Z. angew. Math. Phys. 52 (2001) 421–432 0044-2275/01/030421-12 $ 1.50+0.20/0 c 2001 Birkh¨auser Verlag, Basel Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP A Hartman–Grobman result for retarded functional differential equations with an application to the numerics around hyperbolic equilibria Gyula Farkas Abstract. A weak form of the Hartman–Grobman theorem for retarded functional differential equations around hyperbolic equilibria is presented. Orbits on a center-unstable manifold are compared to orbits on a center-unstable subspace of the linearized equation. The result is applied to obtain a conjugacy between the semidynamical system generated by the functional differential equation and its numerical approximation. A version of the Hartman-Grobman theorem around hyperbolic periodic orbits of functional differential equations is also given. Mathematics Subject Classification (2000). 34K15, 65Q05. Keywords. Hartman-Grobman theorem, discretization, retarded functional differential equa- tion. 1. Introduction and preliminaries The Hartman-Grobman Theorem plays a fundamental role in the local theory of ordinary differential equations. Unfortunately, there are several difficulties in formulating and proving a similar local stability theorem for infinite dimensional differential equations, e.g. for retarded functional equations (RFDE for short) or partial differential equations. For existing results we refer to [1], [10], [11], [12] without claiming completeness. From now on we concentrate on the RFDE case only. One of the difficulties is that the solution map of an RFDE forms a semidynamical system and some of the solutions do not have backward continuations. The usual way to overcome this difficulty is to restrict the semiflow to the attractor. In fact, this approach can succeed, see [10], at the expense of assuming the existence of a compact global attractor on which the semiflow is one-to-one. The aim of the present paper is to formulate a Hartman-Grobman Theorem for RFDE without this structural and “almost global” assumption; indeed, we use only the hyperbolicity of the equilibrium point and compare orbits on an arbitrary but finite dimensional eigenspace of the linearized equation. On the other
Zeitschrift fur angewandteMathematik und Physik ZAMP
A Hartman–Grobman result for retarded functionaldifferential equations with an application to the numericsaround hyperbolic equilibria
Gyula Farkas
Abstract. A weak form of the Hartman–Grobman theorem for retarded functional differentialequations around hyperbolic equilibria is presented. Orbits on a center-unstable manifold arecompared to orbits on a center-unstable subspace of the linearized equation. The result is appliedto obtain a conjugacy between the semidynamical system generated by the functional differentialequation and its numerical approximation. A version of the Hartman-Grobman theorem aroundhyperbolic periodic orbits of functional differential equations is also given.
The Hartman-Grobman Theorem plays a fundamental role in the local theoryof ordinary differential equations. Unfortunately, there are several difficulties informulating and proving a similar local stability theorem for infinite dimensionaldifferential equations, e.g. for retarded functional equations (RFDE for short) orpartial differential equations. For existing results we refer to [1], [10], [11], [12]without claiming completeness.
From now on we concentrate on the RFDE case only. One of the difficultiesis that the solution map of an RFDE forms a semidynamical system and someof the solutions do not have backward continuations. The usual way to overcomethis difficulty is to restrict the semiflow to the attractor. In fact, this approachcan succeed, see [10], at the expense of assuming the existence of a compact globalattractor on which the semiflow is one-to-one.
The aim of the present paper is to formulate a Hartman-Grobman Theoremfor RFDE without this structural and “almost global” assumption; indeed, weuse only the hyperbolicity of the equilibrium point and compare orbits on anarbitrary but finite dimensional eigenspace of the linearized equation. On the other
422 G. Farkas ZAMP
hand we do not conceal that our Theorem 1 and Theorem 2 below are not theultimate answers to the problem, more or less they are only analogues of the finitedimensional case. The formulation of a “genuine” Hartman-Grobman Theoremfor RFDE requires more care as it is demonstrated by the example of Section 3showing that a conjugacy on the whole phase space does not exist without anyfurther assumption; but it is reasonable to expect that investigations in [12] canbe adapted to RFDE as well.
The main result of the present paper applies to linearization about hyperbolicperiodic orbits as well. As a second application we obtain conjugacy between afinite dimensional restriction of the semidynamical system generated by the equa-tion and its numerical approximation. We also show that the obtained conjugacyis O(hp)-close to the identity, where p is the order of the numerical method. De-tails are placed into Subsection 3.3. To the best of our knowledge this is the firstresult concerning conjugacy between true and approximating orbits of retardedfunctional differential equations, similar results for ordinary differential equationswere proved in [2] and [5]. A more general ODE result on numerical structuralstability can be found in [9].
