n × n and f, g ¥ C 2(R n, R n) are bounded functions with bounded derivatives. Our standard reference is [13]. In what follows C(B, E2 ), C 1(B, E2 ), resp. C 2(B, E2 ) denote the Banach spaces of bounded continuous, C 1, resp. C 2 functions with bounded derivatives between the closed subset B of a Banach space E1 and the Banach space E2 equipped with the usual sup, C 1, resp. C 2 norm. Set Ce :=C([ − e, 0], R n) endowed with the sup norm || · ||. Denote the C0 -semigroup on Ce generated by the linear ordinary differential equation x˙(t)=Ax(t) by {Te (t)}t \ 0 . Decompose Ce by s(A) (the spectrum of A) as Ce =Pe À Qe , where Pe =pCe , Qe =(id − p) Ce , and p is defined as (pf)(h) := e Ahf(0), f ¥ Ce , h ¥ [ − e, 0]. Observe that there exists a positive constant M independent of e such that ||p|| [ M and ||(id − p)|| [ M. Thus we have (H0) for all e > 0 there is a Te -invariant splitting Ce =Pe À Qe and a positive constant M independent of e such that the norms of the associated projections p: Ce Q Pe , id − p: Ce Q Qe are bounded by M. Moreover, subspaces Pe have e-independent finite dimension. Note that [(id − p) f](0)=0 for all f ¥ Ce and thus Te (t)(id − p) f=0 for all t \ e. We get that for all b > 0 and for all f ¥ Ce ||Te (t)(id − p) f|| [ Me bee −bt ||f||,

t \ 0.

Set eb =b1 . Thus we have (H1) for all b > 0 there exists eb > 0 such that for all 0 < e [ eb ||Te (t)(id − p) f|| [ 3Me −bt ||f||,

t \ 0.

Small Delay Inertial Manifolds Under Numerics

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Set Fe (f) :=f(f(0))+g(f(−e)), f ¥ Ce . Then (H2) Fe ¥ C 2(Ce , R n) is bounded with bounded derivatives. Moreover, the bounds are independent of e, i.e., ||Fe ||C 2 [ K for all e > 0. Finally, (H3) there exists an w > 0 independent of e such that for all e > 0 ||Te (t) pf|| [ Me w |t| ||f||,

t ¥ R.

These properties will be frequently used in later sections. Finally, the Banach space of bounded linear, resp. bilinear operators between Banach spaces E1 , E2 (endowed with the induced operator norm) will be denoted by BL(E1 , E2 ), resp. BL 2(E1 , E2 ). 3. EXISTENCE AND SMOOTHNESS OF SMALL DELAY INERTIAL MANIFOLDS The main result of this section is the following theorem. Theorem 1. For all e > 0 small enough there exists a C 2-function Fe : Pe Q Qe such that graph(Fe )={f+Fe (f) : f ¥ Pe } is an exponentially attractive invariant manifold for (1). Moreover, ||Fe ||C 2(B, Qe ) Q 0 as e Q 0+ for all closed bounded set B … Pe . Before we turn to the proof of Theorem 1, which is the content of the following subsections, we make some remarks. Remark 1. Our proof works for equations of the form x˙(t)=Le xt +Fe (xt ) possessing properties (H0)–(H3), where Te is the C0 semigroup generated by x˙(t)=Le xt . Remark 2. Note that the reason for the nonexistence of C 2 inertial manifolds for parabolic differential equations is that the stronger gap condition required for the C 2-property is not generally satisfied by such equations. However, for our problem this stronger gap condition can be achieved by choosing sufficiently small time-delay. Thus it is reasonable to expect that if the nonlinearity is smooth then the small delay inertial manifolds will be smoother and smoother as the time-delay goes to zero. This is in fact the case. Details will be published elsewhere.

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Remark 3. For the largest value of e* for which the theorem is still true can be estimated as e* > (72NMK) −1, see the proof below. 3.1. Existence Choose a natural number N ¥ N such that w < NMK holds. Set b > 72NMK, a :=8NMK, d0 :=NMK, w1 :=a − w > 0, w2 :=b − a > 0 and e ¥ [0, eb ]. The fundamental matrix solution X of x˙(t)=Ax(t)

(2)

on Ce is defined to be the (unique) matrix solution of (2) with initial value X0 at zero, where X0 is the n × n matrix function on [ − e, 0] defined by X0 (h)=0 for − e [ h < 0 and X0 (0)=I. Let X P0 e =e A · and X Q0 e =X0 − X P0 e . Then we have the following exponential estimates ||e atTe (t) pf|| [ Me w1 t ||f||

t [ 0,

||e atTe (t)(id − p) f|| [ 3Me −w2 t ||f|| ||e atTe (t) X P0 e || [ Me w1 t

t \ 0,

t [ 0,

and ||e atTe (t) X Q0 e || [ 3Me −w2 t

t \ 0.

Define the Banach space Sg :={Y : Y: R − Q Ce is continuous and sup e gt ||Y(t)|| < .} t ¥ R−

with norm |Y|g :=sup e gt ||Y(t)||. t ¥ R−

Fix an arbitrary f ¥ Pe and for Y ¥ Sg we define t

Te (Y)(t) :=Te (t) f+F Te (t − s) X P0 e Fe (Y(s)) ds 0

+F

t

−.

Te (t − s) X Q0 e Fe (Y(s)) ds,

t [ 0.

Lemma 1. For all f ¥ Pe and d ¥ [0, d0 ] operator Te maps Sa − d into itself and is a uniform 1/3- contraction.

Small Delay Inertial Manifolds Under Numerics

555

Proof of Lemma 1. Let f ¥ Pe and Y ¥ Sa − d be given. Then t

||e (a − d) tTe (Y)(t)|| [ Me (w1 − d) t ||f||+MK F e (w1 − d)(t − s)e (a − d) s ds 0

+3MK F

t

e −(w2 +d)(t − s)e (a − d) s ds

−.

which shows that |Te (Y)|a − d < .. Let Y1 , Y2 ¥ Sa − d be given. Then t

||e (a − d) t(Te (Y1 ) − Te (Y2 ))(t)|| [ MK F e (w1 − d)(t − s)e (a − d) s ||Y1 (s) − Y2 (s)|| ds 0

+3MK F

t

−.

[ MK

e −(w2 +d)(t − s)e (a − d) s ||Y1 (s) − Y2 (s)|| ds

1 w 1− d+w 3+d 2 |Y − Y | 1

1

2 a−d

2

which implies that |Te (Y1 ) − Te (Y2 )|a − d [ MK

1 w 1− d+w 3+d 2 |Y − Y | 1

1

2 a−d

.

2

Note that w1 − d \ 6NMK and w2 +d \ 18NMK whenever d ¥ [0, d0 ], and the desired contraction property follows. i Denote the fixed point of Te by Y de (f). Since Sa − d … Sa we have by uniqueness that the Y de (f)=Y 0e (f). Set Ye (f)=Y 0e (f). Now we can define a mapping Fe : Pe Q Qe by Fe (f) :=[(id − p) Ye (f)](0). This mapping defines our small delay inertial manifold. In what follows we prove that Fe ¥ C 2(Pe , Qe ). In order to do so we prove that Ye ¥ C 2(Pe , S2a ). Choose an arbitrary bounded closed ball B … Pe . Redefine operator Te on the space C(B, Sa − d ) by setting for Y ¥ C(B, Sa − d ), f ¥ B t

(Te (Y))(f, t) :=Te (t) f+F Te (t − s) X P0 e Fe (Y(f, t)) ds 0

+F

t

−.

Te (t − s) X Q0 e Fe (Y(f, t)) ds,

t [ 0.

The proof of the following lemma goes along the same line as the proof of Lemma 1 and thus it is omitted. (The extra continuity of Te (Y) can be seen from the definition.)

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Lemma 2. For all d ¥ [0, d0 ] operator Te maps C(B, Sa − d ) into itself and is a uniform 1/3 contraction. Moreover, the fixed points are independent of d, and equal to Ye | B . Thus we obtain that our manifold is continuous. Moreover, sup ||Fe (f)||=sup f¥B

f¥B

>F

0

−.

[ 3MK F

0

−.

Te (−s) X Q0 e Fe (Ye (f, s)) ds

>

e bs ds=3MKeb Q 0.

