Invertible contractions and asymptotically stable ODE´s that are not C¹-linearizable

INVERTIBLE CONTRACTIONS AND ASYMPTOTICALLY STABLE ODE’S THAT ARE NOT C1-LINEARIZABLE. HILDEBRANDO M. RODRIGUES†AND J. SOL` A-MORALES‡ Abstract. We present an example of a contraction diffeomorphism in infinite dimensions that is not C1-linearizable, and we construct a regular ordinary differential equation in a Hilbert space whose time-one map is that diffeomorphism. With this we have an example of an asymptotically stable ODE that is not C1-conjugate to its linear part. Keywords: linearization, conjugacy, contraction. AMS Classification. Primary: 35B05, 34G20. Secondary: 35B40, 34D05. 1. Introduction and Main Result. In two previous papers, we obtained positive results for smooth linearization in infinite dimensional systems. Under some sufficient conditions, that include a nonresonance condition, in Rodrigues & Sola-Morales [10] we proved that C1-linearization is possible for contractions. The main theorem of that paper extends a classical result for finite dimensional systems by P. Hartman [6] and a result for infinite dimensions by Mora & Sola-Morales [8]. In the works ElBialy [3], Bin Tan [13] and Abbaci [1] results are obtained in the same direction. Date: November 26, 2005. †Departamento de Matem´atica, Instituto de Ciˆencias Matem´aticas e de Computa¸c˜ao, Universidade de S˜ao Paulo, Caixa Postal 668, 13560-970, S˜ao Carlos, SP, Brazil, e-mail: [email protected]. ‡Departament de Matem`atica Aplicada 1, Universitat Polit`ecnica de Catalunya, Av. Diagonal 647, 08028 Barcelona, Spain, e-mail: [email protected], phone: (34)93 4016552. †Partially supported by CNPq Processo: 301994/85-4 and Programa Pronex-Projeto Tematico, CNPqFAPESP no. 2003/10042-0, Brasil. ‡Partially supported by MCyT-MEC, Spain (BMF2002-04613-C03-01) and by DURSI, Generalitat de Catalunya (2005BE00245). 1 2 HILDEBRANDO M. RODRIGUES AND J. SOL ` A-MORALES Also in Rodrigues & Sola-Morales [11], we proved a similar result for a more particular saddle case. It extends a result proved by P. Hartman (see [6]) for two-dimensional systems (see also Aroson, Belitskii and Zhuzhoma, [2] ). In particular in Rodrigues & Sola-Morales [10] we obtained the following theorem: Theorem 1. (The Linearization Theorem For Contractions.) Let Zbe a Banach space with the property that there exists function ρsuch that ρ∈ C1,1(Z,R),with ρ(z) = 1,when |z| ≤ 1/2and ρ(z) = 0,when |z| ≥ 1.(1.1) Suppose that L, L−1∈ L(Z). We assume that there exist real numbers ν− i, ν+ i, i= 1,··· , n such that: 0< ν− n< ν+ n< ν− n−1< ν+ n−1<···ν− 1< ν+ 1<1 ν+ 1ν+ i< ν− i, i = 1,··· , n (nonresonance condition) |σ(L)| ⊂ (ν− n, ν+ n)∪(ν− n−1, ν+ n−1)∪ · · · ∪ (ν− 1, ν+ 1).          (1.2) Let F=F(z)be a C1,1-function in a neighborhood of the origin with values in Z, such that F= 0, ∂zF= 0, at z= 0. Then, for the map T:z7→ z0,z0=Lz +F(z), there exists a C1-map R:z7→ u, u=z+ψ(z), satisfying ψ= 0, ∂zψ= 0, at z= 0, such that RTR−1:u7→ u0has the form u0=Lu in a sufficiently small neighborhood of the origin. In a short Note, Rodrigues & Sola-Morales [12], we presented a first example of an analytic invertible contraction that is not C1-linearizable. This result is interesting because it is in contrast with the finite dimensional case, since as proved by P. Hartman (see [6]), for finite dimensional systems, every C1,1contraction can be linearized in the class C1. Before the existence of this example, the possibility or not of extending Hartman’s result to all infinite dimensional contractions was an open question in this field, as it was said for example in [1]. Of course, the nonresonance condition of (1.2) is not satisfied in the example, but (1.1) is. In the present paper we improve the result of Rodrigues & Sola-Morales [12], by presenting a different example, that is simpler in some respects, and a more complete analysis of the problem. This is stated in Theorem 2 below, that will be proved in Section 2. INVERTIBLE CONTRACTIONS AND ODE’S THAT ARE NOT C1-LINEARIZABLE. 3 Our first example contained in [12] was constructed using a sequence of Jordan blocks of increasing order. The example presented now is somehow simpler because instead, we use only one infinite-dimensional Jordan block. We prove that the conjugation map does not exist not only in the class of C1diffeomorphisms but even in the wider class of local homeomorphisms that are differentiable at the origin, together with their inverses. This possibility was only slightly mentioned in [12]. So, our example shows also the impossibility to extend to infinite dimensions the result of Guysinsky, Hasselblatt and Rayskin, [5]. These authors prove that if the map of the classical Hartman-Grobman theorem is C∞, then the linearizing homeomorphism can be taken to be differentiable at the origin and its derivative at zero being the identity. It