The parameterization method for invariant manifolds III: overview and applications
Abstract
We describe a method to establish existence and regularity of invariant manifolds and, at the same time to find simple maps which are conjugated to the dynamics on them. The method establishes several invariant manifold theorems. For instance, it reduces the proof of the usual stable manifold theorem near hyperbolic points to an application of the implicit function theorem in Banach spaces. We also present several other applications of the method.
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THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS III: OVERVIEW AND APPLICATIONS XAVIER CABR´ E, ERNEST FONTICH, AND RAFAEL DE LA LLAVE Abstract. We describe a method to establish existence and regularity of invariant manifolds and, at the same time to find simple maps which are conjugated to the dynamics on them. The method establishes several invariant manifold theorems. For instance, it reduces the proof of the usual stable manifold theorem near hyperbolic points to an application of the implicit function theorem in Banach spaces. We also present several other applications of the method. Index 1. Introduction 2. The parameterization method 3. Main results Part I: 4. Analytic one-dimensional stable manifolds 5. One-dimensional stable directions around periodic orbits of analytic differential equations 6. A C0invariant stable manifold theorem Part II: 7. Crone-dimensional invariant manifolds 8. A C0slow manifold theorem 9. Non-resonant invariant manifolds for maps 10. Non-resonant invariant manifolds for differential equations Appendix A. Remarks on cohomology equations and nonuniqueness of invariant manifolds Appendix B. Historical remarks and information on the literature on nonresonant invariant manifolds 1. Introduction The goal of this paper is to present a tutorial on “the parameterization method”, a technique recently introduced by the authors [CFdlL03a, CFdlL03b] to study invariant manifolds of dynamical systems. As a first simple application, the method allows to give quick proofs of stable and unstable manifold theorems. 1
2 X. CABR´ E, E. FONTICH, AND R. DE LA LLAVE More importantly, it leads to new results on existence of invariant manifolds, as well as on their regularity and dependence on parameters. To be more precise, the parameterization method allows to establish the existence of smooth invariant manifolds associated to linear subspaces invariant by the linearization and which satisfy some non-resonance conditions. As a novelty with respect to previous works, the invariant linear subspaces need not be spectral subspaces. Even further, they need not have an invariant complement. The parameterization method lends itself to very efficient computer implementations since it provides a global representation of the manifold, and it also allows a very efficient discussion of dependence on parameters. Some extensions of the method to quasi-periodic systems and numerical implementations are presented in [HdlL03b, HdlL03a]. In this paper we highlight the main geometric ideas and invariant objects obtained, as well as the main technical tools (Banach spaces with norms tailored to the problem, differentiability results of composition operators, implicit function and fixed point theorems, cohomology equations). Hence, we have not included the optimal regularity results, neither the technical ideas needed to obtain them. The interested reader may find these in [CFdlL03a, CFdlL03b]. Some variants of the method seem to have appeared in fragmentary form in the work of Poincar´e on automorphic forms [Poi90], later in his research on dynamics, and also in the work on Lyapunov [Lya92]. Of course, modern techniques such as implicit function theorems on Banach spaces were not available at that time, which made these works quite fragmentary and full of restrictions. In some particular applications (specially in relation with numerical calculations), the method seems to have been rediscovered several times, again under extra restrictions. In Appendix B we comment on these historical matters. In Section 2 we describe the basic ideas and objects of the method, both for dynamical systems given by maps and for those given by ordinary differential equations. Section 3 describes the main result of [CFdlL03a], stated both for maps and for differential equations. We have tried that each of the sections after Section 3 could be read independently of each other. Each of them presents a full proof of one result that illustrates some of the main ideas involved with the method. The applications in Part I (sections 4,5, and 6) are simpler, while the results in Part II (sections 7,8,9, and 10) are sharper and more delicate. Finally, we have included two appendices with comments on cohomology equations, nonuniqueness of invariant manifolds (an important point when doing numerical computations), and historical remarks on the literature of the subject, including applications. 2. The parameterization method 2.1. The parameterization method for maps. Given a map F:U⊂Rd→ Rdwith F(0) = 0, where Uis an open set containing the origin, a natural way to try to find a manifold invariant under Fand modeled on a subspace E⊂Rd,
THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS III 3 is to look for an embedding K:U1⊂E→Rdand a map R:U1⊂E→U1in such a way that F◦K=K◦R. (2.1) The fact that the manifold K(U1) passes through the origin is ensured by requiring K(0) = 0.(2.2) The fact that the manifold is tangent at the origin to Eis guaranteed by requiring DK(0)E=E. (2.3) Note that (2.1) ensures that the range K(U1) of Kis invariant under F. We think of Kas giving a parameterization of the invariant manifold K(U1). Moreover, Ris the dynamics of Frestricted to the invariant manifold. Note also that differentiating (2.1) at the origin and using (2.2) and (2.3), we deduce DF(0)E⊂E. Thus, Emust be an invariant subspace under the linearization DF(0) of Fat the origin. The fact that Ris a representation (in some appropriate coordinates) of the dynamics of the map Frestricted to the manifold, tells us that we need to consider it as part of the objects to be determined (or at least to be flexible about its choice) since, depending on the nonlinear terms, the dynamics on the stable manifold (for instance on the classical stable manifold) may belong to different equivalence classes under smooth conjugacy. An important observation is that if we consider T(F, K, R) := F◦K−K◦R, (2.4) and write equation (2.1) as T(F, K, R) = 0, then Tis differentiable in Kwhenever Kis given the topology of Crspaces (provided that Fis sufficiently differentiable). Hence, equation (2.1) can be studied via the standard implicit function theorem in Banach spaces, even in the standard Crspaces. This leads very quickly and painlessly to some results on existence and differentiability with respect to parameters for finitely differentiable maps. This is the approach undertaken in this article (with the exception of Section 5, where we use the fixed point theorem for contractions). In [CFdlL03a] we used instead fixed point theory in jet spaces in order to obtain optimal regularity results. The fact that Tis differentiable is in contrast with the functional equations that one has to deal with in the graph transform method, in which the operator whose fixed point gives the invariant graph is not differentiable in any of the classical Crspaces, even if it is differentiable in spaces of analytic functions (see [Mey75]). The linearized version of (2.4) with respect to Kis formally (and rigorously, under regularity cases characterized below and in more generality in [dlLO99]): D2T(F, K, R)∆ = (DF ◦K)∆ −∆◦R. (2.5)
4 X. CABR´ E, E. FONTICH, AND R. DE LA LLAVE The equations for ∆ obtained setting D2T(F, K, R)∆ = ηare called cohomology equations. We describe some aspects related to them in Appendix A. These equations have a very rich history (see e.g. [BdlLW96]). Once a theory for the linearized equations is established, one can study the full nonlinear equation (2.1) using a variety of methods (contraction mappings, implicit function theorems, deformation methods, etc.) 2.2. The parameterization method for flows. Very similar ideas to those used in the proof of the results for maps can be used to study invariant manifolds for differential equations x0=X(x). Here Xis a vector field in U⊂Rd, where Uis an open set containing the origin, with X(0) = 0. If Eis a subspace of Rdinvariant by DX(0), we look for a parameterization K:U1⊂E−→ Rdand a vector field Rin U1⊂Esuch that X ◦ K=DK ·R. (2.6) That is, we ask the vector field Xon the image of Kto be the pull forward of the vector field Rin E. Equation (2.6) expresses that at the range of K, the vector field Xis tangent to the range of K. Hence, the range of Kis invariant under the flow of X. Moreover, the vector field Ris the representation in parameters of the restriction of Xto the invariant manifold. This direct study, which will be described in Section 10, is illuminating and, for practical calculations advantageous. Nevertheless those interested mainly in existence and regularity results may prefer an abstract argument which shows that the results for differential equations follow from the results for maps. This indirect approach is as follows. If {ϕt}t∈Ris the flow associated to Xand Wis a manifold such that ϕ1(W)⊂ W(that is, Wis an invariant manifold for the map ϕ1), then we have ϕ1(ϕt(W)) = ϕt(ϕ1(W)) ⊂ϕt(W).(2.7) If the invariant manifold theorem for the map ϕ1includes local uniqueness under hypothesis that are invariant under the evolution by ϕt, since from (2.7) we obtain that ϕt(W) is also invariant under ϕ1, we conclude that ϕt(W)⊂W. That is, Wis also invariant under ϕt, for all t > 0. We point out that in all discussions in this section it makes no difference to replace Rdby a Banach space X. In the following sections we work in Rdexcept in sections 6 and 8, where we deal with invariant manifolds in Banach spaces. 3. Main results In this section we present the statement of two theorems on non-resonant invariant manifolds associated to a fixed point. We have selected them as representatives of the results of the paper. However, in the paper we deal with other
THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS III 5 results which range from simpler to more difficult situations, and also with an invariant manifold theorem associated to periodic orbits of vector fields. To state the results, we first recall some standard terminology. The spectrum of a linear operator Ain Rdwill be denoted by Spec(A). We emphasize that Spec(A) denotes the spectrum of the complex extension of A, and hence Spec(A) is a compact subset of C. For j∈Nand S⊂C, we use the notation jS := {a1+· · · +aj|ai∈S}. We use a similar notation for sum, difference, and product of sets. We say that a function is in Cωif it is analytic. 3.1. Manifolds associated to a fixed point of a map. The heuristic idea is that, given a map Fin Rd, with F(0) = 0, to every linear subspace E⊂Rd invariant under DF(0), there should correspond a smooth manifold invariant under the map F, passing through the origin and tangent there to E. Of course, this should be true if the map Fis smoothly linearizable but, as the stable (or strong stable) manifold theorems show, the hypothesis of linearizability is much stronger than needed. Nevertheless, as shown in examples in [dlL97], some nonresonance conditions are necessary for the existence of an invariant manifold. The non-resonance conditions (hypothesis 3) in the following theorem) consist of certain hypotheses on the spectrum of DF(0). They are automatically satisfied when dealing with the stable or strong stable manifold theorems. The following is the result concerning non-resonant invariant manifolds for maps. Its proof is given in Section 9. Theorem 3.1. Let F:U⊂Rd→Rdbe a Cr+1 map in a neighborhood Uof the origin, with F(0) = 0, and r∈N∪ {ω}. Denote A=DF(0). Let L∈N,L≥1. Assume that: 1) There is a linear subspace Eof Rdsuch that A(E)⊂E. Hence there is a decomposition Rd=E⊕Cand, with respect to it, Ahas the form A=AEB 0AC.(3.1) 2) kAEk<1. 3) Spec(AE)j∩Spec(AC) = ∅for j= 2, . . . , L. 4) Ais invertible. 5) (Spec(AE))L+1 Spec(A−1)⊂ {z∈C| |z|<1}. 6) L+ 1 ≤r. Then, there exist a Crmap K:U1⊂E→Rd, where U1is an open neighborhood of 0in E, and a polynomial R:E→Eof degree at most L, such that F◦K=K◦Rin U1, K(0) = 0, DK(0)E=E, R(0) = 0, DR(0) = AE. (3.2)