3.2. Smoothness Choose an arbitrary sequence d0 > d1 > d2 > · · · > 0. With the help of suitably chosen sequences we prove that the C 1 property is preserved under a certain loss of exponential weights when we apply Te . Similar result holds for the second derivative, see Lemma 4 later. When we apply these results in the proof of Lemma 7 we have to be able to control the exponential weights. This motivates the introduction of some constant 0 < D < d0 after the proof of Lemma 4. Lemma 3. If Y ¥ C 1(Pe , Sa − di ) then Te (Y) ¥ C 1(Pe , Sa − di+1 ), for i= 0, 1,... . Proof of Lemma 3. Let Y ¥ C 1(Pe , Sa − di ) be fixed. Then DY ¥ C(Pe , BL(Pe , Sa − di )), where DY denotes the Fréchet derivative of Y. Differentiate formally Te (Y) to obtain (DTe (DY) · [k])(t, f) t

:=Te (t) k+F Te (t − s) X P0 e DFe (Y(f, s)) · [DY(f) · [k]](s) ds 0

+F

t

−.

Te (t − s) X Q0 e DFe (Y(f, s)) · [DY(f) · [k]](s) ds.

We claim that D(Te (Y))=DTe (DY). Let f1 , f2 ¥ Pe be given. Define I :=||e (a − di+1 ) t(Te (Y)(f1 , t) − Te (Y)(f2 , t) − DTe (DY) · [f1 − f2 ](f2 , t))||.

Small Delay Inertial Manifolds Under Numerics

557

Since DTe (DY) is linear and continuous in k ¥ Pe it suffices to show that sup I=o(||f1 − f2 ||)

t ¥ R−

as ||f1 − f2 || Q 0. To this end let g > 0 be given and write I [ I1 +I2 where

>

1

t

I1 = e (a − di+1 ) t F Te (t − s) X P0 e (Fe (Y(f1 , s)) − Fe (Y(f2 , s)) 0

− DFe (Y(f2 , s)) · [DY(f2 ) · [f1 − f2 ]](s)) ds

2>

and

>

1

I2 = e (a − di+1 ) t F

t

−.

Te (t − s) X Q0 e (Fe (Y(f1 , s)) − Fe (Y(f2 , s))

− DFe (Y(f2 , s)) · [DY(f2 ) · [f1 − f2 ]](s)) ds

2>.

We prove that I1 =o(||f1 − f2 ||) as ||f1 − f2 || Q 0. Choose T < 0 so that 2MK ||Y||C 1 (di − di+1 ) T e < g/2. w1 − di+1 There are two cases. Case t \ T. Write

>

t

I1 = e (a − di+1 ) t F Te (t − s) X P0 e

1F

0

1

0

DFe (uY(f1 , s)+(1 − u) Y(f2 , s)

2

· [Y(f1 , s), Y(f2 , s)] du − DFe (Y(f2 , s)) · [DY(f2 ) · [f1 − f2 ](s)) ds

>

t

[ e a − di+1 ) t F Te (t − s) X P0 e

1F

0

1

0

(DFe (uY(f1 , s)+(1 − u) Y(f2 , s)

− DFe (Y(f2 , s))) · [DY(f2 ) · [f1 − f2 ]](s) du) ds

>

t

+ e (a − di+1 ) t F Te (t − s) X P0 e 0

1F

1

0

2

>

DFe (uY(f1 , s)+(1 − u) Y(f2 , s)

· [Y(f1 , s) − Y(f2 , s) − DY(f2 ) · [f1 − f2 ](s)] du) ds

>

2

>

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Farkas

Now we choose o > 0 such that if ||f1 − f2 || < o then ||DFe (uY(f1 , s)+(1 − u) Y(f2 , s)) − DFe (f2 , s))|| [

g(w1 − di+1 ) 2M ||Y||C 1

for all u ¥ [0, 1] and s ¥ [T, 0], and sup {e (a − di ) s ||Y(f1 , s) − Y(f2 , s) − DY(f2 ) · [f1 − f2 ](s)||}

s ¥ R−

[

g(w1 − di+1 ) ||f1 − f2 || 2MK

hold. It is easy to see that in this case I1 [ g ||f1 − f2 ||. Case t < T. Write I1 =I 11 +I 21 where

>

t

I 11 = e (a − d2 ) t F Te (t − s) X P0 e (Fe (Y(f1 , s)) − Fe (Y(f2 , s)) T

− DFe (Y(f2 , s)) · [DY(f2 ) · [f1 − f2 ]](s)) ds)

>

and

>

T

I 21 = e (a − d2 ) t F Te (t − s) X P0 e (Fe (Y(f1 , s)) − Fe (Y(f2 , s)) 0

− DFe (Y(f2 , s)) · [DY(f2 ) · [f1 − f2 ]](s)) ds)

>

We have I 11 [

2MK ||Y||C 1 (d1 − d2 ) T e ||f1 − f2 || < g/2 ||f1 − f2 || w1 − d2

A similar argument as in Case t \ T shows that I 21 [ g/2 ||f1 − f2 || and thus in both cases I1 [ g ||f1 − f2 || whenever ||f1 − f2 || is small enough. The proof of I2 =o(||f1 − f2 ||) is similar and is omitted. i

Small Delay Inertial Manifolds Under Numerics

559

Lemma 4. If Y ¥ C 1(Pe , Sa − di ) 5 C 2(Pe , S2a − di ) then Te (Y) ¥ C 1(Pe , Sa − di+1 ) 5 C 2(Pe , S2a − di+1 ) for i=0, 1,... . Proof of Lemma 4. Let Y ¥ C 1(Pe , Sa − di ) 5 C 2(Pe , S2a − di ) be fixed. Then DY ¥ C(Pe , BL(Pe , Sa − di )) and D 2Y ¥ C(Pe , BL 2(Pe , S2a − di )), where D 2Y is the Fréchet derivative of DY. The previous lemma shows that Te (Y) ¥ C 1(Pe , Sa − di+1 ) and the derivative D(Te (Y))=DTe (DY) where (DTe (DY) · [k])(t, f) t

=Te (t) k+F Te (t − s) X P0 e DFe (Y(f, s)) · [DY(f) · [k]](s) ds 0

+F

t

−.

Te (t − s) X Q0 e DFe (Y(f, s)) · [DY(f) · [k]](s) ds.

Differentiate DTe (DY) to obtain (D 2Te (D 2Y) · [k1 , k2 ])(t, f) t

:=F Te (t − s) X P0 e (D 2Fe (Y(f, s)) 0

· [DY(f) · [k1 ](s), DY(f) · [k2 ](s)] +DFe (Y(f, s)) · [D 2Y(f) · [k1 , k2 ](s)]) ds +F

t

−.

Te (t − s) X Q0 e (D 2Fe (Y(f, s)) · [DY(f) · [k1 ](s), DY(f) · [k2 ](s)]

+DFe (Y(f, s)) · [D 2Y(f) · [k1 , k2 ](s)]) ds. We claim that D 2(Te (Y))=D 2Te (D 2Y). Let f1 , f2 ¥ Pe be given. Define I :=||e (2a − di+1 ) t((DTe (DY) · [k])(f1 , t) − (DTe (DY) · [k])(f2 , t) − (D 2Te (D 2Y) · [f1 − f2 , k])(f2 , t))||. Since D 2Te (D 2Y) is bilinear and continuous in k1 , k2 ¥ Pe it suffices to show that sup

sup I=o(||f1 − f2 ||)

k ¥ Pe , ||k|| [ 1 t ¥ R −

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Farkas

as ||f1 − f2 || Q 0. To this end let g > 0 be given and write I=I1 +I2 where

>

1

t

I1 = e 2a − di+1 ) t F Te (t − s) X P0 e (DFe (Y(f1 , s)) · [DY(f1 ) · [k](s)] 0

− DFe (Y(f2 , s)) · [DY(f2 ) · [k](s)] − D 2Fe (Y(f2 , s)) · [DY(f2 ) · [f1 − f2 ](s), DY(f2 ) · [k](s)]

2 >

− DFe (Y(f2 , s)) · [D 2Y(f2 ) · [f1 − f2 , k](s)]) ds and

>

1

I2 = e (2a − di+1 ) t F

t

−.