We begin with recalling the necessary part of the theory of functional differen-tial equations.
Consider an autonomous functional differential equation of the form
x(t) = f(xt), (1)
where xt(θ) := x(t+θ), θ ∈ [−r, 0], r > 0 fixed, and C := C([−r, 0];Rn). Equippedwith the usual supremum norm ‖·‖, C is a Banach space. Assume that f : C → Rn
is of class Ck (k ≥ 1) and f(0) = 0. Assume that all solutions exist for all t ≥ 0.With xt(ϕ) denoting the solution starting from ϕ define the solution operator of(1) by Φ(t)ϕ := xt(ϕ). It is known that {Φ(t) : t ≥ 0} is a semidynamicalsystem and Φ(t) : C → C is Ck for all t ≥ 0, see [7]. Denote the linearization ofΦ(t) around 0 by T (t), i.e. T (t) = DΦ(t)0. Then {T (t) : t ≥ 0} is a stronglycontinuous semigroup of linear operators and is the solution operator of the linearvariational equation of (1) around 0, see [7].
From now on assume that our equilibrium point 0 is hyperbolic and isolated.Then there exists a T -invariant splitting C = U ⊕ S, a renorming of C, and apositive constant ω such that T (t) extends to a group on U and
‖T (t)ϕ‖ ≤ exp(ωt)‖ϕ‖ for all ϕ ∈ U and t ≤ 0 (2)
and‖T (t)ϕ‖ ≤ exp(−ωt)‖ϕ‖ for all ϕ ∈ S and t ≥ 0. (3)
For renorming we refer to Theorem 5.1. on page 108 in [13].Let C+ be a finite dimensional realified generalized eigenspace of T . Note that
T |C+ extends to a group on C+ and C decomposes as the direct sum C = C+⊕C−
Vol. 52 (2001) Hartman-Grobman theorem and discretization 423
of T -invariant subspaces. For example C+ = U , C− = S but C+ could correspondto any compact spectral subset.
Define η(t) = Φ(t)−T (t), let t0 > 0 be fixed and set BR(0) = {ϕ ∈ C : ‖ϕ‖ ≤R}, R > 0.
Lemma 1. Given any ε > 0 there is a neighborhood V of 0 in C and a positivenumber δ such that the function η(t) is Lipschitz with constant
Lip(η(t)|V ) ≤ εt for t ∈ [0, δ).
Proof. Write f(ϕ) = Df(0) · ϕ + q(ϕ) and let V ∗ be a neighborhood of 0 in Cand κ := ‖Df(0)‖, λ := Lip(q|V ∗). By continuity there is a positive number δ anda neighborhood V of 0 in C such that Φ(t)(V ) ⊂ V ∗ and T (t)(V ) ⊂ V ∗ for allt ∈ [0, δ). Let µ := κ + λ and observe that Lip(f |V ∗) ≤ µ.
First we prove that Lip(Φ(t)|V ) ≤ exp(µt) for all t ∈ [0, δ). For θ ∈ [−r, 0],t + θ ≥ 0, and for all ϕ1, ϕ2 ∈ V we find
‖Φ(t)ϕ1(θ)− Φ(t)ϕ2(θ)‖ ≤ ‖ϕ1(θ)− ϕ2(θ)‖+∫ t+θ
0‖f(Φ(s)ϕ1)− f(Φ(s)ϕ2)‖ds
≤ ‖ϕ1 − ϕ2‖+ µ
∫ t+θ
0‖Φ(s)ϕ1 − Φ(s)ϕ2‖ds
≤ ‖ϕ1 − ϕ2‖+ µ
∫ t
0‖Φ(s)ϕ1 − Φ(s)ϕ2‖ds.
On the other hand, for θ ∈ [−r, 0], t + θ < 0, and for all ϕ1, ϕ2 ∈ V we find
On the other hand, for θ ∈ [−r, 0], t + θ < 0, and for all ϕ1, ϕ2 ∈ V , we find
‖η(t)ϕ1(θ)− η(t)ϕ2(θ)‖ = 0.