6 X. CABR´ E, E. FONTICH, AND R. DE LA LLAVE Theorem 3.1 is a simplified, but quite close, version of the main result of [CFdlL03a]. In that paper, the result was stated in general Banach spaces instead of just in Rd. The regularity results on dependence with respect to parameters were studied in [CFdlL03b]. We remark that the loss of one derivative in the regularity of the manifold as stated in Theorem 3.1 above (in which Fis a assumed to be Cr+1, but the manifold obtained is just Cr), can be improved. In [CFdlL03a] the manifold is proved to have the same finite differentiability as the map, that is, no derivative is lost. To obtain this sharp result one needs to consider equation (2.1) as a fixed point problem enjoying special properties, and study in detail the convergence to the limit —rather than applying the implicit function theorem as we do in the present paper. 3.2. Manifolds associated to a fixed point of a vector field. As in the case for maps, if Xis a vector field in U⊂Rd, with X(0) = 0, and E⊂Rdis a linear subspace invariant under DX(0), there may correspond a smooth manifold invariant under the flow of X, passing through the origin and tangent to Eat it. The analogous remarks of the previous subsection apply in this case. The corresponding result is the following theorem. Its proof is given in Section 10. Theorem 3.2. Let Xbe a Cr+1 vector field on an open set Uof Rdwith 0∈U, such that X(0) = 0 and r∈N∪{ω}. Let A=DX(0) and L∈N,L≥1. Suppose that: 1) There is a linear subspace Eof Rdsuch that A(E)⊂E. Hence there is a decomposition Rd=E⊕Cand, with respect to it, Ahas the form A=AEB 0AC. 2) Spec(AE)⊂ {z∈C|Re z < 0}. 3) jSpec(AE)∩Spec(AC) = ∅for j= 2, . . . , L. 4) Spec(−A)+(L+1) Spec(AE) = {−λ+µ1+· · ·+µL+1 |λ∈Spec(A)and µ1, . . . , µL+1 ∈Spec(AE)} ⊂ {z∈C|Re z < 0}. 5) L+ 1 ≤r. Then, there exist a Crmap K:U1⊂E−→ Rd, where U1is a neighborhood of 0in E, and a polynomial R:E−→ Eof degree at most L, such that X ◦ K=DK ·Rin U1,(3.3) K(0) = 0, DK(0)E=E, (3.4) R(0) = 0, DR(0) = AE.(3.5) Note that (3.3) ensures that the image K(U1) of Kis invariant by the flow of X. Condition (3.4) ensures that K(U1) passes through the origin and it is tangent to Ethere. We emphasize that the subspace Eneed not have an invariant complement. In addition, the spectrum of AEand that of ACneed not be disjoint, since
THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS III 7 condition 3) is only required for indices jbigger or equal than 2. For example, the theorem applies to A= −2 1 0−2 −3 −5 1 0−5 . Then, denoting by Eithe ith coordinate axis, we could associate invariant manifolds to E1, E3, E4,E1⊕E2,E4⊕E5, or to sums of these spaces, e.g., E1⊕E4, E1⊕E2⊕E4, etc. On the other hand, the result does not apply to E1⊕E3since the eigenvalues corresponding to E1and E3when added give an eigenvalue in the complement. Indeed, if we consider a non-linear part e4x1·x3, it is easy to see that there is no C2invariant manifold tangent to E1⊕E3. PART I 4. Analytic one-dimensional stable manifolds In this section we prove a theorem that serves as motivation for several results later. Indeed, the main ideas of future results appear here. As a matter of fact, a very similar theorem had been proved by Poincar´e [Poi90], which used a very different method (the majorant method) from the one used here. Clearly, the methods used here (Banach spaces and implicit function theorem) were not available at the time of [Poi90]. The theorem has been rediscovered in different guises in the literature, specially in relation with numerical calculations (see Appendix B). Theorem 4.1. Let F:U⊂Rd→Rdbe an analytic map in a neighborhood Uof 0, with F(0) = 0. Let λ∈Rbe an eigenvalue of A:= DF(0), and let v∈Rd\ {0}satisfy Av =λv. Assume: 1) Ais invertible. 2) 0 <|λ|<1. 3) λn/∈Spec(A)for every integer n≥2. Then, there exists an analytic map K:U1⊂R→Rd, where U1is an open neighborhood of 0in R, satisfying F(K(x)) = K(λx)in U1,(4.1) K(0) = 0, and K0(0) = v. Therefore, the image of Kis an analytic onedimensional manifold invariant under Fand tangent to vat the origin. Moreover, the dynamics on the invariant manifold is conjugated to the linear map x7→ λx in the space of parameters.
8 X. CABR´ E, E. FONTICH, AND R. DE LA LLAVE In addition, if b Kis another analytic solution of F◦K=K◦λin a neighborhood of the origin, with b K(0) = 0 and b K0(0) = βK0(0) for some β∈R, then b K(t) = K(βt)for tsmall enough. Remark 4.2. In the statement of the theorem, Spec(A) denotes the spectrum of A. Note that, since 0 /∈Spec(A) by 1), we have that Spec(A) excludes a ball of radius ρ. By 2), there is an integer n0such that |λ|n0< ρ. Now, if n≥n0 then condition 3) holds. Hence, hypothesis 3), even if it seems to require infinitely many conditions, all but n0of them are always fulfilled. Note also that n0is constant in open neighborhoods of A. This shows that hypothesis 3) (which is called a nonresonance condition) fails only on a manifold of maps Fof finite codimension. It turns out that condition 3) is necessary for having a manifold as the one claimed in the statement. Indeed, consider the map (x, y)7→ (1 2x, 1 4y+x2). An invariant manifold tangent to the vector (1,0) can be put as a graph of a function ϕ:U1⊂R→Rsatisfying the invariance condition ϕ(x/2) = ϕ(x)/4 + x2. Such ϕcan not be C2because taking two derivatives on both sides of the previous condition we get a contradiction. Remark 4.3. By considering F−1and λ−1in place of Fand λ, hypothesis 2) in Theorem 4.1 can be changed to |λ|>1. Remark 4.4. Note that we do not require λto be a simple eigenvalue. We could have that λhas other eigenvectors, linearly independent of v, or that vis an eigenvector in a non-trivial Jordan block. In the latter case, note that the invariant eigenspace generated by vdoes not have an invariant complement. Remark 4.5. Note that if F−1is entire (for instance, if F−1is a polynomial as it happens for the H´enon map), then Kis an entire function. Indeed, when F−1is entire, if Kis defined on a ball Bρ, (4.1) shows that K=F−1◦K(λ·) is defined on λ−1Bρ=Bλ−1ρ. Repeating the argument, the domain of definition of Kbecomes the whole plane. This was the motivation in [Poi90], namely, to construct entire functions which satisfied polynomial “duplication of angle formulas” similar to the familiar formulas for sin, cos, or for elliptic functions. The argument we present here leads to the construction of functions whose “double angle” values can be expressed —through (4.1)— as a given function of those of the “single angle”. One particularly interesting case of the situations covered by Theorem 4.1 is when kAk<1, λis simple and it is the eigenvalue of Aclosest to the unit circle. Note that, upon iteration of A, the component along vis the one that decays more slowly and hence, the one which controls the asymptotic behavior. The invariant manifold associated to this eigenvalue is a nonlinear analogue and can also be used to study the asymptotic behavior of the iterates of F. It is usually called a slow manifold.
THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS III 9 Proof of Theorem 4.1. Using power series, Fcan be considered as an analytic function in a neighborhood of 0 in Cd. Following the main idea of the parameterization method, we try to find K:D⊂C→Cdsuch that F◦K(z)−K(λz) = 0 (4.2) for z∈D, where Dis the unit disk of C. We write F(ζ) = Aζ +N(ζ) with A=DF(0). If we try to solve (4.2) equating powers of zon both sides, we obtain that K(z) = Pn≥1Knznshould satisfy AK1=λK1 AKn+Rn(K1, . . . , Kn−1) = λnKn, n ≥2,(4.3) where Rnis a polynomial expression obtained expanding the composition in (4.2). The first equation in (4.3) does not determine K1completely, only tells us that K1is an eigenvector of Awith eigenvalue λ. We take K1to be a multiple of v such that |K1|=δ, where δis small enough (we will give the precise smallness conditions on δlater, as they appear in the proof). Once we have chosen K1, (4.3) allows to determine in a unique fashion all the other Kn’s, as Kn=−(A−λn)−1Rn(K1, . . . , Kn−1), n ≥2.(4.4) The inverse used in (4.4) exists by assumption 3). One can indeed show, studying directly the recursion in (4.4), that the Knthus defined lead to an analytic function. This was the approach used in [Poi90]. We will, however, follow another route to study (4.2). We use techniques of functional analysis which can be adapted to other settings, such as dealing with a finitely differentiable F, or treating maps Fdefined in Banach spaces. We write K(z) = K1z+K>(z), and recall that we have already picked K1 and that it is small. The equation for K>(z) reads AK>(z) + N(K1z+K>(z)) −K>(λz) = 0.(4.5) We consider K>belonging to the Banach space Hof analytic functions in the unit disk, vanishing at the origin along with their first derivative, and with the following norm being finite: H=K>:D⊂C→Cd|K>(z) = ∞ X n=2 Knzn,kK>k:= ∞ X n=2 |Kn|<∞. We recall that the analytic functions f:D⊂C−→ Csuch that kfk:= Pn≥0|fn|<∞form a Banach algebra with the previous norm (see, e.g., [Car95] for a straightforward proof). Indeed, it suffices to apply the triangle inequality in the expression for the coefficients of the product (fg)n=Pi+j=nfigjand then sum in n. Of course, the ideal H={f|f0=f1= 0}is also a Banach algebra. We can reformulate (4.5) as an operator equation T(K1, K>) = 0,(4.6)
16 X. CABR´ E, E. FONTICH, AND R. DE LA LLAVE parameterization of the periodic orbit and an eigenvector vof Φ1of eigenvalue eλ, then the first and all higher order terms are uniquely determined. We can now state the result on invariant manifolds around periodic orbits. Theorem 5.4. Let Xbe an analytic vector field on Rdand assume that it admits a periodic orbit γof period T. Let γ=γ(θ),0≤θ≤1, be parameterized according to (5.1). Let Φ1be the monodromy matrix associated to γ. That is, Φ1= Φ(1), where Φθ= Φ(θ)is the fundamental solution of (5.8). Assume that λ∈Rsatisfies: 1) eλ<1and eλ∈Spec(Φ1). Let v∈Rd\ {0}be a solution of Φ1v=eλv. 2) enλ /∈Spec(Φ1)for every integer n≥2. Then, there exists an analytic two-dimensional manifold invariant under the flow of X, containing the periodic orbit γ, and tangent to the space generated by {˙γ(θ), K1(θ)}at γ(θ),0≤θ≤1, where K1(θ) = eµ(1−θ)ΦθΦ−1 1v. In addition, the manifold can be represented as the image of an analytic function Ksatisfying (5.2),K(θ, 0) = γ(θ)and ∂K ∂σ (θ, 0) = K1(θ). Consequently, the motion in the space of parameters is given by (5.3). In numerical applications, it is often enough to follow the procedure indicated previously in this section to obtain very approximate representations of the manifold. In the next subsection, we give a complete proof of Theorem 5.4. In particular, we develop estimates for the solutions of equations of the form (5.16) and we study the convergence of the power series, which will give confidence in the numerical analysis. Remark 5.5 explains how the main existence result is an ‘‘a posteriori” estimate which justifies the results of numerical calculations. 5.2. Convergence of the formal solutions. In this subsection, we show that the solution previously obtained is not just a formal solution, but that it converges and defines an analytic function. 5.2.1. Formulation as a fixed point problem. The proof of Theorem 5.4 consists in rewriting (5.2) as a well-posed fixed point problem. As in many of the proofs presented in the paper, we accomplish this by separating a low part from a high part of the solution. Here we separate the linear part, that we already have found. Then, once we show that there is a true analytic solution, by the uniqueness of the terms of order bigger or equal than 2, we see that the formal order by order calculation has to produce this analytic solution. We write K(θ, σ) = K0(θ) + K1(θ)σ+K>(θ, σ) K≤(θ, σ) = K0(θ) + K1(θ)σ. We assume that K0and K1are already determined (since we computed them just in the first two steps of the iterative procedure), so that the only unknown is K>.
THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS III 17 Equation (5.2) becomes 1 T ∂ ∂θ +λσ T ∂ ∂σ (K≤+K>) = X ◦ (K≤+K>), that we write as 1 T ∂ ∂θ +λσ T ∂ ∂σ (K0+K1σ+K>) = X ◦ K0+DX ◦ K0(K1σ+K>) + H(K>) (5.17) where His the remainder of Taylor’s theorem: H(K>) := X ◦ (K0+K1σ+K>)− X ◦ K0−DX ◦ K0(K1σ+K>).(5.18) Regrouping terms in (5.17) and using (5.5) and (5.6), (5.17) becomes L+λσ ∂ ∂σ K>=TH(K>).(5.19) The plan of the proof is to show that under the non-resonance conditions of Theorem 5.4, the operator L+λσ ∂ ∂σ is boundedly invertible in some appropriately defined spaces. Hence, (5.19) will become K>=L+λσ ∂ ∂σ −1 TH(K>) =: N(K>).(5.20) Equation (5.20) will be analyzed by fixed point methods. We introduce the notation Nto denote the operator for which we will be seeking fixed points. We will show that the Lipschitz constant of H, and hence of N, can be taken to be small if we take K1to be small. This is reasonable since His the second order remainder of the Taylor expansion. If rather than applying the contraction mapping theorem, we apply to (5.19) the implicit function theorem in Banach spaces, we obtain automatically smooth dependence on parameters for the invariant manifolds. Remark 5.5. Note that one of the conclusions of the fixed point method is that if we obtain an approximate solution of (5.20), that is, a function K> ap such that kK> ap − N(K> ap)k ≤ δ, (5.21) then there exists a true solution K>of the equation such that kK>−K> apk ≤ Cδ, where the constant Cwill be rather explicit from the proof. This can be used to justify numerical calculations. A careful numerical implementation of the algorithm discussed at the beginning of the section produces a function K> ap for which δin (5.21) is just the round off error plus the truncation error, which in several practical applications —e.g. [Cap04]— can be made to be a few thousand times the round-off unit. Hence, the proof presented here will ensure that the numerically computed solutions are close to the true one. This procedure is what is usually called “a posteriori estimates” in numerical analysis.
18 X. CABR´ E, E. FONTICH, AND R. DE LA LLAVE Moreover, we establish that the true solution is close to the computed one in an analytic norm defined below. Hence, the computed solution gives information not only about the location of the stable manifold, but also about its derivatives. This makes possible to discuss bifurcations, tangencies, etc. 5.2.2. Norms. In this section we introduce norms that are convenient to carry out the contraction mapping argument. Let αand βbe positive numbers. We consider the sets Sα={θ∈C/Z| |Im θ|< α} Dβ={σ∈C| |σ|< β} Uα,β =Sα×Dβ. Note that Sαcan be considered as a complex extension of the torus T1. Functions in Sαcan be identified with functions of period 1 defined in a complex strip. We consider the space of functions Γα={f:Sα→Cd|fcontinuous and analytic in Sα}. We endow Γαwith the norm kfkΓα= supθ∈Sα|f(θ)|. As it is well known, this norm makes Γαa Banach space. Given a function Kdefined on Uα,β which is analytic in both variables, we write it as K(θ, σ) = ∞ X n=0 Kn(θ)σn, and we denote by Hα,β the space of analytic functions for which kKkHα,β := ∞ X n=0 kKnkΓαβn<∞. It is an easy exercise to check that Hα,β is a Banach space with the norm k·kHα,β . We will use the same notation for norms of functions from these domains taking values in other spaces (e.g., matrices). When we consider functions which take values in spaces for which there is a multiplication (e.g., matrix valued functions and vector valued functions), the spaces Γαand Hα,β inherit Banach algebra properties. For example, if mand Mare d×dmatrix valued functions defined in Sαand Uα,β respectively, and if vand Vare d-dimensional vector valued functions defined in Sαand Uα,β respectively, then we have kmvkΓα≤ kmkΓαkvkΓα kMV kHα,β ≤ kMkHα,β kVkHα,β .(5.22) The first inequality in (5.22) is just that the supremum of the product is less than the product of the suprema. The second inequality is a consequence of the well known product formula for power series: MV = ∞ X n=0 σn n X k=0 MkVn−k.
THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS III 19 Therefore, kMV kHα,β = ∞ X n=0 βn n X k=0 MkVn−kΓα ≤ ∞ X n=0 βkβn−k n X k=0 kMkkΓαkVn−kkΓα =∞ X k=0 βkkMkkΓα∞ X i=0 βikVikΓα. Remark 5.6. There are other norms that we could have used in the proof. For example, we could take the suprema on both variables, which in some respects is more natural. The choice we have made is based on the observation that since the argument is based in solving equations for each coefficient in σ, it is natural to use a norm that emphasizes the role of the coefficients in powers of σ. At the same time, the supremum norm in the angle variables makes simpler to obtain estimates for expressions such as (5.11). The inequality sup (θ,σ)∈Uα,β |K(θ, σ)| ≤ kKkHα,β (5.23) is obvious from the triangle inequality. On the other hand, using Cauchy integral formula Kn(θ) = 1 2πi Z|z|=β z−(n+1)K(θ, z)dz, we obtain kKnkΓα≤β−nsup (θ,σ)∈Uα,β |K(θ, σ)|.(5.24) This immediately gives kKkHα,β−δ≤βδ−1sup (θ,σ)∈Uα,β |K(θ, σ)|.(5.25) Inequalities (5.23) and (5.25) allow us to use supremum to estimate derivatives. It is also convenient to think of functions in Hα,β as analytic functions on Dβ with values in the space Γα. Recall that K(θ, σ) = K0(θ) + K1(θ)σ+K>(θ, σ) and that we consider K in the space Hα,β. We therefore take the function K>in the space H2 α,β ={K>∈Hα,β |K>(θ, 0) ≡∂σK>(θ, 0) ≡0}. Equivalently, we are requiring that the two first coefficients (K>)0= (K>)1≡0 in the expansion of K>in powers of σ. We endow H2 α,β with the norm k·kH2 α,β = k·kHα,β inherited from Hα,β. Finally, we consider the operators Hand N, defined in (5.18) and (5.20), acting on functions K>in the space H2 α,β.
20 X. CABR´ E, E. FONTICH, AND R. DE LA LLAVE 5.2.3. Estimates for the linearized equation. We start establishing estimates in the space H2 α,β for solutions of the equation L+λσ ∂ ∂σ Ψ(θ, σ) = η(θ, σ).(5.26) This will be an easy task since (5.26) is equivalent, matching coefficients in σn, to (L+nλ)Ψn(θ) = ηn(θ), n ≥2.(5.27) The estimates for each of these coefficients can be readily obtained from Proposition 5.2. Lemma 5.7. Assume that, for some α > 0and some constants C1and C2, 1) kΦkΓα≤C1and kΦ−1kΓα≤C1, where Φθis the fundamental solution of the linearized equation. 2) µ < 0and eµis not an eigenvalue of the monodromy matrix Φ1. 3) k(Φ1−eµId)−1k ≤ C2. Then, the solution ∆of the linearized equation (5.10) satisfies k∆kΓα≤CkRkΓα,(5.28) where C=C2 1(1 + C2). The proof follows from formulas (5.11) and (5.12), using that µ < 0. From now on, we take α > 0 small enough so that γ=K0, Φ and Φ−1all belong to the space Γα, as in the previous proposition. Lemma 5.8. Let α > 0be small as indicated above, and let β > 0. Assume that λ < 0is such that enλ is not an eigenvalue of the monodromy matrix Φ1for every integer n≥2. Let η(θ, σ) = P∞ n=2 η(θ)σnbe a function in H2 α,β. Then, there is one and only one Ψ∈H2 α,β solving (5.26). In addition, it satisfies kΨkH2 α,β ≤CkηkH2 α,β , for some constant C. Proof. We use that (5.26) is equivalent to the system of equations (5.27), n≥2. Next, under the assumptions that λ < 0 and eλn is not an eigenvalue of Φ1 for n≥2, we can bound k(Φ1−eλn Id)−1kuniformly in n. Here we have used the non-resonance conditions together with the fact that Φ1−eλn Id →Φ1, which is invertible, as n→+∞. Hence, the bounds we obtain applying Lemma 5.7 to each of the coefficients Ψnare uniform in n. Therefore kΨnkΓα≤CkηnkΓαfor some constant C independent of n, from which the desired result follows. Now we turn to show that the maps Hand Ndefined in (5.18) and (5.20) are indeed well defined in the space H2 α,β. Moreover, we will show that if K1is chosen small enough, we can get a ball centered at 0 mapped by Ninto itself, and on which the Lipschitz constant of Nis small.
THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS III 21 Proposition 5.9. Let K0and K1be chosen as throughout this section. Let α > 0 be small as indicated above, and let β, r and ρbe positive numbers. Assume that: 1) The vector field Xis analytic and bounded in a domain that includes the complex ball of radius raround each point K0(θ),θ∈Sα. Let a:= sup θ∈Sα sup |z−K0(θ)|≤r |X(z)|. 2) kK1(θ)σkHα,β ≤ρ. 3) ρ≤r/4. Then: a) The map K>∈Bρ(0) ⊂H2 α,β 7→ X(K0+K1σ+K>)∈Hα,β is well defined, where Bρ(0) is the ball in H2 α,β of radius ρcentered at the origin. b) If K>∈Bρ(0) ⊂H2 α,β then H(K>) = X(K0+K1σ+K>)− X ◦ K0−DX ◦ K0(K1σ+K>) (5.29) belongs to H2 α,β. c) For every K>and ˜ K>in Bρ(0) ⊂H2 α,β, we have kH(K>)kH2 α,β ≤8ar−2ρ2,(5.30) kH(K>)− H(˜ K>)kH2 α,β ≤6ar−2ρkK>−˜ K>kH2 α,β .(5.31) Proof. By the analyticity properties of Xand hypothesis 1), we see that if X(K0(θ) + z) = X n≥0 Xn(θ)zn, we then have |Xn(θ)| ≤ ar−n; this is proved using Cauchy integral formula as in (5.24) before. Since the above inequality is true for any θ∈Sα, we have kXnkΓα≤ar−n. Using the Banach algebra properties of our spaces of functions, we see that X(K0(θ) + K1(θ)σ+K>(θ, σ)) = ∞ X n=0 Xn(θ)K1(θ)σ+K>(θ, σ)n(5.32) is well defined and converges uniformly, for K>∈Bρ(0). Indeed, the Hα,β-norm of each term in the series is bounded by kXnkHα,β (kK1(θ)σkHα,β +kK>kHα,β )n≤ ar−n(2ρ)n, which converges by assumption 3). Next, from (5.32) and the definition (5.29) of H, we see that H(K>)(θ, σ) = ∞ X n=2 Xn(θ)K1(θ)σ+K>(θ, σ)n.
22 X. CABR´ E, E. FONTICH, AND R. DE LA LLAVE The same argument as before establishes that this series converges, and hence H(K>) is well defined in H2 α,β. Moreover, using assumptions 2) and 3), we obtain kH(K>)kH2 α,β ≤ ∞ X n=2 kXnkHα,β kK1(θ)σkHα,β +kK>kHα,β n ≤ ∞ X n=2 ar−n(2ρ)n=a(2r−1ρ)21 1−2r−1ρ ≤8a(r−1ρ)2, as claimed in (5.30). Finally, we also have kH(K>)− H(˜ K>)kH2 α,β ≤ ∞ X n=2 kXnkHα,β (K1(θ)σ+K>(θ, σ))n−(K1(θ)σ+˜ K>(θ, σ))nHα,β ≤ ∞ X n=2 ar−nkK>−˜ K>kH2 α,β n(2ρ)n−1 =ar−1kK>−˜ K>kH2 α,β ∞ X n=2 n(2r−1ρ)n−1≤6ar−1r−1ρkK>−˜ K>kH2 α,β . This establishes (5.31). Note that assumptions 2) and 3) in the previous proposition can be accomplished either by taking K1small enough (recall that K1is defined up to a multiplicative constant), or by taking βsmall enough. Since N= (L+λσ ∂ ∂σ )−1TH, we see from estimates (5.30), (5.31) and Lemma 5.8 that if we choose ρsmall enough, we obtain a ball that gets mapped into itself by Nand on which the map Nis a contraction. Therefore, Nhas a unique fixed point in such ball. This finishes the proof of the result of this section. Remark 5.10. In [Mey75] one can find that the operator Nis actually analytic in the indicated spaces. Hence, we could use the implicit function theorem and obtain automatically smooth dependence on parameters. Remark 5.11. When one considers a discrete time dynamical system (i.e., a map) whose inverse is entire (e.g. polynomial), we argued in Remark 4.5 that the solutions Kis entire. In contrast, the solutions of polynomial differential equations usually are not entire, and very often they present essential singularities. Hence, when working with differential equations, one should not expect the coefficients to decay fast. Choosing the bas indicated in Remark 5.3 is quite important for numerical applications.
THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS III 23 6. AC0invariant stable manifold theorem In this section we prove a version of the stable manifold theorem. It is not optimal in terms of the regularity obtained, but its proof is simple and obtains differentiability with respect to parameters. In contrast to other sections in this paper, we formulate the results for maps in general Banach spaces. We also call attention to some rather subtle technicalities such as the fact that, in infinite dimensional spaces, the existence of smooth cut-off functions cannot be taken for granted. To state it, we use the following terminology. We say that a map is C1 uif it is of class C1and has uniformly continuous derivative. Recall that for a function defined in a finite dimensional space, if the function is C1in a neighborhood of a point then it is C1 uin a smaller neighborhood —since continuous functions in compact sets are uniformly continuous. Recall also that if a function is C1+εfor some ε > 0, then it is C1 u, even in infinite dimensions. Theorem 6.1. Let Xbe a Banach space, and F:U⊂X→Xbe a C1map in a neighborhood Uof 0, with F(0) = 0. Let A:= DF(0). Assume: 1) Ais an invertible operator. 2) There exists a decomposition X=Xs⊕Xu such that: 2.1) It is invariant under A. That is, AXs⊂Xs, AXu⊂Xu. 2.2) Let As:= πsA|Xsand Au:= πuA|Xu, where πsand πuare the projections onto Xsand Xu, respectively. Suppose that kAsk<1 and kA−1 uk<1. 3) If Xis infinite dimensional, assume that Xadmits smooth cut-off functions, and that Fis C1 u(U)(that is, DF is uniformly continuous in U). Then, there exists a continuous map K:U1⊂Xs→X, where U1is a neighborhood of 0, such that a) K(0) = 0. b) F◦K=K◦Asin U1. Moreover, assume that Fλis a C1family of C1 umaps (i.e., the map λ∈ V7→ Fλ∈C1 u, where Vis a neighborhood of 0in Λ, is C1when the maps Fλ are given the C1topology) with Fλ(0) = 0, and that F0satisfies the hypotheses above. Then, for λsmall enough, there exists a continuous map Kλsatisfying Kλ(0) = 0 and Fλ◦Kλ=Kλ◦A0,s in a neighborhood of the origin, where A0,s =πsA0|Xsand A0=DF0(0). In addition, the map λ7→ Kλis C1in a neighborhood of 0when the maps Kλare given the C0topology.
24 X. CABR´ E, E. FONTICH, AND R. DE LA LLAVE Remark 6.2. The hypothesis on cut-off functions, made in 3) above, means that there exists a C∞function ξ:X→Rwhich is identically 1 in the unit ball centered at 0 and identically 0 outside of the ball of radius 2 centered at 0. Such function always exists if Xis finite dimensional or a Hilbert space. It suffices to take Ψ(|x|), where Ψ is a C∞real valued function over the reals with the indicated properties. Perhaps surprisingly, the existence of smooth cut-off functions is not true for arbitrary Banach spaces. For example, C0[0,1] does not admit a C2cut-off function. We refer to [DGZ93]. The previous result, Theorem 6.1, is far from optimal in several respects. For example, the regularity can be improved, and the existence of cut-off functions can be eliminated (see the following remark). Nevertheless, we point out the naturalness and the speed of the proof. Remark 6.3. For some ρ > 0 small enough, the local stable invariant manifold K(Bρ(0)) can be characterized by the following dynamical property: K(Bρ(0)) ∩V={x∈V|F(i)(x)∈Vfor all i≥0} for every sufficiently small neighborhood Vof the origin; see [PdM82]. This fact automatically implies the uniqueness of the local invariant manifold. The above characterization (and some extra work) also allows to establish that Kis indeed differentiable at 0 and that DK(0) = (Id,0). That is, K(Bρ(0)) is tangent to Xs at the origin. We refer to [PdM82] for all these matters. Proof of Theorem 6.1. We write A:= DF(0) and N(x) := F(x)−Ax. A preliminary reduction, standard in the field, is to consider the function Fδ(x) := Ax +Nδ(x),where Nδ(x) := ξ(x)N(δx) δ. for δ > 0 small enough. Here ξ:X→Ris a smooth cut-off function (ξ≡1 in the unit ball centered at 0 and ξ≡0 outside of the ball of radius 2 centered at 0). Even if Nis only defined in a neighborhood of the origin, for δsmall enough we may consider the nonlinearity Nδto be defined and be C1 uin the whole X, since ξhas bounded support. We use here that, in finite dimensions, every C1 function in a neighborhood of 0 is automatically C1 uin a smaller neighborhood (since continuous functions in compact sets are uniformly continuous). It is important to note that if Wδis a manifold through 0 invariant under Fδ, then δWδis invariant under Fin a neighborhood of 0. It suffices therefore to find an invariant manifold for Fδfor δsmall enough. This will be accomplished using the implicit function theorem. For this, we consider the nonlinearity Nδ as belonging to the following Banach space. We work in the space C1 0,u of bounded C1maps M:X→Xwith bounded and uniformly continuous derivative DM in the whole X, and such that M(0) = DM(0) = 0. We equip this space with the standard C1norm.
THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS III 25 Note that Nδ∈C1 0,u for all δsmall enough. Using that N(0) = DN(0) = 0, it is easy to check that, by choosing δsmall enough, we can assume that kNδkC1 is as small as we want. Abusing of notation, we work with maps F=A+Nwith N∈C1 0,u —where in reality, all what follows is applied to Fδ=A+Nδ, for which Nδdoes truly belong to C1 0,u. We write the parameterization as K= (Id,0) + K>, and look for K>in the Banach space C0 0=C0 0(Xs;X) of bounded continuous maps K>:Xs→Xwith K>(0) = 0. We have F◦K−K◦As= (A+N)◦((Id,0) + K>)−((Id,0) + K>)◦As =A◦K>+N◦((Id,0) + K>)−K>◦As. Theorem 6.1 follows by application of the implicit function theorem in Banach spaces to the operator T(N, K>):=A◦K>+N◦((Id,0) + K>)−K>◦As, considered as an operator from C1 0,u ×C0 0to C0 0. We want to solve T(N, K>) = 0 and obtain K>as a function of N, for N near 0. Note that T(0,0) = 0 —this corresponds to the linear map F=A, for which the invariant manifold is Xs. Recall also that, by taking δsmall, we may assume that N=Nδis as small in C1 0,u as needed. It is easy to verify (see [dlLO99]) that the operator Tis C1, and that D2T(N, K>)∆ = A∆ + DN ◦((Id,0) + K>)∆ −∆◦As. It suffices to verify that the expression above satisfies the definition of derivative. The verification of the definition of derivative is where we use that DN is uniformly continuous. More details on the verification and examples that show that uniform continuity of DN is needed to get differentiability of Tcan be found in [dlLO99]. To complete the proof, we only need to show that the operator S:= D2T(0,0), given by S∆ = A∆−∆◦As, is invertible from C0 0to C0 0. This amounts to, given η∈C0 0, find ∆ ∈C0 0such that A∆−∆◦As=η(6.1) and show that k∆ukC0≤CkηkC0.
32 X. CABR´ E, E. FONTICH, AND R. DE LA LLAVE Proof of Lemma 8.3. Let ξ:R+−→ R+be a C1cut-off function such that 0≤ξ(t)≤1, ξ(t) = 1 if 0 ≤t≤1, and ξ(t) = 0 if t≥2. To prove the existence of a right inverse S−1, for any given η= (η1, η2)∈Γ we have to find ∆ = (∆1,∆2)∈Γ such that S∆ = η. The formulas ∆1= ∞ X i=0 A−(i+1) 1η1◦Ai 1(8.4) ∆2=− ∞ X i=0 Ai 2η2◦A−(i+1) 1(8.5) give a formal solution. It will be a true solution provided the series converge uniformly and η2is globally defined in order that the formula for ∆2makes sense. To deal with the second difficulty we only have to extend η2in a continuous way, which is always possible in a Banach space as follows. We substitute η2by the extension ˜η2defined by ˜η2(x) = η2(x),if |x|<1 ˜η2(x) = ξ(|x|)η2(x/|x|),if |x| ≥ 1. Note that if |A−(i+1) 1x|<1 then |˜η2(A−(i+1) 1x)| ≤ kηkΓ(|A−(i+1) 1x|)1+µ. Also, if |A−(i+1) 1x| ≥ 1 then |˜η2(A−(i+1) 1x)| ≤ ξ(|A−(i+1) 1x|) sup|y|=1 |η(y)| ≤ kηkΓ≤ kηkΓ(|A−(i+1) 1x|)1+µ. By hypothesis 3) we have |A−(i+1) 1η1(Ai 1(x))| ≤ kA−1 1ki+1kηkΓ|Ai 1x|1+ν ≤ kηkΓkA−1 1kkA−1 1k kA1k1+νi|x|1+ν and |Ai 2η2(A−(i+1) 1(x))| ≤ kA2kikηkΓ(kA−1 1ki+1|x|)1+µ ≤ kηkΓkA−1 1k1+µkA2k kA−1 1k1+µi|x|1+µ, which prove that the series in (8.4) and (8.5) are uniformly convergent and therefore their sums define continuous functions. From these bounds it is also clear that ∆ so obtained indeed belongs to Γ and k∆kΓ≤CkηkΓ. We consider the operator N:C1+ε 0×Γ−→ Γ defined by N(N, K>) = N◦((Id,0) + K>). We recall that C1+ε 0is the subspace of C1+εconsisting of the functions vanishing at the origin together with their derivative. Even that there are several available results on the differentiability of the composition operator, for the given topology of Γ we need to provide a proof.
THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS III 33 Proposition 8.5. Under the above conditions, we have that Nis C1and D2N(N, K>)∆ = DN ◦((Id,0) + K>) ∆. Proof. Let α= min(ν, µ) and β= max(ν, µ). Note that (1+α)(1+ε)≥1+α+ε≥ 1 + ε≥1 + β. To shorten the notation we will write K= (Id,0) + K>.Nis linear with respect to N. We claim that N7→ N (N, K>) is continuous. Indeed, since |N(K(x))| ≤ Z1 0 |DN(sK(x))K(x)|ds ≤kNkC1+ε 1 + ε|x|+kK>kΓ|x|1+α1+ε we see that kT (N, K>)kΓ≤CkNkC1+ε. Now we prove that Tis C1with respect to K>. We first show that it is differentiable. This follows easily from the bound |N(K(x) + ∆(x)) −N(K(x)) −DN(K(x))∆(x)| =Z1 0hDN(K(x) + s∆(x)) −DN(K(x))i∆(x)ds ≤Z1 0 kNkC1+εsε|∆(x)|1+εds ≤1 1 + εkNkC1+εk∆k1+ε Γ|x|(1+α)(1+ε). Finally we show that D2Nis continuous. We have to bound kD2N(N, K>)−D2N(N, K>)kL(Γ,Γ) = sup k∆kΓ≤1 k[DN ◦K−DN ◦K]∆kΓ. The continuity follows from hkDN(K(x)) −DN(K(x))k+kDN(K(x)) −DN(K(x))ki|∆(x)| ≤[kDN −DNkCε|K(x)|ε+kDNkCε|K>(x)−K>(x)|ε]k∆kΓ|x|1+α ≤hkN−NkC1+ε|x|+kK>kΓ|x|1+αε +kNkC1+εkK>−K>kε Γ|x|(1+α)εik∆kΓ|x|1+α ≤h1 + kK>kΓ|x|αεkN−NkC1+ε+kNkC1+εkK>−K>kε Γi· ·k∆kΓ|x|1+α+ε. Now we can easily finish the proof of Theorem 8.1. We define T:C1+ε 0× Γ−→ Γ by T(N, K>) = K>+S−1N◦((Id,0) + K>). We have that T(0,0) = 0. By Lemma 8.3 and Proposition 8.5 the operator T is C1and D2T(0,0) = Id. Then we can apply the implicit function theorem to T(N, K>) = 0 and obtain a neighborhood Vof 0 in C1+ε 0and a C1function K>,∗= (K>,∗ 1, K>,∗ 2) : V⊂C1+ε 0−→ Γ such that T(N, K>,∗(N)) = 0 for all N∈V.