Te (t − s) X Q0 e (DFe (Y(f1 , s)) · [DY(f1 ) · [k](s)]

− DFe (Y(f2 , s)) · [DY(f2 ) · [k](s)] − D 2Fe (Y(f2 , s)) · [DY(f2 ) · [f1 − f2 ](s), DY(f2 ) · [k](s)]

2 >

− DFe (Y(f2 , s)) · [D 2Y(f2 ) · [f1 − f2 , k](s)]) ds

We prove that I1 =o(||f1 − f2 ||) as ||f1 − f2 || Q 0. Choose T < 0 so that 2MK(||Y||C 1 +||Y||C 2 ) (di − di+1 ) T e < g/2. w1 +a − di There are two cases. Case t \ T. Write

>

t

I1 [ e (2a − di+1 ) t F Te X P0 e DFe (Y(f2 , s)) 0

· [DY(f1 ) · [k](s) − DY(f2 ) · [k](s) − D 2Y(f2 ) · [f1 − f2 , k](s)] ds

>

t

+ e (2a − di+1 ) t F Te X P0 e 0

1F

1

0

>

D 2Fe (uY(f1 , s)+(1 − u) Y(f2 , s))

· [DY(f1 ) · [k](s) − DY(f2 ) · [k](s), Y(f1 , s) − Y(f2 , s)

2 >

− DY(f2 ) · [f1 − f2 ](s)] du ds

>

t

+ e (2a − di+1 ) t F Te X P0 e 0

1F

1

0

D 2Fe (uY(f1 , s)+(1 − u) Y(f2 , s))

2 >

− D 2Fe (Y(f2 , s)) · [DY(f2 ) · [k](s), DY(f2 ) · [f1 − f2 ](s)] du ds

Small Delay Inertial Manifolds Under Numerics

561

Now choose o > 0 such that if ||f1 − f2 || < o then sup ||e (a − di ) s(DY(f1 ) − DY(f2 )) · [k](s)|| [ ||k||

s ¥ R−

||D 2Fe (uY(f1 , s)+(1 − u) Y(f2 , s)) − D 2Fe (Y(f2 , s))|| <

g(w1 +a − di ) 3M ||Y|| 2C 1

for all u ¥ [0, 1] and s ¥ [T, 0], sup ||e (a − di ) s(Y(f1 , s) − Y(f2 , s) − DY(f2 ) · [f1 − f2 ](s)||

s ¥ R−

<

g(w1 +a − di ) ||f1 − f2 || 3MK

sup ||e (2a − di ) s(DY(f1 ) · [k](s) − DY(f2 ) · [k](s) − D 2Y(f2 ) · [f1 − f2 , k](s))||

s ¥ R−

<

g(w1 +a − di ) ||k|| · ||f1 − f2 || 3MK

hold. Then we have that I1 [ g ||k|| · ||f1 − f2 ||. Case t < T. Write I1 =I 11 +I 21 where

>

1

t

I 11 = e (2a − di+1 ) t F Te (t − s) X P0 e (DFe (Y(f1 , s)) · [DY(f1 ) · [k](s)] T

− DFe (Y(f2 , s)) · [DY(f2 ) · [k](s)] − D 2Fe (Y(f2 , s)) · [DY(f2 ) · [f1 − f2 ](s), DY(f2 ) · [k](s)]

2 >

− DFe (Y(f2 , s)) · [D 2Y(f2 ) · [f1 − f2 , k](s)]) ds and

>

1

T

I 21 = e (2a − di+1 ) t F Te (t − s) X P0 e (DFe (Y(f1 , s)) · [DY(f1 ) · [k](s)] 0

− DFe (Y(f2 , s)) · [DY(f2 ) · [k](s)] − D 2Fe (Y(f2 , s)) · [DY(f2 ) · [f1 − f2 ](s), DY(f2 ) · [k](s)]

2 >

− DFe (Y(f2 , s)) · [D 2Y(f2 ) · [f1 − f2 , k](s)]) ds . We have I 11 [

2MK(||Y||C 1 +||Y||C 2 ) (di − di+1 ) T e ||k|| · ||f1 − f2 || < g/2 ||k|| · ||f1 − f2 ||. w1 +a − di

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Farkas

A similar argument as in Case t \ T shows that I 21 [ g/2 ||k|| · ||f1 − f2 || and thus supt ¥ R− I1 [ g ||k|| · ||f1 − f2 || whenever ||f1 − f2 || is small enough. The proof for I2 is similar and is omitted. i Let B … Pe be a closed bounded ball and fix 0 < D < d0 . For Y ¥ C(B, Sa − D ) define the operator DTe, Y on C(B, BL(Pe , Sa − d )), d ¥ [0, D] by setting (DTe, Y (Y) · [k])(t, f) t

:=Te (t)+F Te (t − s) X P0 e DFe (Y(f, s)) · [Y(f) · [k](s)] ds 0

+F

t

−.

Te (t − s) X Q0 e DFe (Y(f, s)) · [Y(f) · [k](s)] ds

It is easy to see (e.g., Lemma 2) that DTe, Y maps C(B, BL(Pe , Sa − d )) into C(B, BL(Pe , Sa − d )) for all d ¥ [0, D] and is a uniform 1/3-contraction. Denote the fixed points by Y dY . Lemma 5. The fixed points Y dY are independent of d and their norms are uniformly bounded, i.e., there exists a constant W1 independent of Y, d and e such that ||Y dY ||C(B, BL(Pe , Sa − d )) [ W1 . Proof of Lemma 5. Since C(B, BL(Pe , Sa − d )) … C(B, BL(Pe , Sa )) by uniqueness we have that the fixed points are independent of d. With norm || · ||=|| · ||C(B, BL(Pe , Sa − d )) we have that ||Y dY ||=||DTe, Y (Y dY )|| [ sup f¥B

sup

sup I,

k ¥ Pe , ||k|| [ 1 t ¥ R −

where I=||e (a − d) t Ie (t) k||

> +> e

t

+ e (a − d) t F Te (t − s) X P0 e DFe (Y(f, s)) · [Y dY (f) · [k](s)] ds

>

0

(a − d) t

F

t

−.

Te (t − s) X Q0 e DFe (Y(f, s)) · [Y dY (f) · [k](s)] ds

>

Small Delay Inertial Manifolds Under Numerics

563

By a simple calculation MK 3MK I [ M ||k||+ ||Y dY || · ||k||+ ||Y dY || · ||k|| w1 − d w2 +d [ M ||k||+(1/3) ||Y dY || · ||k||. Hence ||Y dY || [ M+(1/3) ||Y dY || i

and the Lemma is proved by setting W1 =(3/2) M. 0 Y

Set YY =Y . Similarly, for Y ¥ C 1(B, Sa − D/2 ) define the operator D 2Te, Y on C(B, BL 2(Pe , S2a − d )), d ¥ [0, D] by setting (D 2Te, Y (Y) · [k1 , k2 ])(t, f) t

:=F Te (t − s) X P0 e (D 2Fe (Y(f, s)) · [DY(f) · [k1 ](s), DY(f) · [k2 ](s)] 0

+DFe (Y(f, s)) · [Y(f) · [k1 , k2 ](s)]) ds +F

t

−.

Te (t − s) X Q0 e (D 2Fe (Y(f, s)) · [DY(f) · [k1 ](s), DY(f) · [k2 ](s)]

+DFe (Y(f, s)) · [Y(f) · [k1 , k2 ](s)]) ds. It is easy to see (e.g., Lemma 2) that D 2Te, Y maps C(B, BL 2(Pe , S2a − d )) into C(B; BL 2(Pe , S2a − d )) for all d ¥ [0, D] and is a uniform 1/3-contraction. Denote the fixed points by k dY . Lemma 6. The fixed points are independent of d and there is constant W2 independent of d and e such that ||Y dY ||C(B, BL 2(Pe , S2a − d )) [ W2 ||Y|| 2C 1(B, Sa − D/2 ) . Proof of Lemma 6. Since C(B, BL 2(Pe , S2a − d )) … C(B, BL 2(Pe , S2a )) by uniqueness we have that the fixed points are independent of d. With norm || · ||=|| · ||C(B, BL 2(Pe , S2a − d )) we have that ||Y dY ||=||D 2Te, Y (Y dY )|| [ sup f¥B

sup

sup I,

k1 , k2 ¥ Pe , ||k1 ||, ||k2 || [ 1 t ¥ R −

564

Farkas

where I=e (2a − d) t

> F T (t − s) X t

e

0

Pe 0

(D 2Fe (Y(f, s)) · [DY(f)

· [k1 ](s), DY(f) · [k2 ](s)] +DFe (Y(f, s)) · [Y dY (f) · [k1 , k2 ](s)]) ds +e (2a − d) t

>F

t

−.