Therefore,
‖η(t)ϕ1 − η(t)ϕ2‖ ≤ κ
∫ t
0‖η(s)ϕ1 − η(s)ϕ2‖ds +
λ
µ(exp(µt)− 1)‖ϕ1 − ϕ2‖,
and by Gronwall’s inequality,
‖η(t)ϕ1 − η(t)ϕ2‖ ≤ (exp(µt)− exp(κt))‖ϕ1 − ϕ2‖.
Finally, observe that by shrinking the neighborhood V ∗, we can make the Lipschitzconstant of η arbitrarily small. This completes the proof of Lemma 1. ¤
Now we are in a position to formulate our main result.
Theorem 1. There exists a neighborhood W of 0 in C+ and a homeomorphismH of W onto H(W ) ⊂ C such that
Φ(t0) ◦H(ϕ) = H(T (t0)ϕ)
whenever ϕ, T (t0)ϕ ∈ W .
Remark 1. From Theorem 1 it follows that we can choose a neighborhood W ⊂W of 0 in C+ such that Φ(t0) is one-to-one from H(W ) onto (Φ(t0) ◦ H)(W )and can be defined for t = −t0 by setting Φ(−t0) = H ◦ T (−t0) ◦ H−1 whereT (−t) = (T (t)|C+)−1.
Remark 2. Note that Theorem 1 (and its proof) applies to linearizing noninvert-ible mappings of a Banach space as well.
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2. Proof of Theorem 1
First we consider the following functional equation on C+
Φ(t0) ◦ (id + g) = (id + g) ◦ T (t0), (4)
where g : C+ → C is the unknown function. For brevity, we write L = T (t0), a =exp(−ωt0) and e = η(t0). With a slight abuse of notation, (L|C+)−1 is abreviatedby L−1. Note that a ∈ (0, 1) and choose a constant c > 1 with ‖L−1‖ ≤ c.
Then (4) is equivalent to
g = L ◦ g ◦ L−1 + e ◦ (id + g) ◦ L−1. (5)
Let ϕ = ϕU + ϕS be the decomposition of ϕ in U ⊕ S. Let C(C+, C) bethe space of bounded continuous functions from C+ into C. Note that everyg ∈ C(C+, C) decomposes as g = gU + gS where gU (ϕ) = (g(ϕ))U and gS(ϕ) =(g(ϕ))S . Equipped with norm ‖g‖ = max{‖gU‖, ‖gS‖}, C(C+, C) is a Banachspace.
Define the linear operator L : C(C+, C) → C(C+, C) by L(g) := L ◦ g ◦ L−1.
Lemma 2. The operator id− L is invertible and ‖(id− L)−1‖ ≤ (1− a)−1.
Proof. It is easy to see that ‖Lϕ‖ ≥ a−1‖ϕ‖ for all ϕ ∈ U and ‖Lϕ‖ ≤ a‖ϕ‖ forall ϕ ∈ S. Since C(C+, C) = C(C+, U)⊕C(C+, S) is an L-invariant splitting and
‖Lg‖ ≥ a−1‖g‖ for all g ∈ C(C+, U)
and‖Lg‖ ≤ a‖g‖ for all g ∈ C(C+, S)
by the exponential estimates (2) and (3), the usual Neumann series argument, seeLemma 7.2. on page 160 in [13], shows that (id−L)−1 exists and ‖(id−L)−1‖ ≤(1− a)−1. ¤
We solve equation (5) on a small neighborhood of 0 in C+. With constantK > 0 specified later, we set V = BK(0) ∩ C+ and define
CK(V,C) := {g : V → C : g continuous , ‖g‖ ≤ K}
andCK,a(V,C) := {g : V → C : Lip(g) ≤ a/(2− a), ‖g‖ ≤ K},
respectively.Our strategy is to apply the contraction mapping principle in CK(V,C). In
order to overcome the difficulties caused by L−1(V ) 6⊂ V , we introduce the radial
426 G. Farkas ZAMP
extension of functions of class CK(V,C) and CK,a(V,C). Define for g ∈ C(V,C)the function g ∈ C(C+, C) by letting g(ϕ) = g(ϕ) if ‖ϕ‖ ≤ K and g(ϕ) :=g(Kϕ/‖ϕ‖) if ‖ϕ‖ > K. It is routine to verify (V is a ball) that the operator · islinear, ‖g‖ = ‖g‖ and Lip(g) ≤ 2KLip(g). For g ∈ CK(V,C) define
Θ(g) = [(id− L)−1(e ◦ (id|V + g) ◦ L−1)]|V .