34 X. CABR´ E, E. FONTICH, AND R. DE LA LLAVE We note that the fact that |K>,∗ 1(x)| ≤ C|x|1+νand |K>,∗ 2(x)| ≤ C|x|1+µ implies that Kis differentiable at 0, DK>,∗ 1(0) = 0, DK>,∗ 2(0) = 0. Therefore, the invariant manifold obtained is tangent to X1at 0, simply because K= (Id +K>,∗ 1, K>,∗ 2). 9. Non-resonant invariant manifolds for maps The goal of this section is to study some non-resonant invariant manifolds of maps. The non-resonant invariant manifolds were introduced in [dlL97], and they include as particular cases the stable or strong stable manifolds. For optimal results concerning differentiability and also in the setting of Banach spaces we refer to [CFdlL03a, CFdlL03b]. An exposition of results that can be obtained using the graph transform is in [dlL03]. The concrete result that we prove in this section is Theorem 3.1, stated in Section 3. We will present first a proof in the analytic case, which is simpler, and then a proof in the finitely differentiable case. 9.1. Overview of the proof. We look for Kof the form K=K≤+K> where K≤(x) = PL i=1 Kix⊗iis a polynomial of degree L. We recall that Kiis a symmetric i-linear operator in E⊗itaking values in Rd. Similarly, we will write R(x) = PL i=1 Rix⊗iwhere Riis a symmetric i-linear operator in E⊗itaking values in E. In Subsection 9.2, we will show that, under appropriate non-resonance conditions, it is possible to find K≤and Rjust matching powers of x. Then, the search for a K>that leads to an invariant parameterization will be reduced to solving a nonlinear equation in a Banach space, which will be discussed in Subsection 9.3 for the analytic case, and in Subsection 9.4 for the finite differentiable case. 9.2. Solution of the formal problem. Lemma 9.1. Assume that (Spec(AE))i∩Spec(AC) = ∅for i= 2, . . . , L and that r≥L. Then, we can find polynomials K≤, R as before in such a way that Dj(F◦K≤−K≤◦R)(0) = 0, j = 0, . . . , L, (9.1) K≤(0) = 0, DK≤(0) = (Id,0),(9.2) R(0) = 0, DR(0) = AE.(9.3) Moreover, if we assume that N=F−Ais sufficiently small, then K≤−(Id,0) and R−AEwill be arbitrarily small. This lemma is a simplified version of the normal form calculations which often appear in dynamical systems. A good reference for related results is [Nel69].
THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS III 35 Of fundamental importance in the proof of the lemma are the operators (sometimes called Sylvester operators) defined on the space Si=Si(X, Y ) of symmetric i-multilinear operators from Xto Y(X,Ybeing vector spaces) by Li A,BM=AM −MB⊗i,(9.4) where A,Bare linear maps. A key result in the study of these operators is the following. Proposition 9.2. Spec(Li A,B) = Spec(A)−(Spec(B))i ={λ−µ1·. . . ·µi|λ∈Spec(A), µ1, . . . , µi∈Spec(B)}.(9.5) A proof of this proposition can be found in [Nel69]. In [CFdlL03a] (and also in [BK98]), one can find another proof and an analogue for Banach spaces. We note that in the generality of Banach spaces one has the inclusion ⊂instead equality in (9.5). Proof of Proposition 9.2. Note that since the left and the right hand sides of (9.5) are continuous with respect to the matrices Aand B, it suffices to prove (9.5) for a dense set of matrices Aand B. Hence, it suffices to establish the result when Aand Bare diagonalizable over the complex. In such a case, we see that if (λj, ej), (µj, vj) are eigenvalues and eigenvectors for A, B respectively, given a set of indices σ1, . . . , σi, and jwe can consider the multilinear operator defined by Γj σ1,...,σi(vα1⊗ · · · ⊗ vαi) = (ejif the sets {α1, . . . , αi},{σ1, . . . , σi}are equal, 0 otherwise. Clearly, Γi σ1,...,σiis symmetric and we have Li A,BΓj σ1,...,σi= (λj−µσ1·. . . ·µσi)Γj σ1,...,σi. Hence, Γj σ1,...,σiis an eigenvector of Li A,B of eigenvalue λj−µσ1·. . .·µσi. Therefore we have the inclusion Spec(Li A,B)⊃Spec(A)−(Spec(B))i. To prove the opposite inclusion, we note that the Γ’s obtained for different (σ, j)’s are linearly independent. Hence, since their number equals the dimension of the space of symmetric i-linear operators, we see that they are a complete set of eigenvectors. Hence, we have found all the spectrum of Li A,B. Remark 9.3. Note that, as a particular case of Proposition 9.2, we obtain that, if iis such that (max |µj|)i<min |λj|(which happens for all i > log min |λj| log max |µj| when max |µj|<1), then Li A,B is invertible.
36 X. CABR´ E, E. FONTICH, AND R. DE LA LLAVE Proof of Lemma 9.1. The case j= 0 of the conclusions is satisfied when K≤(0) = 0, R(0) = 0. Hence, we pick K0= 0, R0= 0. The case j= 1 amounts to AK1=K1R1. Hence, we take R1=AEand K1=IE, where IE= (Id,0) is the immersion of Einto Rd. In this way, we ensure (9.2) and (9.3). The previous choice of R1and K1will only affect the smallness conditions that we have to impose on the nonlinear terms, which as in previous sections, can be adjusted by scaling. This is, of course, analogous to our choice of a multiple of the eigenvector for the one dimensional invariant manifolds of previous sections. For j≥2, equating terms of order jin F◦K−K◦R= 0 we obtain AKj−KjA⊗j E−K1Rj+Pj(K1, . . . , Kj−1, R1, . . . , Rj−1) = 0,(9.6) where Pjis a polynomial expression in its arguments. Taking projections over the spaces Eand C, denoting by KE j= ΠEKj, etc., and using the block notation for Aas in (3.1), formula (9.6) becomes AEKE j+BKC j−KE jA⊗j E−Rj+PE j= 0 ACKC j−KC jA⊗j E+PC j= 0,(9.7) where we have used that ΠCK1= 0. Using the operators Lwe can write (9.7) as Lj AE,AEKE j=Rj−BKC j−PE j(9.8) Lj AC,AEKC j=−PC j.(9.9) Note that, by Proposition 9.2, the hypotheses of Lemma 9.1 imply that Lj AC,AE is invertible. Now we follow the next iterative algorithm to solve (9.7). Assuming we already know Ki,Rifor 1 ≤i < j, 1) Since at this stage, the right hand side of (9.9) is known and Lj AC,AEis invertible, we can obtain one and only one solution KC jof it. 2) We choose Rjin such a way that the right hand side of (9.8) is in the range of Lj AE,AE. 3) We solve equation (9.8) for KE j. A particular way to do 2) and 3) above, is to choose Rj:= BKC j+PE jand then KE j= 0. Of course other procedures are possible. We can add to Rjterms in the range of Lj AE,AE, and we can add to KE jterms in the kernel of Lj AE,AE. The last statement of the lemma just follows from observing that the polynomials PE jand PC Jvanish when N≡0 and are continuous in the Taylor coefficients of N. This finishes the proof of Lemma 9.1. Remark 9.4. Hypotheses 2), 3) and 5) of Theorem 3.1 imply that if j > L the operator Lj AE,AEis invertible. In particular, when j > L we can take Rj= 0.
THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS III 37 9.3. Proof of Theorem 3.1 when r=ω.We discuss first the case r=ω, that is when all the considered functions are analytic. We take a norm in Rdsuch that the associated operator norms kAEkand kA−1kverify kAEk< ρ(AE) + εand kA−1k< ρ(A−1) + εfor some εsmall enough, where ρstands for the spectral radius (see Proposition A1 in [CFdlL03a] for details). Then, by hypothesis 5), kA−1k kAEkL+1 <1.(9.10) Note that the equation F◦K=K◦Rcan be written as T(N, K>) := AK≤+AK>+N◦(K≤+K>)−K≤◦R−K>◦R= 0,(9.11) where we need to consider K≤and Ras functionals of Nwhich are computed precisely through the algorithm that we have indicated in the proof of Lemma 9.1. To prove Theorem 3.1 it suffices to consider small N, since we can always reduce to this case by scaling. This scaling technique will be considered in detail later. Let Hk δ={G:¯ Bδ(0) ⊂E→Rd|G= ∞ X i=k Gix⊗i, ∞ X i=k |Gi|δi<∞} endowed with the norm kGk:= P∞ i=k|Gi|δi. We consider T:H2 3×HL+1 2→ HL+1 2(here the space H2 3corresponds to maps Nfrom Rdto Rd, instead of maps from Eto Rd). Note that if N= 0 then K≤=IEand R=AE, and hence T(0,0) = 0. Proposition 9.5. We have: 1) The operator T:V⊂H2 3×HL+1 2→HL+1 2is analytic in a neighborhood Vof (0,0). 2) D2T(0,0)∆ = A∆−∆◦AE. Proof. The fact that K≤, R are analytic in Nis a consequence of the fact that they are algebraic expressions in a finite number of coefficients of N. Also from Lemma 9.1, if Nis small enough, Ris a contraction. Therefore the operator Tis well defined. Now, we use that on the set Hk 3× {G∈H` δ| kGk<2}, the map (H, G)7→ H◦Gis analytic (see [Mey75] for more details). The idea is that if we interpret the series H◦G=PiHiGias a series in the Banach algebra of analytic functions, we can bound the norm of the general term by kHiGik ≤ |Hi|kGki, and hence the series obtained summing HiGiconverges as a series of elements in the Banach algebra. Applying this fact to H=N, G =K≤+K>, we obtain that the third term in (9.11) is analytic. Applying it when H=K>, G =Rwe obtain the analyticity of the last term in (9.11). Lemma 9.6. Under hypothesis 5) of Theorem 3.1, we have that D2T(0,0) from HL+1 2to HL+1 2is boundedly invertible.