>

Te (t − s) X Q0 e (D 2Fe (Y(f, s))

· [DY(f) · [k1 ](s), DY(f) · [k2 ](s)]

>

+DFe (Y(f, s)) · [Y dY (f) · [k1 , k2 ](s)]) ds . By a simple calculation I[

MK 3MK 2 1 w +a + ||Y|| ||k || · ||k || − d w − a+d 3MK 2 MK +1 + ||Y || ||k || · ||k || w +a − d w − a+d 2 C 1(B, Sa − D/2 )

1

1

2

2

d Y

1

1

2

2

[ (1/3)(||Y|| 2C 1(B, Sa − D/2 ) +||Y dY ||) ||k1 || · ||k2 || Hence ||Y dY || [ (1/3)(||Y|| 2C 1(B, Sa − D/2 ) +||Y dY ||) and the lemma is proved by setting W2 =1/2.

i

Set YY =Y 0Y . Now we are in a position to prove the C 2 smoothness of the small delay inertial manifold. Lemma 7. The small delay inertial manifold is C 2 smooth, i.e., Ye ¥ C 1(B; Sa ) 5 C 2(B; S2a ). Moreover, DYe =YYe and D 2Ye =YYe . Proof of Lemma 7. Set Y 0 — 0 and Y n+1 :=Te (Y n). Fix a sequence d0 > d1 > d2 > · · · > D > 0. By Lemma 2, Y n Q Ye in C(B, Sa − D ). Moreover, by Lemma 3, Y n ¥ C 1(B, Sa − dn ) and DY n+1=DTe, Y n (DY n). Thus {Y n}n \ 0 … C 1(B, Sa − D ). In what follows we show that {Y n}n \ 0 is a Cauchy sequence in C 1(B, Sa − D/2 ). Clearly, it is enough to prove that {DY n}n \ 0 is a Cauchy

Small Delay Inertial Manifolds Under Numerics

565

sequence in C(B, BL(Pe , Sa − D/2 )). In the estimates below || · || stands for the norm of C(B, BL(Pe , Sa − D/2 )): ||DY n − YY n || [ 1/3 ||DY n − 1 − YY n − 1 ||+||YY n − YY n − 1 ||. With L ¥ N we set eL :=supn, m \ L ||YY n − YY m ||. By an inductive application of the above estimate we have ||DY n − YY n || [ (1/3) n − L ||DY L − YY L ||+3/2eL and thus for m \ n \ L ||DY m − DY n|| [ 2(1/3) n − L ||DY L − YY L ||+3eL . It remains to prove that eL Q 0 as L Q .. Since ||YY n − YY m || [ 1/3 ||YY n − YY m ||+||DTe, Y n (YY n ) − DTe, Y m (YY n )|| it is enough to prove that sup ||DTe, Y n (YY n ) − DTe, Y m (YY n )|| Q 0 n, m \ L

as L Q .. By a simple calculation we have ||DTe, Y n (YY n ) − DTe, Y m (YY n )|| [ sup f¥B

sup

sup (I1 +I2 ),

k ¥ Pe , ||k|| [ 1 t ¥ R −

where

>

t

I1 = e a − D/2) t F Te (t − s) X P0 e (DFe (Y n(f, s)) 0

− DFe (Y m(f, s))) · [YY n (f) · [k](s)] ds

>

and

>

I2 = e a − D/2) t F

t

−.

Te (t − s) X Q0 e (DFe (Y n(f, s))

>

− DFe (Y m(f, s))) · [YY n (f) · [k](s)] ds . Consider I1 . Let g > 0 be given. Choose T < 0 so that 2KMW1 (D/2) T e < g/2. w1 − D/2

566

Farkas

There are two cases. Case t \ T. Choose L so large such that ||DFe (Y n)(f, s)) − DFe (Y m(f, s))|| [

g(w1 − D/2) 2MW1

holds for n, m \ L, f ¥ B and s ¥ [T, 0]. Then I1 [ g/2 ||k||. Case t < T. Write I1 =I 11 +I 21 where

>

t

I 11 = e (a − D/2) t F Te (t − s) X P0 e (DFe (Y n(f, s)) T

− DFe (Y m(f, s))) · [YY n (f) · [k](s)] ds

>

and

>

T

I 21 = e (a − D/2) t F Te (t − s) X P0 e (DFe (Y n(f, s)) 0

>

− DFe (Y m(f, s))) · [YY n (f) · [k](s)] ds . It is easy to see that I 11 [ g/2 ||k|| while I 21 can be handled as I1 in Case t \ T, so we have that I1 [ g ||k||. The proof for I2 < g ||k|| is similar and is omitted. Hence eL Q 0 as L Q . and {Y n}n \ 0 … C 1(B, Sa − D/2 ) is a Cauchy sequence and DYe =YYe . Let us turn to the C 2 property. By Lemma 4, Y n ¥ C 1(B, Sa − dn ) 5 2 C (B, S2a − dn ) and D 2Y n+1=D 2Te, Y n (DY n). We prove that {Y n}n \ 0 is a Cauchy sequence in C 2(B, S2a ). Clearly, it is enough to prove that {D 2Y n}n \ 0 is a Cauchy sequence in C(B, BL 2(Pe , S2a )). Note that {Y n}n \ 0 is bounded in C 1(B, Sa − D/2 ), i.e., there exist a constant W3 such that ||Y n||C 1(B, Sa − D/2 ) [ W3

for n=0, 1, 2,... .

By Lemma 6 there exists a constant W4 independent of d ¥ [0, D], n \ 0 and e such that ||YY n ||C(B, BL 2(Pe , S2a − d )) [ W4 .

Small Delay Inertial Manifolds Under Numerics

567

with norm || · ||=|| · ||C(B, BL 2(Pe , S2a )) we have that ||D 2Y n − YY n || [ 1/3 ||D 2Y n − 1 − YY n − 1 ||+||YY n − YY n − 1 ||. For L ¥ N we set eL :=supn, m \ l ||YY n − YY m ||. By an inductive application of the estimate above we have that ||D 2Y n − YY n || [ (1/3) n − L ||D 2Y L − YY L ||+3/2eL and thus for m \ n \ L ||D 2Y m − D 2Y n|| [ 2(1/3) n − L ||D 2Y L − YY L ||+3/2eL . It remains to prove that eL Q 0 as L Q .. Since ||YY n − YY m || [ 1/3 ||YY n − YY m ||+||D 2Te, Y n (YY n ) − D 2Te, Y m (YY n )|| it is enough to prove that sup ||D 2Te, Y n (YY n ) − D 2Te, Y m (YY n )|| Q 0 n, m \ L

as L Q .. Write ||D 2Te, Y n (YY n ) − D 2Te, Y m (YY n )|| [ sup f¥B

sup (I1 +I2 +I3 +I4 +I5 +I6 ),

sup

k1 , k2 ¥ Pe , ||k1 ||, ||k2 || [ 1 t ¥ R −

where

>

t

I1 = e 2at F Te (t − s) X P0 e (D 2Fe (Y n(f, s)) 0

· [(DY n(f) − DY m(f)) · [k1 ](s), (DY n(f) − DY m(f)) · [k2 ](s)]) ds

>

t

I2 = e 2at F Te (t − s) X P0 e (D 2Fe (Y n(f, s)) − D 2Fe (Y m(f, s)) 0

· [DY n(f) · [k1 ](s), DY m(f) · [k2 ](s)] ds

>

t

>

I3 = e 2at F Te (t − s) X P0 e (DFe (Y n(f, s)) − DFe (Y m(f, s))) 0

· [YY n (f) · [k1 , k2 ](s)] ds

>

>

568

Farkas

>

I4 = e 2at F

t

−.

Te (t − s) X Q0 e (D 2Fe (Y n(f, s))

· [(DY n(f) − DY m(f)) · [k1 ](s), (DY n(f) − DY m(f)) · [k2 ](s)]) ds

>

I5 = e 2at F

t

−.

>

Te (t − s) X Q0 e (D 2Fe (Y n(f, s)) − D 2Fe (Y m(f, s)))

· [DY m(f) · [k1 ](s), DY m(f) · [k2 ](s)] ds

>

and

>

I6 = e 2at F

t

−.

Te (t − s) X Q0 e (DFe (Y n(f, s))

>

− DFe (Y m(f, s))) · [YY n (f) · [k1 , k2 ](s)] ds . Let g > 0 be given. Consider first I1 . Since {Y n}n \ 0 is a Cauchy sequence in C 1(B, Sa − D/2 ) we can choose L so large such that for n, m \ L ||DY n − DY m||C(B, BL(Pe , Sa )) [

g(w1 +a) 3MK

holds and thus I1 [ g/3 ||k1 || · ||k2 || Let us estimate I2 . Choose T < 0 so that 2MKW 23 DT e [ g/3. w1 +a There are two cases. Case t \ T. Choose L so large such that ||D 2Fe (Y n(f, s)) − D 2Fe (Y m(f, s))|| [ holds for all n, m \ L, f ¥ B and s ¥ [T, 0]. Then I2 [ g/3 ||k1 || · ||k2 ||.

g(w1 +a) 3MKW 23

Small Delay Inertial Manifolds Under Numerics

569

Case t < T. Write I2 =I 12 +I 22 where

>

t

I 12 = e 2at F Te (t − s) X P0 e (D 2Fe (Y n(f, s)) − D 2Fe (Y m(f, s))) T

· [DY m(f) · [k1 ](s), DY m(f) · [k2 ](s)] ds

>

and

>

T

I 22 = e 2at F Te (t − s) X P0 e (D 2Fe (Y n(f, s)) − D 2Fe (Y m(f, s))) 0

>

· [DY m(f) · [k1 ](s), DY m(f) · [k2 ](s)] ds . We have I 12 [ g/3 ||k1 || · ||k2 || and a similar estimate for I 22 goes along the same line as in Case t \ T. Now we turn to I3 . Recall that ||YY n ||C(B, L 2(Pe , S2a − D )) [ W4 . Choose T < 0 so that 2MKW4 DT e [ g/3. w1 +a There are two cases. Case t \ T. Choose L so large such that ||DFe (Y n(f, s)) − DFe (Y m(f, s))|| [

g(w1 +a) 3MKW4

holds for n, m \ L, f ¥ B and s ¥ [T, 0]. Then I3 [ g/3 ||k1 || · ||k2 ||. Case t < T. Write I3 =I 13 +I 23 where