(If C+ = U then L−1(V ) ⊂ V and the proof can be made simpler.) SinceDη(t0)0 = 0, there is a ball B2K(0) such that Lip(η(t0)|B2K(0)) ≤ ((1− a)/(4c))a.Moreover we can assume that K ≤ 1.
Lemma 3. The operator Θ has a fixed point g in CK,a(V,C) which is the uniquefixed point in CK(V,C).
Proof. First we prove that Θ : CK,a(V,C) → CK,a(V,C). On the one hand
Observe that CK,a(V,C) is a Θ-invariant closed subset of CK(V,C). It remainsto prove that Θ is a contraction on CK(V,C). To this end let g1, g2 ∈ CK(V,C).It is readily obtained that
Let H := id|V + g and choose a ball V1 around 0 in C+ such that V1, L−1(V1),L(V1) ⊂ V .
Lemma 4. H : V → H(V ) is a homeomorphism, and
Φ(t0) ◦H(ϕ) = H(Lϕ)
for all ϕ ∈ V1.
Proof. Lip(g) ≤ a/(2 − a) < 1 thus g is a contraction. But id+ contraction isalways a homeomorphism.
Let G := (id − L)−1(e ◦ (id|V + g) ◦ L−1). Then G − L ◦ G ◦ L−1 + id − id =e◦(id|V + g)◦L−1 and for all ϕ ∈ V1, (id+G)(Lϕ) = L◦(id+G)(ϕ)+e◦(id|V +g)(ϕ).Since G(ϕ) = g(ϕ) and G(Lϕ) = g(Lϕ) we obtain that Φ(t0) ◦ (id|V + g)(ϕ) =(id|V + g)(Lϕ). ¤
3. Applications
We use conditions and notation introduced in Sections 1 and 2.
3.1. A Hartman-Grobman result around hyperbolic equilibria
Theorem 2. There exists a neighborhood W of 0 in C+, an ε > 0, and ahomeomorphism H of W onto H(W ) ⊂ C such that H(0) = 0 and
Φ(t) ◦H(ϕ) = H(T (t)ϕ)
whenever t ∈ [0, ε) and ϕ, T (t)ϕ ∈ W .
Proof. Let β : R → [0, 1] be a C∞ bump function with β(x) = 1 if |x| ≤ 1 andβ(x) = 0 if |x| > 2. Then there is a uniform M ≥ 1 such that |β′(x)| ≤ M for allx ∈ R.
First we prove that given any ε1 > 0, there is a neighborhood V of 0 in C anda function f : C → Rn with f |V = f |V , and Lip(f −Df(0)) < ε1. Moreover, wecan assume that the only equilibrium point in V is 0.
Write f = Df(0) + q. Take R > 0 such that ‖Dq(ϕ)‖ < ε1/(4M) for allϕ ∈ B2R(0). Finally, let q(ϕ) = β(‖ϕ‖/R)q(ϕ) and f(ϕ) = Df(0) · ϕ + q(ϕ).
It remains to check the Lipschitz bound of q. Since q = 0 outside B2R(0), wecan assume ϕ1, ϕ2 ∈ B2R(0) in the following calculation of the Lipschitz constant:
Denote the solution operator of the RFDE with right hand side f by Φ andlet η(t) := Φ(t)− T (t). Note that the only equilibrium point of Φ is 0. The proofof Lemma 1 shows that Lip(η(t)) ≤ exp((κ + ε1)t)− exp(κt). Thus, if ε1 is smallenough we can achieve that Lip(η(t0)) < ((1− a)/(4c))a. Set e = η(t0).
We solve the functional equation Φ(t0) ◦ (id|C+ + g) = (id|C+ + g) ◦ L onC+. Define the operator Θ : C(C+, C) → C(C+, C) by letting Θ(g) = (id −L)−1(e ◦ (id|C+ + g) ◦ L−1). Then our functional equation is equivalent to thefixed point setting g = Θ(g). An argument similar to the one used in Section2 shows that Θ possesses a fixed point g in Ca(C+, C) = {g : C+ → C :g is bounded and Lip(g) ≤ a/(2 − a)} which is unique in C(C+, C). Moreover,H = id|C+ + g is a homeomorphism.