38 X. CABR´ E, E. FONTICH, AND R. DE LA LLAVE Proof. If we consider the equation A∆−∆◦AE=η(9.12) with η=P∞ i=L+1 ηix⊗i, we see that ∆i−A−1∆iA⊗i E=A−1ηi, i ≥L+ 1.(9.13) By hypothesis 2) and (9.10) we have that kA−1k kAEki<1, for i≥L+ 1. Therefore (9.13) defines ∆ uniquely and we have |∆i| ≤ kA−1k 1− kA−1k kAEki|ηi| ≤ kA−1k 1− kA−1k kAEkL+1 |ηi|, and hence ∞ X i≥L+1 |∆i|2i≤C ∞ X i≥L+1 |ηi|2i. We can now finish the proof of Theorem 3.1 when r=ω. Proposition 9.5 and Lemma 9.6 give that the hypotheses of the implicit function theorem for T(N, K>) = 0 near (0,0) are satisfied, and therefore we have established Theorem 3.1 for Nsmall enough. Theorem 3.1 for a general Ncan be obtained by observing that if we consider Fδ= (1/δ)F(δx) for δsmall enough, then Nδis small in the sense needed by the result hitherto proved. Hence, we obtain Fδ◦K=K◦R for some K, R analytic. It is immediate to verify that K1/δ(x) = δK(1 δx), R1/δ(x) = δR(1 δx) verify F◦K1/δ =K1/δ ◦R1/δ in a neighborhood of 0, as well as the other claims. 9.4. Proof of Theorem 3.1 when r∈N.The method used in the previous subsection does not directly work in this case because the term (K>, R)7→ K>◦R ceases to be differentiable with respect to Rwhen we give K>and K>◦Rthe Crtopology with r∈N. Simply note that when differentiating K>◦Rformally with respect to R, the term DK>appears and does not belong to Cr, but just to Cr−1. We write N=N≤+N>, where N≤is the Taylor polynomial of Nup to order L, and we denote by K0,R0the solution of (A+N≤)◦K0=K0◦R0which is obtained applying the analytic result —already established— to F≤=A+N≤ in place of F=A+N. Then we look for Kin the form K=K0+K>. Let Σ be the space of polynomials Q:Rd→Rdof degree less than or equal to Lsuch that Q(0) = 0 and DQ(0) = 0 and, for r≥L+ 1, let Cr L(¯ Bρ(0)) = {f∈Cr(¯ Bρ(0)) |Dif(0) = 0,0≤i≤L}
THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS III 39 endowed with the topology given by |||f||| := max L+1≤i≤rsup |x|≤ρ kDif(x)k. It is clearly a Banach space. We consider the operator N:V⊂Σ×Cr+1 L(¯ B2(0)) ×Cr L(¯ B1(0)) −→ Cr L(¯ B1(0)) in a neighborhood Vof (0,0,0) defined by N(N≤, N>, K>) =AK0+AK>+N≤◦(K0+K>) +N>◦(K0+K>)−K0◦R0−K>◦R0,(9.14) where K0and R0are considered as analytic functions of N≤. With these notations, the original problem becomes N(N≤, N>, K>) = 0. Since N(0,0,0) = 0, again we try to construct the solution by using the implicit function theorem. Proposition 9.7. The operator Ndefined in (9.14) is continuous with respect to all three variables and C1with respect to K>. Moreover, D3N(0,0,0)∆ = A∆−∆◦AE.(9.15) Proof. We note that the linear operator H1 2→Cr+1 0(¯ B1(0)) sending a map to itself is continuous and thus differentiable. Then we can consider Nas being the composition of the maps V⊂Σ×Cr+1 L(¯ B1(0)) ×Cr L(¯ B1(0)) →Σ×H1 2×H1 2×Cr+1 L(¯ B1(0)) ×Cr L(¯ B1(0)) sending (N≤, N>, K>) to (N≤, K0, R0, N>, K>), where K0and R0are the maps obtained applying the analytic result to F≤=A+N≤, which depend analytically on N≤, the injection Σ×H1 2×H1 2×Cr+1 L(¯ B1(0)) ×Cr L(¯ B1(0)) −→ Σ×Cr+1 0(¯ B1(0)) ×Cr+1 0(¯ B1(0)) ×Cr+1 L(¯ B1(0)) ×Cr L(¯ B1(0)) and finally the map ˜ V⊂Σ×Cr+1 0(¯ B1(0)) ×Cr+1 0(¯ B1(0)) ×Cr+1 L(¯ B1(0)) ×Cr L(¯ B1(0)) →Cr(¯ B1(0)) (9.16) sending (N≤, K0, R0, N>, K>) to N(N≤, N>, K>). We restrict Vand ˜ Vin order to have the maps well defined. Since the space Cr Lis a closed linear subspace of Crand the norm there is the restriction of the Crnorm, we obtain that the differentiability results in [dlLO99] also hold for Cr L. Then the map (9.16) is continuous (we are working with sets of maps defined in spaces of finite dimension) and it is C1with respect to K>. Hence Nis continuous with respect to its three variables and C1with respect to K>. Moreover, by the definitions of K0and R0the range of Nis contained in Cr L(¯ B1(0)). Actually we have that D3N(N≤, N>, K>)∆ = [A+DN≤◦(K0+K>)+DN>◦(K0+K>)]∆−∆◦R0.
40 X. CABR´ E, E. FONTICH, AND R. DE LA LLAVE The following is slightly weaker than Lemma 5 of [BdlLW96]. The proof, however is simpler. Lemma 9.8. The operator D3N(0,0,0) is boundedly invertible as an operator from Cr L(¯ B1(0)) to itself. Proof. We have to solve A∆−∆◦AE=ηwhich is equivalent to ∆−A−1∆◦AE=A−1η. We introduce the operator Adefined by A∆ = A−1∆◦AE. We claim that the norm of Aconsidered as an operator from Cr L(¯ B1(0)) to itself is strictly smaller than 1. Indeed, if L+ 1 ≤i≤r kDi(A−1∆◦AE)kC0≤ kA−1k kDi∆kC0kAEki≤ kA−1k kDi∆kC0kAEkL+1 (9.17) and hence |||A∆||| ≤ kA−1k kAEkL+1|||∆|||. Therefore, we have that ∆ = (Id −A)−1(A−1η). Proposition 9.7 and Lemma 9.8 establish the hypotheses of the generalized version of the implicit function theorem, which assumes continuity of the map and being C1with respect to the variable that one wants to isolate, but only provides continuity of the implicit function that it defines (see [Nir01]). Applying this to N(N≤, N>, K>) = 0 near (0,0,0) we get a continuous map K>=M(N≤, N>) defined in a neighborhood of (0,0). Now, given a map Fsatisfying the hypotheses of Theorem 3.1, we scale it to Fδ=A+N≤,δ +N>,δ with δso small that (N≤,δ, N>,δ) belongs to the domain of M. The parameterization K=K0+K>thus obtained is the solution of Fδ◦K=K◦R0, where K0and R0are the analytic maps depending on F≤,δ =A+N≤,δ provided by the proof in the analytic case. 10. Non-resonant invariant manifolds for differential equations The results we have proved in the previous section translate to results for flows using the argument mentioned at the end of Section 2. Nevertheless, it is interesting, specially from the point of view of implementing algorithms, to give direct proofs of the results for differential equations. We will see that the leading ideas and methods are very similar to those for maps. The result we deal within this section is Theorem 3.2, which we prove following the parameterization method. For differential equations x0=X(x) such that X(0) = 0, if we have an invariant subspace Eby DX(0), we look for a parameterization Kand a polynomial Rdefined in Esuch that X ◦ K=DK ·R, (10.1) that is, we ask the vector field Xon the image of Kto be the pull forward of a vector field Rin E.
THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS III 41 Following the same strategy as for the case of maps we look for Kof the form K=K≤+K> where K≤is a polynomial of degree Land K>is a function vanishing at the origin together with its first Lderivatives. The polynomials K≤and Rwill be found matching powers in (10.1). Then, we will write a functional equation for K>whose solution will be found by applying the implicit function theorem in an appropriate Banach space. 10.1. Formal solution. We summarize the formal calculations needed to find K≤and Rin the following result. Lemma 10.1. Given X:U⊂Rd−→ Rd,X(0) = 0,X ∈ CL, satisfying hypotheses 3) of Theorem 3.2, we can find polynomials K≤and Rof degree not bigger than Lsuch that Dj(X ◦ K≤−DK≤·R)(0) = 0,0≤j≤L, (10.2) K≤(0) = 0, DK≤(0)E=E, (10.3) R(0) = 0, DR(0) = AE.(10.4) Moreover, if we assume that N=F−Aand Bare sufficiently small, then K≤−(Id,0) and R−AEare arbitrarily small. For the calculations in Lemma 10.1 we will use the operators e Li A,B from the space Siof symmetric i-linear operators in Ewith values in E(or Rd), defined by (e Li A,BK)(x) = AK(x)−DK(x)Bx, (10.5) for x∈E. These operators are similar to the ones used in Section 9, but their spectrum is different. Proposition 10.2. Spec( e Li A,B) = Spec(A)−iSpec(B) (10.6) := {λ−(µ1+µ2+· · · +µi)|λ∈Spec(A), µ1, . . . , µi∈Spec(B)}. Proof. The proof is completely analogous to the one of Proposition 9.2. Because of the continuity of the objects of (10.6) with respect to Aand B, it is sufficient to prove (10.6) for the dense subset of diagonalizable matrices Aand B. Assuming that Aand Bare diagonalizable, let (λj, ej) and (µj, vj) be the eigenvalues and eigenvectors of Aand Brespectively. Given indices σ1, . . . , σi and `, let the i-linear symmetric form Γ` σ1,...,σi∈Sibe defined by Γ` σ1,...,σi(vα1⊗ · · · ⊗ vαi) = (e`if the sets {α1, . . . , αi} {σ1, . . . , σi}are equal, 0 otherwise. Note that the Γ0sso defined are linearly independent and form a basis of Si. An easy calculation gives that e Li A,BΓ` σ1,...,σi= (λ`−(µσ1+· · · +µσi))Γ` σ1,...,σi.