>

t

I 13 = e 2at F Te (t − s) X P0 e (DFe (Y n(f, s)) − DFe (Y m(f, s))) T

· [YY n (f) · [k1 , k2 ](s)] ds

>

570

Farkas

and

>

T

I 23 = e 2at F Te (t − s) X P0 e (DFe (Y n(f, s)) − DFe (Y m(f, s))) 0

>

· [YY n (f) · [k1 , k2 ](s)] ds . We have I 13 [ g/3 ||k1 || · ||k2 || and I 23 [ g/3 ||k1 || · ||k2 || as in Case t \ T. The proof of I4, 5, 6 [ g ||k1 || · ||k2 || is similar and is omitted. Hence eL Q 0 as L Q . and {Y n}n \ 0 is a Cauchy sequence in C 2(B, S2a ) and i D 2Ye =YYe , and the lemma is proved. 2 We end this section with estimating the C norm of the small delay inertial manifolds. Note that by Lemmata 6 and 7, there exists a constant W5 independent of e such that ||Ye ||C 2(B, S2a ) [ W5 . We have ||DFe ||C(B, BL(Pe , Qe )) =sup f¥B

[ sup f¥B

sup

>F

0

Te (−s) X Q0 e DFe (Ye (f, s)) · [DYe (f) · [k](s)] ds

k ¥ Pe , ||k|| [ 1

−.

sup

3MK F

k ¥ Pe , ||k|| [ 1

0

−.

e bs ||DYe (f) · [k](s)|| ds [

>

3MKW5 Q0 w2 − a

as b Q . (or eb Q 0). Similarly, ||D 2Fe ||C(B, BL 2(Pe , Qe )) =sup f¥B

sup k1 , k2 ¥ Pe , ||k1 ||, ||k1 || [ 1

>F

0

−.

Te (−s) X Q0 e (D 2Fe (Ye (f, s))

· [DYe (f) · [k1 ](s), DYe (f) · [k2 ](s)] +DFe (Ye (f, s)) · [D 2Ye (f) · [k1 , k2 ](s)]) ds [ sup f¥B

sup k1 , k2 ¥ Pe , ||k1 ||, ||k2 || [ 1

3MK F

0

−.

>

e bs (||DYe (f) · [k1 ](s)||

· ||DYe (f) · [k2 ](s)||+||D 2Ye (f) · [k1 , k2 ](s)||) ds [ as b Q ..

3MK(W 25 +W5 ) Q0 w2 − a

Small Delay Inertial Manifolds Under Numerics

571

It remains to prove the invariance and the exponential attractivity of the small delay inertial manifolds.

3.3. Invariance Denote the solution of (1) starting from f=x0 ¥ Ce by xt , t \ 0. Observe that Ye (px0 , s), s [ 0 is a backward (in time) solution of (1) on the small delay inertial manifold starting from px0 +Fe (px0 ). Thus Fe (pxt )=F

0

Te (−s) X Q0 e Fe (pxs+t +Fe (pxs+t )) ds

−.

=F

t

Te (t − u) X Q0 e Fe (pxu +Fe (pxu )) du

−.

1

=Te F

0

−.

Te (−s) X Q0 e Fe (pxs +Fe (pxs )) ds

2

t

+F Te (t − s) X Q0 e Fe (pxs +Fe (pxs )) ds 0

t

=Te (t) Fe (px0 )+F Te (t − s) X Q0 e Fe (pxs +Fe (pxs )) ds. 0

Now let x0 ¥ graph Fe , i.e., (id − p) x0 : Fe (px0 ). By the variation of constants formula we have that t

(id − p) xt =Te (t) Fe (px0 )+F Te (t − s) X Q0 e Fe (xs ) ds. 0

Hence t

(id − p) xt − Fe (pxt )=F Te (t − s) X Q0 e (Fe (xs ) − Fe (pxs +Fe (pxs ))) ds 0

and t

||(id − p) xt − Fe (pxt )|| [ 3MK F e −b(t − s) ||(id − p) xs − Fe (pxs )|| ds 0

from which it follows that ||(id − p) xt − Fe (pxt )|| — 0.

572

Farkas

3.4. Exponential attractivity Let xt be an arbitrary solution of (1). Define v(t) :=(id − p) xt − Fe (pxt ). By a simple calculation t

v(t)=Te (t) v(0)+F Te (t − s) X Q0 e (Fe (xs ) − Fe (pxs − Fe (pxs ))) ds, 0

hence t

||v(t)|| [ 3Me −bt ||v(0)||+3MK F e −b(t − s) ||v(s)|| ds 0

from which it follows (by using the Gronwall inequality) that ||v(t)|| [ 3Me −mt ||v(0)|| where m=b − 3MK > 0. This proves exponential attractivity and the proof of the theorem is now complete. i 4. EXISTENCE OF ASYMPTOTIC PHASES In this section we prove that our small delay inertial manifolds have asymptotic phases. Namely, we have the following Theorem 2. For all solution xt of (1) there exists a solution xt of (1) on the small delay inertial manifold such that ||xt − xt || [ const · e −mt, where m=b − 3MK. Proof. Let xt , t \ 0 be an arbitrary solution of (1). Let xt be the unknown solution of (1) on the small delay inertial manifold, i.e., t

xt =Te (t) x0 +F Te (t − s) X P0 e Fe (pxs +Fe (pxs )) ds. 0

Recall that v(t)=(id − p) xt +Fe (pxt ) and ||v(t)|| [ 3Me −mt ||v(0)||

Small Delay Inertial Manifolds Under Numerics

573

by the exponential attractivity. Set w(t)=xt − pxt . A simple calculation yields that t

w(t)=Te (t) w(0)+F Te (t − s) X P0 e (Fe (pxs +w(s)+Fe (pxs +w(s))) 0

− Fe (pxs +v(s)+Fe (pxs ))) ds. Define the Banach space + mt S+ m :={w: R Q Pe : w is continuous and sup ||w(t)|| e < .} t ¥ R+

with norm |w|m :=sup ||w(t)|| e mt. t ¥ R+

Define operator F on S+ m by setting .

F(w)(t) :=F Te (t − s) X P0 e (Fe (pxs +w(s)+Fe (pxs +w(s))) t

− Fe (pxs +v(s)+Fe (pxs ))) ds. It is easy to see that |F(w)|m [

MK((1+||Fe ||C 1 ) |w|m +3M ||v(0)||) <. m−w

and |F(w1 ) − F(w2 )|m [

MK(1+||Fe ||C 1 ) |w1 − w2 |m [ 1/2 |w1 − w2 |m . m−w

Denote the fixed point of F by w*. Then .

w*(t)=F Te (t − s) X P0 e (Fe (pxs +w*(s)+Fe (pxs +w*(s))) t

− Fe (pxs +v(s)+Fe (pxs ))) ds

1

.

=Te (t) F Te (−s) X P0 e (Fe (pxs +w*(s)+Fe (pxs +w*(s))) 0

− Fe (pxs +v(s)+Fe (pxs ))) ds

2

574

Farkas t

× F Te (−s) X P0 e (Fe (pxs +w*(s)+Fe (pxs +w*(s))) 0

− Fe (pxs +v(s)+Fe (pxs ))) ds t

=Te (t) w*(0)+F Te (t − s) X P0 e (Fe (pxs +w*(s)+F(pxs +w*(s))) 0

− Fe (pxs +v(s)+Fe (pxs ))) ds. Hence xt :=pxt +w*(t) is a solution on the small delay inertial manifold such that ||pxt − pxt || [ |w*|m e −mt. Finally, the result follows by observing that ||xt − xt || [ ||pxt − pxt ||+||v(t)||+||Fe (pxt ) − Fe (pxt )||.

i

5. DISCRETIZATION OF SMALL DELAY INERTIAL MANIFOLDS For N ¥ N set h=e/N and consider the Euler-discretization of (1) with step-size h, i.e., we consider the map on R n defined by yk+1 =(I+hA) yk +h(f(yk )+g(yk − N )),

(3)

where yk is the approximating value of the exact solution x(kh). An initial value f ¥ Ce of (1) gives rise to an initial value fh :=(y0 ,..., y−N ) ¥ R n × (N+1) of (3) by setting yi :=f(ih), i=0,..., −N. Identify the space R n × (N+1) (endowed with the usual max norm) with a subspace of Ce consisting of piecewise linear continuous functions defined on the mesh-points {ih: i=0,..., −N}. Denote this subspace by C he . Define the projection p he : Ce Q C he by p he f= piecewise linear continuous extension from the values on the mesh points. The map (3) gives rise to a map on C he xk+1 =Th, e xk +hE0 Fh, e (xk ), where

r

I+hA I Th, e = 0 0

0 0 ··· ···

E0 =p he X0 and Fh, e =f(xk (0))+g(xk (−e)).