Observe that
Φ(t0) ◦ Φ(t) ◦H ◦ T (−t) = Φ(t) ◦ Φ(t0) ◦H ◦ T (−t)
Φ(t) ◦H ◦ L ◦ T (−t) = Φ(t) ◦H ◦ T (−t) ◦ L
on C+. The conjugacy H is unique among maps for which H−id|C+ is a continuousbounded map. To see that Φ(t) ◦H ◦ T (−t)− id|C+ is bounded, note that
Φ(t) ◦H ◦ T (−t)− id|C+ = η(t) ◦H ◦ T (−t) + T (t) ◦ g ◦ T (−t).
The second quantity on the right hand side is bounded because g is bounded. Bycontinuity we can choose a positive ε such that η(t) is bounded for all t ∈ [0, ε), sothe first quantity on the right hand side is bounded for t ∈ [0, ε). By uniquenesswe obtain that
Φ(t) ◦H = H ◦ T (t) for t ∈ [0, ε)
on C+. From Φ(t)◦H(0) = H(T (t)0) = H(0) it follows that H(0) is an equilibriumpoint and thus H(0) = 0.
By restricting to a sufficiently small neighborhood of 0, we have proved Theo-rem 2. ¤
It is natural to ask whether we can take C+ = C. The following simple exampleshows that the answer is negative without any further hypotheses. Consider thescalar delay equation
x(t) = x(t) + x2(t− 1).
The zero solution is a hyperbolic equilibrium point. Assume that there exists ahomeomorphism H of a 0-neighborhood W onto H(W ) such that H(0) = 0 and
Vol. 52 (2001) Hartman-Grobman theorem and discretization 429
Φ(t) ◦H = H ◦ T (t) for t small enough. It is easy to see that if ϕ ∈ C such thatthere is a τ > 0 with Φ(τ)ϕ = 0 then ϕ = 0. On the other hand, for all functionswith ϕ(0) = 0 we have T (t)ϕ = 0 for some t > 0. Choose a function ϕ ∈ W suchthat ϕ 6= 0 and ϕ(0) = 0. Then
Φ(t) ◦H(ϕ) = H(T (t)ϕ) = H(0) = 0
and thus H(ϕ) = 0 = H(0) is a contradiction.
3.2. A Hartman–Grobman result around hyperbolic periodic orbits
Let Γ = {γα : α ∈ R+} be a hyperbolic periodic orbit of (1) with period τ > r,i.e.
C = E(α)⊕ U(α)⊕ S(α)
where E(α) is the one-dimensional eigenspace of DΦ(τ)γα to eigenvalue 1, U(α)and S(α) are closed subspaces of C that are invariant under DΦ(τ)γα and thespectrum of the restriction of DΦ(τ)γα to U(α) and S(α) is strictly outside andinside the unit disk, respectively, see [7]. Let C+(α) be a finite dimensional realifiedgeneralized eigenspace of DΦ(τ)γα.
Around a periodic orbit we must allow reparameterization which means thatwe obtain topological equivalence instead of conjugacy. Details are contained in
Corollary 1. Let Γ be a hyperbolic periodic orbit of (1). Then for all α ∈ [0, τ)there exists a neighborhood Wα of 0 in C+(α), a homeomorphism Hα of Wα ontoH(Wα) ⊂ U(α)⊕ S(α), and a function σα : γα + Wα → R+ such that
Proof. Fix an α ∈ [0, τ). Let P (α) := U(α) ⊕ S(α) + γα. The global chartϕ 7→ ϕ− γα makes P (α) a C1-submanifold of C and for all ϕ ∈ P (α), TϕP (α) =U(α) ⊕ S(α) where TϕP (α) denotes the tangent space of P (α) at ϕ. Denote thePoincare map corresponding to P (α) by πα : D(πα) ⊂ P (α) → P (α). Then γα
is a fixed point of πα and there exist a function σα : D(πα) → R+ such thatπα(ϕ) = Φ(σα(ϕ))ϕ. As a special case of Proposition 3.2. on page 370 in [3], weobtain that Dπα(γα) ·χ = DΦ(τ)γα ·χ for all χ ∈ U(α)⊕S(α). Thus the Lipschitzconstant of ([Φ(σα(γα + ·)) ◦ (γα + ·)− γα]−DΦ(τ)γα)|V can be made arbitrarilysmall by shrinking the neighborhood V of 0 in U(α) ⊕ S(α). Since DΦ(τ)γα isinvertible on C+(α), the proof of Theorem 1 shows the desired result. ¤
430 G. Farkas ZAMP
3.3. Conjugacy between true and approximate orbits
Let Φh : Ch → Ch be a Ck-function for 0 < h ≤ h0 where Ch are closed subspacesof C with projection Ph : C → Ch. We assume that ∪0<h≤h0