48 X. CABR´ E, E. FONTICH, AND R. DE LA LLAVE Proposition 10.8. Under the previous conditions, we have: a) The operator V:V⊂Σ×Cr+1 L(¯ B2(0)) ×Cr L(¯ B1(0)) −→ Cr L(¯ B1(0)) defined by (10.30) is continuous with respect to all the three variables and C1with respect to K>in a neighborhood of Vof (0,0,0). b) We have V(0,0,0) = 0 and D3V(0,0,0)∆ = ∆. Proof. The operator Vis the identity plus W. The latter can be written as the composition W1(W2(N≤),W3(N≤, N>, K>)) where W3(N≤, N>, K>) = (A+N≤+N>)◦(K0+K>)−(A+N≤)◦K0−AK>, W2sends N≤to the flow ϕ(t, x) of x0=AEx+N≤(x), and W1(ϕ, g) = Z∞ 0 ψ(s)g(ϕ(s, x)) ds. (10.31) We recall that, as in the previous section, Σ →H1 3sending N≤to K0is analytic, H1 3→Cr 0(¯ B1(0)) sending K0to itself is C∞, and Σ →Cr+1 0(¯ B2(0)) sending N≤ to itself is C∞. By the results of [dlLO99] and the argument in Proposition 9.7 the operator f W3:V3⊂Cr+1 1(¯ B2(0)) ×Cr+1 L(¯ B2(0)) ×Cr 0(¯ B1(0)) ×Cr L(¯ B1(0)) →Cr(¯ B1(0)) defined by f W3(N≤, N>, K0, K>) = (A+N≤+N>)◦(K0+K>)−(A+N≤)◦ K0−AK>is of class C1. Then W3is C1, takes values in Cr L(¯ B1(0)) and D3W3(N≤, N>, K>)∆ = DX(K0+K>)∆ −A∆. In particular, D3W3(0,0,0) = 0. The proposition will be proved once we establish the regularity of W1and W2, that we do in the next lemmas. To study the regularity of W1and W2we introduce the space Γr={ϕ: [0,∞)ׯ B1(0) →E|ϕ∈C0, ϕ(t, ·)∈Cr, max 0≤j≤rsup t,x e−(µ++ε)t|Dj xϕ(t, x)|<∞} with the norm kϕkΓ:= max0≤j≤rsupt,x e−(µ++ε)tkDj xϕ(t, x)k. It is a Banach space. Lemma 10.9. The map W2:V2⊂Σ→Γrwhich sends RNto ϕ, where ϕ(t, x) is the solution of x0=AEx+RN(x)is well defined in a neighborhood V2of 0 and it is continuous. From the basic theory of ordinary differential equations we know that ϕ(t, x) depends continuously on RN, in the sense that if ais the vector of the coefficients of RN,ϕ(t, x, a) is continuous. However the lemma states that the continuity holds with respect to the norm in Γr. We will use the following version of
THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS III 49 Gronwall’s lemma: if u: [0, b)→Ris continuous v: [0, b)→Ris differentiable, with v(0) = 0, α≥0 and u(t)≤v(t) + αRt 0u(s)ds for t∈[0, b) then u(t)≤Zt 0 v0(s)eα(t−s)ds, t ∈[0, b). Proof of Lemma 10.9. First we note that the map Σ →Σ which sends N≤to RN is analytic. From Lemma 10.7 we know that if kRNkCr+1 ≤ε/2 then |ϕ(t, x)| ≤ e(µ++ε)t|x|,kDxϕ(t, x)k ≤ e(µ++ε)tand, for j≥2, kDj xϕ(t, x)k ≤ Mje(µ++ε)t, for all x∈¯ B1(0) and t≥0. Let R, ˜ R∈¯ Bε/2(0) ⊂Cr+1 1(B1(0)) and let ϕ, ˜ϕbe the associated flows. Assume that kR−˜ RkCr+1 ≤δ. We write ϕ(t, x) = eAEtx+Zt 0 eAE(t−s)R(ϕ(s, x)) ds and the analogous formula for ˜ϕ. Subtracting both equations we have ˜ϕ(t, x)−ϕ(t, x) = Zt 0 eAE(t−s)[˜ R( ˜ϕ)−R( ˜ϕ) + R( ˜ϕ)−R(ϕ)] ds. Multiplying both sides by e−(µ++ε/2)t, using that |˜ R( ˜ϕ(s, x)) −R( ˜ϕ(s, x))| ≤ ck˜ R−RkC2|ϕ(s, x)|2and taking norms we get e−(µ++ε/2)t|˜ϕ(t, x)−ϕ(t, x)| ≤Zt 0 cδe(µ++(3/2)ε)sds + (ε/2) Zt 0 e−(µ++ε/2)s|˜ϕ(s, x)−ϕ(s, x)|ds and by Gronwall’s lemma, e−(µ++ε/2)t|˜ϕ(t, x)−ϕ(t, x)| ≤ cδe(ε/2)t(−1/(µ++ε)). Hence |˜ϕ(t, x)−ϕ(t, x)| ≤ cδ −(µ++ε)e(µ++ε)t. Proceeding in the same way from Dxϕ(t, x) = eAEt+Zt 0 eAE(t−s)DR(ϕ(s, x))Dxϕ(s, x)ds and the analogous formula for Dx˜ϕ(t, x) we arrive at kDx˜ϕ(t, x)−Dxϕ(t, x)k ≤ δ+ (ε/2)k˜ϕ(t, x)−ϕ(t, x)kΓ −(µ++ε)e(µ++ε)t. If k≥2 we proceed inductively starting from Dk xϕ(t, x) = Zt 0 eA(t−s) k X j=1 X i0s cDjR(ϕ(s, x))Di1 xϕ(s, x). . . Dij xϕ(s, x)ds and using the analogous manipulations as in the cases k= 0,1 we get kDk x˜ϕ(t, x)−Dk xϕ(t, x)k ≤ δcke(µ++ε)t, where ckare positive constants independent of ϕand ˜ϕ.
50 X. CABR´ E, E. FONTICH, AND R. DE LA LLAVE Lemma 10.10. The map W1:V1⊂Γr×Cr L(¯ B1(0)) →Cr L(¯ B1(0)) defined by (10.31) is well defined in a neighborhood V1of (0,0), is continuous with respect to both variables, is C1with respect to g, and D2W1(ϕ, g)∆ = Z∞ 0 ψ(s)∆(ϕ(s, x)) ds. (10.32) Proof. Throughout the proof Cwill mean a constant independent on the functions, which may take different values in different places. We write W1( ˜ϕ, ˜g)− W1(ϕ, g) =Z∞ 0 ψ(s)[˜g( ˜ϕ(s, x)) −g( ˜ϕ(s, x))] ds +Z∞ 0 ψ(s)[g( ˜ϕ(s, x)) −g(ϕ(s, x))] ds. Let L+ 1 ≤k≤r. The Dk xderivative of ψ(s)[˜g( ˜ϕ(s, x)) −g( ˜ϕ(s, x))] is bounded by ψ(s) k X i=1 X 1≤`1,...,`i≤k `1+···+`i=k C[Di˜g( ˜ϕ(s, x)) −Dig( ˜ϕ(s, x))]· ·D`1 x˜ϕ(s, x)· · · D`i x˜ϕ(s, x) ≤Me(−λ−+ε)s k X i=1 X `0s Ck˜g−gkCk|˜ϕ(s, x)|(L−i+1)+· ·kD`1 x˜ϕ(s, x)k · · · kD`i x˜ϕ(s, x)k ≤Me(−λ−+ε)s L X i=1 X `0s Ck˜g−gkCke(µ++ε)(L−i+1)+s|x|(L−i+1)+· ·M`1e(µ++ε)s· · · M`ie(µ++ε)s ≤Ce[−λ−+ε+(µ++ε)(L+1)]sk˜g−gkCk, since (L−i+ 1)++i≥L+ 1. The Dk xderivative of ψ(s)[g( ˜ϕ(s, x)) −g(ϕ(s, x))] is bounded by kψ(s) k X i=1 X 1≤`1,...,`i≤k `1+···+`i=k C[Dig( ˜ϕ(s, x))D`1 x˜ϕ(s, x)· · · D`i x˜ϕ(s, x) −Dig(ϕ(s, x))D`1 xϕ(s, x)· · · D`i xϕ(s, x)]k. Now, by adding and subtracting appropriate terms, we get the desired bounds. We have to deal with terms [Dig( ˜ϕ(s, x))−Dig(ϕ(s, x))]D`1 x˜ϕ(s, x)· · · D`i x˜ϕ(s, x) which are bounded by kgkCL+1 e(µ++ε)(L+1−i−1)s|˜ϕ(s, x)−ϕ(s, x)| kD`1 x˜ϕ(s, x)k. . . kD`i x˜ϕ(s, x)k ≤Ce(µ++ε)(L+1)sk˜ϕ−ϕkΓ
THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS III 51 if i < L, by kgkCi+1 |˜ϕ(s, x)−ϕ(s, x)| kD`1 x˜ϕ(s, x)k. . . kD`i x˜ϕ(s, x)k ≤Ck˜ϕ−ϕkΓe(µ++ε)se(µ++ε)is if L≤i < r, and by Ce(µ++ε)rsω(k˜ϕ−ϕkC0), where ωis the modulus of continuity of Drg, by the uniform continuity of Drgon ¯ B1(0). We also need to control the terms of the form Djg(ϕ)D`1 x˜ϕ· · · [D`m x˜ϕ−D`m xϕ]· · · D`i xϕ, which are bounded by kgkCi+1 |ϕ(s, x)|L+1−i−1Ce(µ++ε)(i−1)skD`m x˜ϕ−D`m xϕkΓe(µ++ε)s ≤CkgkCi+1 e(µ++ε)(L+1)sk˜ϕ−ϕkΓ, if i < L and analogous bounds in the other cases, as we have got for the previous terms. When we integrate from 0 to ∞, we take the supremum over xand the maximum over k, we obtain the continuity in the topologies we are working with. To prove that W1is differentiable with respect to gwe only have to check that it is a bounded linear operator in g. This follows immediately taking ˜g= 0 and ˜ϕ= 0 in the previous estimates. In such a way we get kW1(g)kCr L≤CkgkCr L. To study the continuity of D2W1(ϕ, g), in view of formula (10.32) we have to do the same kind of estimates as we have done when dealing with ψ(s)[g( ˜ϕ(s, x))− g( ˜ϕ(s, x))] but changing gby ∆ ∈Γr. The end of the proof of Theorem 3.2 in the differentiable case follows in a completely analogous way as in the end of the proof of Theorem 3.1. Applying the generalized version of the implicit function theorem (see [Nir01]) to V(N≤, N>, K>) = 0 near (0,0,0) we get a continuous map K>=V∗(N≤, N>) defined in a neighborhood of (0,0). Given a vector field Xsatisfying the hypotheses of Theorem 3.2, we scale it to Xδ=A+N≤,δ +N>,δ with δso small that (N≤,δ, N>,δ) belongs to the domain of V∗. The parameterization K=K0+K>thus obtained is the solution we are looking for. Remark 10.11. Notice the remarkable similarities between the proofs of Theorem 3.1 and Theorem 3.2. Also the analogy of the argument in Proposition 9.2 which computes the spectrum of the operators Li A,B defined in (9.4) and the one in Proposition 10.2 which computes the spectrum of the operators e Li A,B defined in (10.5). Similarly, their use of the recursive solution to the hierarchy of equations for the low order terms is completely analogous.