··· ··· 0 I

s

0 0 , 0 0

(4)

Small Delay Inertial Manifolds Under Numerics

575

From now on we assume that e > 0 is so small such that (I+eA) is invertible. Let us decompose the space C he by s(I+hA) as C he =P he À Q he , where P he =ph C he , Q he =(id − ph ) C he and the projection ph is defined by setting ph fh (ih) :=(I+hA) i fh (0) for i=0,..., −N. Note that the above splitting is Th, e -invariant for all e and h. The proof of the properties below is straightforward and thus is omitted. (H0)h

For all e and h we have that dim P he =n, there exists a constant M independent of e and h such that ||p he ||, ||ph ||, ||id − ph || [ M,

(H1)h

for all b > 0 there exists eb > 0 such that for all e ¥ (0, eb ] and for all h ||T kh, e (id − ph ) x|| [ 3Me −bkh ||x||,

k \ 0,

(H0)h

for all h the function Fh, e ¥ C 2(C he , R n) is bounded with bounded derivatives. Moreover, the bounds are independent of e and h, i.e., ||Fh, e ||C 2 [ K for all e and h,

(H0)h

there exists an w > 0 independent of e and h such that for all e and h ||T kh, e ph x|| [ Me w |kh| ||x||,

k ¥ Z.

The following theorem can be proved exactly the same way as Theorem 1 (every integral is replaced by the corresponding sum). Details are left to the reader. Theorem 3. For all e small enough and for all h there exists a C 2-function F he : P he Q Q he such that graph(F he )={x+F he (x) : x ¥ P he } is an exponentially attractive invariant manifold for (4) with an asymptotic phase. Note that there exists an approximating small delay inertial manifold for all step-sizes. This is due to the fact that a is small, and thus the ODE part alone is well approximated by the Euler method. In what follows we study the behavior of F he with fixed e as h Q 0+. Let {f1 ,..., fn } be the usual orthonormal basis in R n. Then {f1 ,..., fn } and {f hi ,..., f hn } are bases of Pe and P he , respectively, where fi (h)=e Ahfi for h ¥ [ − e, 0], i=1,..., n and f hi (jh)=(I+hA) j fi for j=0,..., −N, i=1,..., n. For sake of simplicity we write fi =e A · fi and f hi =(I+hA) · fi . Let us define the linear bijection Ph : Pe Q P he by setting Ph f :=; ni=1 ai f hi whenever f=; ni=1 ai fi .

576

Farkas

Lemma 8. (i) There exist a constant M1 independent of e and h such that ||Ph || [ M1 for all e and h, (ii) there exists a continuous function l: R+ Q R+ such that l(0)=0 and ||Ph Te (−h) − T h,−1e Ph ||BL(Pe , P he ) [ l(h) h, (iii) for all T > 0 ||p he Te (kh) X Q0 e − T kh, e (id − ph ) E0 || Q 0

sup kh ¥ [0, T]

as h Q 0+, (iv) ||Ph − p he |Pe ||BL(Pe , C he ) Q 0 as h Q 0+, (v) Fe (f)=Fh, e (p he f), DFe (f) · [k]=DFh, e (p he f) · [p he k], (vi) for all h ph E0 =Ph X P0 e . Proof of Lemma 8. Let f ¥ Pe , ||f|| [ 1. By definition 1 \ ||f||= suph ¥ [ − e, 0] ||e Ah ; ni=1 ai fi || \ ||; ni=1 ai fi || which shows that ; ni=1 |ai | [ n. Thus ||Ph f||=||; ni=1 ai f hi || [ n maxi=1,..., n ||f hi ||. Observe that (I+hA) · Q e A · uniformly on the interval [ − e, 0] and e A · is uniformly bounded in e ¥ [0, eb ]. Thus there exists a constant K1 independent of h, e such that maxi=1,..., n ||f hi || [ K1 . This proves (i). Let us prove (ii). Let, as before, f ¥ Pe , ||f|| [ 1. Then f=e A · ; ni=1 ai fi and Te (−h) f=e A · e −Ah ; ni=1 ai fi . On the other hand T h,−1e Ph f=(I+hA) · (I+hA) −1 ; ni=1 ai fi . Combining these formulas we get that ||Ph Te (−h) f − T h,−1e Ph f|| [ n

max j=0, −1,..., −N

||(I+hA) j|| · ||e −Ah − (I+hA) −1||.

Small Delay Inertial Manifolds Under Numerics

577

Since ||(I+hA) j|| is uniformly bounded with some constant K2 (independent of h, e) and ||e −Ah − (I+hA) −1|| [ K3 h 2 with some constant K3 (independent of h, e) the desired result follows with l(h)=nK2 K3 h. Recall that X Q0 e (h)=−e Ah, if − e [ h < 0 and X Q0 e (0)=0. Then

˛ 0− e

Te (kh) X Q0 e (jh)=

if − kh [ jh [ 0 if jh < − kh

A(jh+kh)

for j=0, −1,..., −N which shows that Te (kh) X Q0 e =0 if kh \ e. Similarly, (id − ph ) E0 (jh)=−(I+hA) j, if j=−1,..., −N and (id − ph ) E0 (0)=0. Then

˛ 0− (I+hA)

T kh, e (id − ph ) E0 (j)=

j+k

if − k [ j [ 0 if j < − k

for j=0, −1,..., −N which shows that T kh, e (id − ph ) E0 =0 if k \ N. Without loss of generality we assume that T [ e. Then sup

||p he Te (kh) X Q0 e − T kh, e (id − ph ) E0 ||

kh ¥ [0, T]

= sup

|| − e A(jh+kh)+(I+hA) j+k|| [ K4 h

sup

kh ¥ [0, T] j=0, −1,..., −N

with some constant K4 (independent of h, e) which proves (iii). Let f ¥ Pe , ||f|| [ 1. Then ||Ph f − p he f|| [ n

sup

||(I+hA) j − e Ajh|| [ nK5 h

j=0, −1,..., −N

with some suitable constant K5 (independent of h, e) and (iv) follows. Finally, it is straightforward to check (v)–(vi) and the lemma is proved. i Now we are in a position to formulate the main result of this section. Theorem 4. Let e be fixed such that (1) has a small delay inertial manifold Fe and (4) has an approximating small delay inertial manifold F he . Then for all B … Pe closed ball we have that ||p he Fe − F he p Ph ||C 1(B, C he ) Q 0 as h Q 0+. Proof. Without loss of generality we assume that constants M, w, a, b, K, d0 , D have the same value as in Section 2.

578

Farkas

Define the Banach space S hg :={Yh : Yh : Z − Q C he is continuous and sup e gkh ||Yh (k)|| < .} k ¥ Z−

with norm |Yh |g = sup e gkh ||Yh (k)||. k ¥ Z−

Let B … P he be an arbitrary bounded closed ball. We define operator T on C(B, S hg ) by setting h e

(T he (Yh ))(fh , k) k

=T kh, e fh − C T kh,−e 1 − i ph hE0 Fh, e (Yh (fh , i)) i=−1 k−1

+ C T kh,−e 1 − i (id − ph ) hE0 Fh, e (Yh (fh , i)),

fh ¥ B, k ¥ Z −.

i=−.

Then for all d ¥ [0, d0 ], T he : C(B, S ha − d ) Q C(B, S ha − d ) is a uniform 1/3 contraction. Then the approximating small delay inertial manifold is defined as F he =(id − ph ) Y he (0), where Y he is the fixed point of T he . We define the operator DT he, Y he on C(B, BL(P he , S ha − d )), d ¥ [0, D] by setting (DT he, Y he (Yh )) · [kh ])(fh , k) k

:=T kh, e kh + C T kh,−e 1 − i ph hE0 DFh, e (Y he (fh , i)) · [Yh (fh ) · [kh ](i)] i=−1 k−1

+ C T kh,−e 1 − i ph hE0 DFh, e (Y he (fh , i)) · [Yh (fh ) · [kh ](i)]. i=−.