Ch is dense in Cand ‖Ph‖ ≤ M for all h. The family (Φh)0<h≤h0
is called a numerical methodof order p for Φ if for every bounded subset of C there is a constant K1 (whichmay depend on the bounded set) such that dC1(ΦhPh,Φ(h)) ≤ K1h
p+1, wheredC1 is the usual distance between C1-functions defined on the given bounded set.The parameter h is called the step-size of the method. These conditions can besatisfied if Φh comes from a p-th order Runge–Kutta method applied to a delayequation, for details we refer to [8]. We assume that 0 is a fixed-point of ΦhPh.We note that in the usual definition of the order of a numerical method the dC0
distance appears instead of the dC1 distance, see [6]. We have
Corollary 2. Let p ≥ 1. For every h small enough there exist a neighborhood Wof 0 in C+ which is independent of h, and homeomorphisms Hh : W → Hh(W )such that
Proof. We only have to rewrite the proof of Theorem 1 with parameter h. Forthe reader’s convenience we recollect symbols used in Section 2 and we display thedependence on h: ah := exp(−ωh), η(h) = Φ(h) − T (h), and for g ∈ C(C+, C)we define Lh(g) = T (h) ◦ g ◦ T (h)−1. The operator (id − Lh) is invertible and‖(id − Lh)−1‖ ≤ (1 − ah)−1, see the proof of Lemma 2. Note that there existsa constant K2 such that for all h small enough ‖(id − Lh)−1‖ ≤ K2/h. Choosea constant c > 1 such that ‖(T (h)|C+)−1‖ ≤ c for all h small enough. FromLemma 1 it follows that there is a ball B2K(0) which is independent of h andLip(η(h)|B2K(0)) ≤ ((1−ah)/(8c))ah. Set V = BK(0)∩C+ and define the operatorΘ(h)(g) := (id−Lh)−1(η(h)◦ (id|V + g)◦T (h)−1). The proof of Theorem 1 showsthat there exist a neighborhood W1 of 0 in C+ which is independent of h, and ahomeomorphism Hh such that
Φ(h) ◦Hh(ϕ) = Hh(T (h)ϕ)
whenever ϕ, T (h)ϕ ∈ W1.Write ΦhPh = Φ(h) + eh = T (h) + η(h) + eh and define the corresponding
p+1 we obtain forall h small enough that Lip(η(h)+eh)|B2K(0) ≤ ((1−ah)/(8c))ah+K1h
p+1 ≤ ((1−ah)/(4c))ah. Now the proof of Theorem 1 shows that there exist a neighborhoodW2 of 0 in C+ which is independent of h, and a homeomorphism Gh : W2 →Gh(W2) such that
ΦhPh ◦Gh(ϕ) = Gh(T (h)ϕ)
Vol. 52 (2001) Hartman-Grobman theorem and discretization 431
whenever ϕ, T (h)ϕ ∈ W2.Set W3 = W1 ∩ W2 and define Hh : Hh(W3) → Gh(W3) by setting Hh(ϕ) =
Gh((Hh)−1(ϕ)). Observe that ΦhPh ◦Gh ◦ (Hh)−1(ϕ) = Gh ◦T (h) ◦ (Hh)−1(ϕ) =Gh ◦ (Hh)−1 ◦ Φ(h)(ϕ) whenever ϕ,Φ(h)ϕ ∈ Hh(W3) and T (h)(Hh)−1(ϕ) ∈ W3.This identity proves the first part of Corollary 2.
It remains to show that ‖Gh − Hh‖ = O(hp). In order to compare the fixed-points we compare the fixed-point equations (we use the Parameterized Contrac-tion Mapping Principle) as
Note that with C+ = U a discretized unstable manifold theorem can be ob-tained as a special case of Corollary 2. A more direct method deriving such aresult can be found in [4].
Acknowledgement
The author would like to express his thanks to Dr. B.M. Garay for his valuableremarks on an earlier version of this paper. The author is grateful to the refereefor his suggestions to improve the presentation of the paper.
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Gyula FarkasDepartment of MathematicsTechnical University of BudapestH-1521 Budapest, Hungarye-mail: [email protected]
(Received: October 13, 1999; revised: February 11, 2000)