52 X. CABR´ E, E. FONTICH, AND R. DE LA LLAVE Appendix A. Remarks on cohomology equations and nonuniqueness of invariant manifolds Since cohomology equations play an important role in this theory, it is interesting to give a heuristic guide to their solution. A cohomology equation is an equation for ∆ of the form M(x)∆(x)−∆◦R(x) = η(x) (A.1) where ∆ and ηare vector valued functions, Mis a function taking values on the space of linear operators and Ris a diffeomorphism. When facing an equation of the form (A.1), it is natural to try to isolate ∆ explicitly from one term and iterate the resulting expression. If we isolate ∆ from the second term in (A.1), we are left with ∆(x) = −η◦R−1(x) + M◦R−1(x) ∆ ◦R−1(x) (A.2) which, upon iteration, leads to ∆ = −η◦R−1−M◦R−1η◦R−2−M◦R−1M◦R−2η◦R−3− · · · −[M◦R−1M◦R−2· · · M◦R−n]η◦R−n−1 + [M◦R−1M◦R−2· · · M◦R−n−1]∆ ◦R−n−1. (A.3) If we isolate ∆ from the first term in (A.1), we are left with ∆(x) = M−1(x)η(x) + M−1(x)∆ ◦R(x) (A.4) which, upon iteration, leads to ∆ = M−1η+M−1M−1◦R η ◦R+· · · + [M−1M−1◦R· · · M−1◦Rn]η◦Rn + [M−1M−1◦R· · · M−1◦Rn]∆ ◦Rn+1. (A.5) Note that the general term in both (A.3) and (A.5) consists of the multiplication by a large number of linear operators applied to ηcomposed by the right with a high iterated of Ror of R−1. These sums can be shown to converge as n→ ∞ in two different cases by two different arguments. In the first argument we use that if Mis a contraction, then (A.3) will converge in the k·kC0norm. In the problems considered in this paper the point is that if M=DF ◦Kis a contraction then R, which agrees at first order with it close to the fixed point, will also be a contraction. Hence, R−1is expansive. Since (A.3) involves composing with R−1, this will require that the functions are defined everywhere. If it is not the case it requires performing extensions, etc. We note that performing extensions of Rcauses that the resulting manifold may depend on the extension procedure, so that the local results will be quite non-unique. This study is the basis of the results in Section 6. The second argument uses that the Rn(x) converges to a point as n→+∞. This happens when kDRkC0<1 in a neighborhood of the origin. In this case, M−1will be an expansion. To have convergence of the right hand side of (A.5)
THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS III 53 we will need that the operator ∆ 7→ ∆◦Ris a strong enough contraction to overcome the expansion caused by M−1. The basic idea to obtain contraction that gives sense to (A.5) is to work in a space of functions ∆ defined on Usuch that supx∈U|x|−L|∆(x)|<∞and to use the weighted norm k∆k:= sup x∈U |x|−L|∆(x)|. With this norm, we have k∆◦Rk= sup x∈U |x|−L|R(x)|L|R(x)|−L|∆◦R(x)| ≤sup x∈U |x|−L|R(x)|L·sup x∈U |R(x)|−L|∆◦R(x)| ≤ kDRkL C0k∆k (A.6) For low L, to work in such spaces is quite natural and we easily have the contractive property. This is the basis of the results in Section 8. For high L, we have to resort to other methods. We observe that, under appropriate conditions, one can obtain the low order terms of the problem matching derivatives and then, obtain the remainder using this method. This is the basis of the results in Section 9. Note that solving the problem for the low order terms requires non-resonance conditions. We note that in both cases, we obtain uniqueness of the solution in the corresponding space (after having fixed an extension of R−1in the first case). Nevertheless, it is quite important to note that, even in the case that both solutions (A.3), (A.5) make sense, they may fail to be the same. Note that, even if one had equal solutions for a certain η, adding an small bump to ηcauses perturbations that go towards the origin in (A.3), and that go towards infinity in (A.5) and hence, for this perturbed ηthe solutions given by (A.3) and (A.5) will be different. One important difference between the methods of solution is that, if we take derivatives of the general term in (A.3), we pick factors DR−n, which are growing. However in (A.5), we obtain factors DRnwhich are decreasing. Hence, if the series of the k-derivatives of the terms in (A.5) converge, the series of the j-derivatives also converge for values of jfrom kto the degree of differentiability of η. The solutions produced by (A.5), as soon as they start converging, they have all the derivatives that ηhas. On the other hand, those produced by (A.3) only have a finite number of derivatives that cannot be improved by assuming more differentiability of η. Unfortunately, the regularity that can be produced automatically by (A.3) is always smaller than that allowed by bootstrap using (A.5). All the above phenomena have a correspondence in the theory of invariant manifolds. The fact that slow manifolds with low regularity are not unique has been in the literature for a certain time. If one makes hypotheses that imply that there is certain growth at infinity, one can readily show uniqueness. This
54 X. CABR´ E, E. FONTICH, AND R. DE LA LLAVE was also known using Irwin’s method [dlLW95]. On the other hand, under nonresonance assumptions, one can get uniqueness under moderate differentiability assumptions. That is, among all the rough invariant manifolds tangent to the space, there is one which is moderately differentiable, and this moderately differentiable manifold is as smooth as the map. Examples that show that the manifolds with good behavior at infinity do not agree with the moderately smooth ones have been constructed in [dlL97]. Appendix B. Historical remarks and information on the literature on non-resonant invariant manifolds In this section we have collected some references on the problem of invariant manifolds associated to subspaces in the stable part of the spectrum. In contrast to the very vast literature on stable manifolds —for which a similar attempt would be beyond the capacity of the authors—, the literature on invariant manifolds associated to smaller sets of the spectrum is much more limited. Of course, this attempt cannot be considered a definitive effort (for instance, we have not been able to trace the work of Darboux, which is mentioned by Poincar´e and Lyapunov). We can only hope that our modest search can inspire others to do a more thorough job. B.1. Early history. It seems to us that one-dimensional invariant submanifolds were more or less known in the analytic case, and with resonance conditions somewhat stronger than those considered in the present paper. It seems well accepted that some versions of invariant manifold theory, at least for the analytic case, were known to Darboux, Poincar´e and Lyapunov. Unfortunately, we have not been able to locate the works of Darboux, but we will comment on some works of Poincar´e and Lyapunov. B.2. Two results of Poincar´e. One of us (R.L.) learned about the existence of [Poi90] (reproduced in [Poi50]) from conversations with D. Ruelle in the early 80’s. The motivation for [Poi90] was the theory of special functions. When Fis a polynomial and E=C, the equation F◦K(t) = K(λt) (B.1) can be interpreted as saying that the system of functions given by the components of Kadmits a multiplication rule (th´eoreme de multiplication). Examples of such systems of functions (or systems satisfying the closely related addition rules) are the trigonometric functions and the elliptic functions. For instance, K(θ) = (sin θ, cos θ) satisfies K(2θ) = F(K(θ)), where F(x, y) = (2xy, y2−x2). Note that F(0,1) = (0,1). Similar formulas for the duplication of the argument are known for elliptic integrals. The fact that there are duplication formulas is related to the solvability of the quintic using elliptic functions and their inverses.
THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS III 55 The paper [Poi90] shows that, given a map Fand provided that λ,|λ|>1, is a simple eigenvalue of DF(0) and that there are no eigenvalues of DF(0) which are powers of λ, one can find a formal series for K. Moreover, using the majorant method, one can show that the formal series for Kconverges. The paper also contains the interesting observation (see page 541 in [Poi50]) that when Fis a polynomial, every function Ksatisfying (B.1) is entire. The reason is that, when Fis a polynomial, the functional equation (B.1) forces the domain of definition of Kto be invariant under multiplication by λ. Hence, if it contains a ball, it is the whole complex plane. We note that this observation generalizes without difficulty to the situation when Fis an entire function and we are working on a Banach space. In [Poi90], Poincar´e also studies the case when F−1is a rational transformation, that he calls Cremona. In this case, he makes some dynamical observations. For instance, in the bottom half of page 561 of [Poi50], he relates the question of existence of solution to whether the iterates of the transformation converge to a fixed point —this is indeed the dynamical characterization of invariant manifold. From a more dynamical point of view, similar series were considered in [Poi87], where all chapter VII is devoted to asymptotic expansions around periodic solutions of periodic vector fields. Taking time-Tmaps, this problem reduces to the setting about maps that we have considered in this paper. The logarithms of the eigenvalues of the time-Tmap are called exposants charact´eristiques. In modern language, they are the Floquet exponents. Note that what we would call today Lyapunov exponents (which can be considered in more general settings than periodic systems) are, in the case of periodic systems, the real part of Poincar´e’s exposants charact´eristiques. More confusingly, in the translation of Lyapunov that we have used, the name characteristic exponent refers to the negative of what we call now Lyapunov exponent. This would be, of course, the negative of the real part of the exposant charact´eristique for the particular case of periodic systems. In [Poi87] the crucial paragraphs dealing with stable and unstable manifolds are 104 and 105. In paragraph 104, under the assumption that there are no resonances (the non-resonance condition is the last formula of paragraph 104), it is shown that one can obtain a formal power series expansion of exponentials with arbitrary constants. The convergence of the series is studied in 105. The first paragraph asserts the convergence of the series of expansions in powers of the exponential under the assumption that the eigenvalues belong to what we now call the Poincar´e domain (i.e., when the convex hull of the eigenvalues does not include zero). Of course, the reason why this condition enters is that, for eigenvalues satisfying these conditions, the small divisors that appear are bounded away from zero. We note that, even if it is not said explicitly, the condition that the eigenvalues are different is indeed assumed. The proof of convergence is rather succinct. Nevertheless, it should have been quite clear to Poincar´e and his contemporaries
56 X. CABR´ E, E. FONTICH, AND R. DE LA LLAVE since it is very similar to arguments that had been done in detail in his thesis [Poi79] (reproduced in [Poi16]). From the point of view of invariant manifold theory, the last paragraph of page 339 is quite interesting. Here, Poincar´e discusses the case when there are stable and unstable characteristic exponents at the same time. He observes that the series for Kremains convergent if one sets to zero the constants corresponding to coordinates along the expanding or neutral eigendirections. The arguments here are somewhat skimpy, but a modern mathematician can supply the missing details without too much trouble. One is left with a set of solutions which tend to zero parameterized by as many constants as stable directions. This is, of course, our modern stable manifold. A similar construction works for the unstable solutions. Poincar´e called these solutions solutions asymptotiques. The rest of chapter VII contains a variety of expansions of these sets of solutions. It includes, quite notably, the expansions in terms of a slow parameter, which are then shown to be divergent. Of course, much modern work is still being done in these slow perturbations and related areas. B.3. The work of Lyapunov. In chapters 11-33 [Lya92] (see also the summary in chapter 3 and the proofs of convergence in chapter 23), Lyapunov introduces the method of arbitrary constants, which consists in finding exponential solutions with arbitrary constants. Since the constants do not evolve in time, this is closely related to the problem of linearization; (compare the expansions of the system studied and those of the linear systems). Invariant manifolds can be obtained by setting some of the constants to zero. One important difference between [Lya92] and [Poi90], [Poi87] is that [Lya92] considers systems which are regular (roughly, the definition is that the forward and backward Lyapunov exponents agree). This is a more general setting than that of periodic systems. In the case of regular systems, [Lya92] contains expansions of the solutions in terms of arbitrary constants. The derivation of the formal expansions in [Lya92] does not need non-resonance conditions. In chapter 23 of [Lya92], the question of convergence of these formal expansions is studied. This is done under the condition that there are no resonances and no repeated eigenvalues, and that all the eigenvalues are stable or unstable. Here one can find a note giving credit to [Poi79] for dealing with the more general case of the Poincar´e domain. In particular, we call attention to Theorem II of section 24, which is a complete statement of the strong stable manifold theorem for analytic systems (see also Theorem II of chapter 13). One interesting remark of Lyapunov in chapter 11 is that one can consider families that correspond to any subset of eigenvalues. This amounts to setting to zero a subset of the arbitrary constants used in the expansion. This is hard to interpret from the dynamical point of view since the arbitrary constants do not have a dynamical interpretation. In particular, the set obtained setting them
THE PARAMETERIZATION METHOD FOR INVARIANT MANIFOLDS III 57 to zero does not need to be invariant. Of course, setting to zero all the nondecreasing modes is an invariant set, as pointed out by Poincar´e. With modern insight, setting to zero all the modes that are non-decreasing or decreasing more slowly than a certain rate is invariant. Indeed, it is the strongly stable manifold. As it was shown in examples in [dlL97], in general one cannot get invariant manifolds tangent to a subspace if there are resonances of the type we have excluded in the present paper. Overall, one cannot be but surprised by the enormous similarities in the problems and in the results between the contents of these chapters and the corresponding ones of the book by Poincar´e, which appeared in the same year. Of course, there are big differences in style and in the methods as well as in the way that proofs are presented. A modern exposition of some of the convergence results of Lyapunov can be found in [Lef77] V.4. It contains a statement and a proof of the expansion in arbitrary constants under non-resonance assumptions and provided that all eigenvalues are stable, and that the linearization is a constant (we remark that using Floquet theory, one can reduce the periodic case to the constant case). We have not been able to locate in any of these classical works the consideration of resonant terms. B.4. Modern work. It seems that the particular case of one dimensional stable invariant manifolds (when there are no resonances) has appeared several times in the modern literature. The papers [FR81] and [FG92] use the parameterization method for one dimensional manifolds, specially in conjunction with numerical analysis. They establish not only convergence of the series involved, but they also estimate the errors incurred when using a numerical approximation. Indeed, both papers have taken care of estimating actually the roundoff error so that a finite calculation can establish facts about transversality of intersections, etc. It seems to us that similar results could be obtained using the functional equations (2.1) and the theory developed in the present article. Numerical work for higher dimensional maps has been studied in [BK98], which undertook the task of systematically computing Taylor expansions of invariant manifolds. This could be considered one implementation of our result in Lemma 9.1 for finite dimensional systems. The authors of [BK98] indeed made the observation that the calculations can be carried out to any order provided that there are no resonances but they leave open the issue of whether these formal calculations are the jet of an invariant object. We note that our formalism could be used to provide an a posteriori estimate of the error of these numerical calculations. Once a polynomial satisfies (2.1) quite accurately, then it is close to being a fixed point of a map N(which is a solution of T= 0). Since Nis a contraction, there is a fixed point at a distance that can be estimated by the error of the numerical approximation. This is the usual a posteriori estimates of numerical analysis.