Then DT he, Y he maps C(B, BL(P he , S ha − d )) into C(B, BL(P he , S ha − d )) for all d ¥ [0, D] and is a uniform 1/3 contraction. Its fixed point is DY he , the derivative of Y he . Define the bounded linear operator Ph : C(B, Sm ) Q C(Ph (B), S hm ) by setting for Y ¥ C(B, Sm ) (Ph Y)(Ph f)(k) :=p he Y(f)(kh).

i

Small Delay Inertial Manifolds Under Numerics

579

Lemma 9. For all Y ¥ C(B, Sa − d ), d ¥ (0, D] ||Ph Te Y − T he Ph Y||C(Ph (B), S ha ) Q 0 as h Q 0+. Proof of Lemma 9. Let Y ¥ C(B, Sa − d ) be fixed. We have that ||Ph Te Y − T he Ph Y||C(Ph (B), S ha ) [ sup sup (I1 +I2 +I3 ), f¥B

k ¥ Z−

where I1 =e akh ||p he Te (kh) f − T kh, e Ph f||,

>

I2 =e akh p he F

kh

0

Te (kh − s) X P0 e Fe (Y(f, s)) ds

k

− C T kh,−e 1 − i ph hE0 Fh, e (p he Y(f, ih))

>

i=−1

and

>

I3 =e akh p he F

kh

−.

Te (kh − s) X Q0 e Fe (Y(f, s)) ds

>

k−1

− C T kh,−e 1 − i (id − ph ) hE0 Fh, e (p he Y(f, ih)) . i=−.

We estimate each component separately. First we estimate I1 as I1 [ e akh(||Ph Te (kh) f − p he Te (kh) f||+||T kh, e Ph f − Ph Te (kh) f||) [ Me w1 kh ||Ph − p he | Pe || · ||f|| |k| − 1 k+j+1 Ph Te (−(j+1) h) f|| +e akh C ||T k+j h, e Ph Te (−jh) f − T h, e j=0

1

|k| − 1

[ M ||Ph − p he | Pe ||+e akh C Me −w(k+j+1) h j=0

2

× ||T h,−1e Ph − Ph Te (−h)||BL(Pe , P he ) Me wjh ||f|| [ (M ||Ph − p he | Pe ||+M 2 |k| he w1 khl(h)) ||f||

1

[ M ||Ph − p he | Pe ||+M 2

2

l(h) ||f|| w1

which shows that supf ¥ B supk ¥ Z− I1 Q 0 as h Q 0+.

580

Farkas

Now we estimate I2 as

>C k

I2 [ e akh

T kh,−e 1 − i (ph E0 − Ph X P0 e ) hFh, e (p he Y(f, ih))

>

i=−1

>C k

+e akh

(T kh,−e 1 − i Ph X P0 e − Ph Te ((k − 1 − i) h) X P0 e ) hFe (Y(f, ih))

>

i=−1

>C k

+||Ph − p he | Pe || e akh

Te ((k − 1 − i) h) X P0 e hFe (Y(f, ih))

>

i=−1

>C k

+e akh ||p he || ·

Te ((k − 1 − i) h) X P0 e hFe (Y(f, ih))

i=−1

−F

kh

0

Te (kh − s) X P0 e Fe (Y(f, s)) ds

>

=I 12 +I 22 +I 32 +I 42 . Further, we have that I 12 =0, k

M 2l(h) M 2Kl(h) h , Kh [ w1 w1 (1 − e −ah)

I 22 [ C e a(i+1) h i=−1

(here we used the estimates obtained for I1 previously) k

I 32 [ ||Ph − p he | Pe || e akh C MKhe −w(k − 1 − i) h i=−1 k

[ ||Ph − p he | Pe || MKh C e w1 (k − 1 − i) h [ i=−1

||Ph − p he | Pe || MKh . 1 − e −w1 h

It is clear from these estimates that supf ¥ B supk ¥ Z− (I 12 +I 22 +I 32 ) Q 0 as h Q 0+. It remains to check this property for I 42 . To this end let g > 0 be given. We choose T < 0 such that MK aT e [ g/2. w1

Small Delay Inertial Manifolds Under Numerics

581

There are two cases. Case kh \ T. Since the integral is now on a compact interval it follows simply that I 42 [ g/2 for all h small enough. Case kh < T. Write

>C

[T/h]

I 42 [ +M1 e akh

Te ((k − 1 − i) h) X P0 e hFe (Y(f, ih))

i=−1

−F

[T/h] h

Te (kh − s) X P0 e Fe (Y(f, s)) ds

0

+M1 e akh

>

>

k

Te ((k − 1 − i) h) X P0 e hFe (Y(f, ih))

C i=[T/h] − 1

−F

kh

Te (kh − s) X P0 e Fe (Y(f, s)) ds

[T/h] h

>

=I 4,2 1 +I 4,2 2 . As in Case kh \ T the desired result follows for I 4,2 1 while k

I 4,2 2 [

Me w1 (k − 1 − i) hKhe a(i+1) h+F

C

Me w1 (kh − s)Ke as ds

[T/h] h

i=[L/h] − 1

[

kh

MK 2 1 1 MKh + e −e w −w1 h

aT

.

1

Let us turn to estimating I3 as

>1 C

k−1

I3 [ e akh

k−1

T kh,−e 1 − i (id − ph ) E0 − C p he Te ((k − 1 − i) h) X Q0 e

i=−.

i=−.

× hFh, e (p he Y(f, ih))

>C

>

k−1

+e akh

p he Te ((k − 1 − i) h) X Q0 e hFe (Y(f, ih))

i=−.

−F

kh −.

p he Te (kh − s) X Q0 e Fe (Y(f, s)) ds

=I 13 +I 23 .

>

2

582

Farkas

First we note that I 23 can be handled as I 42 and thus we omit the details. It remains to prove that I 13 tends to 0 as h tends to 0. To this end let g > 0 be given. We choose a T < 0 such that 6M(1+M1 ) K aT e [ g/2. w2 There are two cases. Case kh < T. In this case k−1

k−1

I 13 [ 3M C e −w2 (k − 1 − i) hKhe a(i+1) h+3MM1 C e −w2 (k − 1 − i) hKhe a(i+1) h i=−.

1

i=−.

2

3MKh 3MM1 Kh aT [ + e < g/2 1 − e −w2 h 1 − e −w2 h for all h sufficiently small. Case kh \ T. In this case I 13 [ |T| sup

||T jh, e (id − ph ) E0 − p he Te (jh) X Q0 e || e akhK

jh ¥ [0, |T|] [T/h]

+ C 3M(1+M1 ) e −w2 (k − 1 − i) hKhe a(i+1) h i=−.

[ |T| sup jh ¥ [0, |T|]

3M(1+M1 ) Kh aT ||T jh, e (id − ph ) E0 − p he Te (jh) X Q0 e || K+ e
for all h small enough. The proof of the lemma is now complete.

i

With norm || · ||=|| · ||C(Ph (B), S ha ) we have that ||Ph Ye − Y he || [ ||Ph Te (Ye ) − T he (Ph Ye )||+(1/3) ||Ph Ye − Y he || which shows that ||Ph Ye − Y he || Q 0 as h Q 0+. Since the derivatives of Ye and Y he have a similar integral representation, the same type of argument can be used to finish the proof of the theorem. We only sketch the main ideas, the rest is left to the reader. Redefine the bounded linear operator Ph : C(B, BL(Pe , Sa − d )) Q C(Ph (B), BL(P he , S ha − d )) by setting (Ph Y)(Ph f) · [Ph k](k) :=p he Y(f) · [Y](kh).

Small Delay Inertial Manifolds Under Numerics

583

Lemma 10. For all Y ¥ C(B, BL(Pe , Sa − d )), d ¥ (0, D/2] ||DT he, Y he (Ph Y) − Ph DTe, Ye (Y)||C(Ph (B), BL(P he , S ha )) Q 0 as h Q 0+. Proof of Lemma 10. Let Y ¥ C(B, BL(Pe , Sa − d )) be fixed. We have that ||DT he, Y he (Ph Y) − Ph DTe, Ye (Y)||C(Ph (B), BL(P he , S ha )) [

sup k ¥ Pe , ||k|| [ 1

sup sup (I1 +I2 +I3 ), f¥B

k ¥ Z−

where I1 =e akh ||p he Te (kh) k − T kh, e Ph k||,

>

I2 =e akh p he F

kh

0

Te (kh − s) X P0 e DFe (Ye (f, s)) · [Y(f) · [k](s)] ds

>

k

− C T kh,−e 1 − i ph hE0 DFh, e (Y he (Ph f, i)) · [p he Y(f) · [k](ih)] . i=−1

and

>

kh

I3 =e akh p he F

−.

Te (kh − s) X Q0 e DFe (Ye (f, s)) · [Y(f) · [k](s)] ds

>

k−1

− C T kh,−e 1 − i (id − ph ) hE0 DFh, e (Y he (Ph f, i)) · [p he Y(f) · [k](ih)] . i=−.

Notice that I1 was already treated in the previous lemma. For I2 we write

>C k

I2 [ e akh

T kh,−e 1 − i ph E0 (DFh, e (Y he (Ph f, i))

i=−1

− DFh, e (Ye (f, ih))) · [p he Y(f) · [k](ih)]

>C

>

k

+e akh

T kh,−e 1 − i ph E0 h DFh, e (Ye (f, ih)) · [p he Y(f) · [k](ih)]

i=−1

− p he F

kh

0

Te (kh − s) X P0 e DFe (Ye (f, s)) · [Y(f) · [k](s)] ds

>

=I 12 +I 22 . The second term can be estimated as I2 in the previous lemma. Let us estimate the first term. Let g > 0 ge given. We choose T < 0 so that MKh ||Y|| dT e [ g/2 1 − e −w2 h

584

Farkas

There are two cases. Case kh \ T. If h is sufficiently small then ||DFh, e (Y he (Ph f, i)) − DFh, e (Ye (f, ih))|| [

(1 − e w2 h) g 2MKh ||Y||

holds for all f ¥ B, ih ¥ [T, 0]. Hence I 12 [ g/2 Case kh < T. Write I 12 [ I 1,2 1 +I 1,2 2 , where I 1,2 1 =e akh

>

k

T kh,−e 1 − i ph E0 h(DFh, e (Y he (Ph f, i))

C i=[T/h] − 1

− DFh, e (Ye (f, ih))) · [p he Y(f) · [k](ih)]

>

and

>C

[T/h]

I 1,2 2 =e akh

T kh,−e 1 − i ph E0 h(DFh, e (Y he (Ph f, i))

i=−1

>

− DFh, e (Ye (f, ih))) · [p he Y(f) · [k](ih)] . The second term can be estimated as I 12 in Case kh \ T while I 1,2 1 [ g/2. By putting an extra term in the estimate of I3 as well we have

>C k

I3 [ e akh

T kh,−e 1 − i (id − ph ) E0 h(DFh, e (Y he (Ph f, i))

i=−.

− DFh, e (Ye (f, ih))) · [p he Y(f) · [k](ih)]

>C

>

k

+e akh

T kh,−e 1 − i (id − ph ) E0 h DFh, e (Ye (f, ih)) · [p he Y(f) · [k](ih)]

i=−.

− p he F

kh

−.

Te (kh − s) X Q0 e DFe (Ye (f, s)) · [Y(f) · [k](s)] ds

and the proof can be finished in a similar way.

> i

Small Delay Inertial Manifolds Under Numerics

585

We end this section with two remarks concerning the rate of the convergence of approximating small delay inertial manifolds and the convergence of higher order derivatives. Remark 4. In the proof of (iii) we observed that T(t) X Q0 e =0, resp. T (id − ph ) E0 =0 if t \ e, resp. k \ N. Moreover, the proof of Lemma 8 shows that l(h) and the convergences in (iii) and (iv) are bounded by O(h). Thus our manifolds have the form k h, e

0

Fe (f)=F Te (−s) X Q0 e Fe (Ye (f, s)) ds −e

and −1

F he (Ph f= C

T 1h,−e i (id − ph ) E0 Fh, e (Y he (Ph f, i))

i=−N − 1

and the speed of the convergence can be estimated by O(h) if the speed of the convergence of Y he (Ph , · ) to Ye (f, · ) can be estimated by O(h) on the compact interval [ − e, 0]. It is easy to see that in this case every term of the estimates in the proof of Lemma 9 are bounded by O(h). Similar convergence result holds true for the derivatives as well. Remark 5. Since the second derivatives of Ye and Y he have similar integral/sum representations the same type of arguments used in Lemmata 9 and 10 can be repeated. Moreover, if f, g are of class C k with bounded derivatives then (via the same procedure and by Remark 2) the convergence of the higher order derivatives can be proved as well. Since our application requires only C 2 smoothness and C 1 closeness we omit the details. 6. A NUMERICAL STRUCTURAL STABILITY RESULT The solution flow of Eq. (1) on the small delay inertial manifold is given by the ordinary differential equation y˙(t)=Ay(t)+f(y(t))+H(y(t)),

(5)

where y(t) ¥ R n and H(y)=g(y+Fe (e A · y)(−e)). The solution flow of (5) is denoted by j t1, e .

586

Farkas

On the other hand, the Euler method on the approximating small delay inertial form takes the form yk+1 =(I+hA) yk +h(f(yk )+Hh (yk ))=: Eh (yk ),

(6)

where yk ¥ R n and Hh (y)=g(y+F he ((I+hA) · y)(−N)). Observe that (6) is the Euler discretization of y˙(t)=Ay(t)+f(y(t))+Hh (y(t)).

(7)

The solution flow of (7) is denoted by j t2, h . Finally, denote the solution flow of the ‘‘limiting ODE’’, i.e., y˙(t)=Ay(t)+f(y(t))+g(y(t)) by j t. Assume that j t flows into a bounded closed ball B along the boundary. Assume further that the chain recurrent set of j t is hyperbolic and j t satisfies the strong transversality condition. (For definitions we refer to [25]). It is known that there is an g > 0 such that if ||H − g||C 1(B) < g then there exist a homeomorphism G1 on B and a continuous function y1 : B Q R such that for all y ¥ B G1 p j y1 (y)(y)=j t1, e p G1 (y). Since ||H − g||C 1(B) Q 0 as e Q 0 by Theorem 1, we can choose an e > 0 such that ||H − g||C 1(B) < g/2. (Structural stability with respect to delay.) Moreover, for all h small enough we find (by Theorem 3) that ||H − Hh ||C 1(B) < g/2 and thus a similar result holds for j t2, h , i.e., there exist a homeomorphism G2 on B and a continuous function y2 : B Q R such that for all y ¥ B G2 p j y2 (y)(y)=j t2, h p G2 (y) Finally, by using that Hh is C 2 we can apply the main result of [20]. Corollary 1. For all h small enough there is a homeomorphism Gh on B and a continuous function yh : B Q R such that for all y ¥ B 1/h p Gh (y) Gh p j 2,yh (y) h (y)=E h

and Gh (y) Q y.

Small Delay Inertial Manifolds Under Numerics

587

Combining these result we obtain conjugacies between the Euler method and the ‘‘limiting ODE’’ as well as between the Euler method and the solutions of (1) on the small delay inertial manifold. We note that similar results are valid when we assume that j t is a Morse–Smale gradient-like dynamical system, however in this case there is no re-parameterization needed, see [20]. Finally we mention that beyond numerical structural stability results one may apply results concerning the persistence of invariant sets under discretization studied for finite-dimensional systems directly to delay equations with small delay. ACKNOWLEDGMENTS Supported by DAAD project 323-PPP, Qualitative Theory of Numerical Methods for Evolution Equations in Infinite Dimensions. This work was done while the author was a visitor at the University of Bielefeld. The author would like to thank Prof. W.-J. Beyn for the stimulating discussions. The author is also grateful to the referee for his/her valuable comments. REFERENCES 1. Boutet de Monvel, L., Chusov, I. D., and Rezounenko, A. V. (1998). Inertial manifolds for retarded semi-linear parabolic equations, Nonlinear Anal. TMA 34, 907–925. 2. Bruschi, S. M., Carvalho, A. N., and Ruas-Filho, J. G. (2000). The dynamics of a onedimensional parabolic problem versus the dynamics of its discretization, J. Differential Equations 168, 67–92. 3. Chen, X.-Y., Hale, J. K., and Tan, B. (1997). Invariant foliations for C 1 semigroups in Banach spaces, J. Differential Equations 139, 283–318. 4. Chen, Y., Wu, J., and Krisztin, T. (2000). Connecting orbits from periodic solutions to phase-locked periodic solutions in a delay differential system, J. Differential Equations 163, 130–173. 5. Chow, S.-N., and Lu, K. (1988). Invariant manifolds for flows in Banach spaces, J. Differential Equations 74, 285–317. 6. Chow, S.-N., Lu, K., and Sell, G. R. (1992). Smoothness of inertial manifolds, J. Math. Anal. Appl. 169, 283–321. 7. Farkas, G. (2001). A Hartman–Grobman result for retarded functional differential equations with an application to the numerics around hyperbolic equilibria, Z. Angew. Math. Phys. 52, 421–432. 8. Farkas, G. Conjugacy in the discretized fold bifurcation, Comput. Math. Appl. (in press). 9. Farkas, G. On approximations of center-unstable manifolds for retarded functional differential equations, Discrete Contin. Dynam. Systems (submitted). 10. Garay, B. M. (1996). On structural stability of ordinary differential equations with respect to discretization methods, Numer. Math. 72, 449–479. 11. Garay, B. M. (2001). Estimates in discretizing normally hyperbolic compact invariant manifolds of ordinary differential equations, Comput. Math. Appl. 42, 1103–1122.

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