will also be shown in Remark 2 below, that our example fits in the hypotheses of Cabr´e, Fontich and de la Llave [4] as a sharp example of both existence and non-existence of some invariant non-hyperbolic manifolds. Besides, as a meaningful contribution of the present paper, we prove that the stated contraction is the time one map of an asymptotically stable ordinary differential equation defined in the space `2, of the square summable sequences, with the usual inner product. This property is based on a careful spectral analysis of some operators involved. This property was not considered in [12], and we believe that it is quite significant. However, with the new ideas developed in the present paper it would be possible to prove that the first example of [12] is also a time-one map of an ODE. To embed a diffeomorphism into the flow of an autonomous ODE is not always possible, even in finite dimensions, but we succeeded in proving that for our example thanks to its special form. See the works of Palis [9] and Li, Llibre and Zhang [7] as some references concerning this problem. Another meaningful contribution of the present paper will be the construction of a family of examples that approach the nonresonance condition (1.2). They can be seen as modifications of the example of Theorem 2, and will be constructed in Section 3. The meaning and the interest of these families of examples will be explained in the Remark 1, at the end of the present Section. The next result will play an important role in the construction of our example. 4 HILDEBRANDO M. RODRIGUES AND J. SOL ` A-MORALES Proposition 1. Let a, ε, δ be positive real numbers, with a < 1. Consider the map, (x, y1, y2,··· , yn,···)T∈`27→ (x0, y0 1, y0 2,··· , y0 n,···)T∈`2,defined as: x0=ax y0 1=ay1+εx2 y0 2=ay2+δy1 . . . y0 n+1 =ayn+1 +δyn . . . (1.3) If ~y = (y1, y2,··· , yn,···)T= Φ(x) = (φ1(x), φ2(x),··· , φn(x),· · · )Tdefines a local invariant manifold for the above map, differentiable at x= 0, such that Φ(0) = 0, ∂xΦ(0) = 0 then necessarily, φ1(x) = 1 a−a2εx2, φ2(x) = δ (a−a2)2εx2,··· , φn+1(x) = δn (a−a2)n+1 εx2,··· . Let us now introduce some notations. We define the infinite matrix Jand the nonlinear function ~ f(x) as: J:=         000··· ··· 100··· ··· 010··· ··· . . .. . .. . .. . .. . .         ,~ f(x) :=         x2 0 0 . . .         .(1.4) For the scalars a, δ, ε, we consider the following infinite matrix Land the nonlinear function F, acting in the Hilbert space `2. L:=   a0 0δJ +aI , F(z) :=   0 ε~ f(x) , z :=   x ~y (1.5) In the next theorem, that is the main result of the present paper, we use the above notations. Theorem 2. Let ε6= 0 and 0< a < 1. Under the hypothesis, a−a2≤δ < min{1−a, a}(1.6) INVERTIBLE CONTRACTIONS AND ODE’S THAT ARE NOT C1-LINEARIZABLE. 5 the operator Lis an invertible contraction on `2, its spectrum σ(L)is the closed disk of center aand radius δand the local analytic diffeomorphism defined in `2by, z0=Tz := Lz +F(z) does not conjugate, even locally, with its linear part L, through a conjugation R, with Rand R−1differentiable at z= 0. Also, this map Tis the time-one map of an ordinary differential equation in `2of the form ˙z=Az +G(z) (1.7) where A:= log Lis a bounded linear operator and G:`2→`2is defined by G(z) :=   0 x2~ β  for some ~ β∈`2. Remark 1. Observe that |σ(L)|= [a−δ, a +δ]and that (1.6) implies that (a+δ)2> a2≥a−δ, so the nonresonance condition (1.2) is not satisfied. Observe also that the spectrum of Lconsists of a single block (that is, nis equal to 1in the notation of Theorem 1). It would have been very satisfactory to find examples of this kind but with the set of moduli of the spectrum of the linear part being equal to [b2, b], and to do that for all b∈(0,1). This kind of examples would fill up the complementary of the condition (1.2) for the case of a single block. Unfortunately, we are not able to construct a family of examples of this kind. We can only construct an approximate version of them, and this will be the purpose of the last section below. Acknowledgment. The second author wants to express its thanks to the Department of Applied Mathematics of the Universidad Complutense de Madrid for its support and hospitality while this research was being carried out. 6 HILDEBRANDO M. RODRIGUES AND J. SOL ` A-MORALES 2. Proof of Theorem 2 Lemma 1. If 0< a < 1and r∈Rthe functional equation, φ(ax) = aφ(x) + rx2(2.1) has a unique local solution φthat is differentiable at x= 0 such that φ(0) = 0, ∂xφ(0) = 0. This solution is given by φ(x) = r a2−ax2. Proof: (For the proof see [12]). The proof essentially follows from two facts. First, that the above function is indeed a solution of equation (2.1) and second, that the unique solution in the above class of the homogeneous equation, φ(ax) = aφ(x), is the zero function. Proof of Proposition 1 Suppose that ~y = (y1, y2, y3,··· , yn,···)T= Φ(x) = (φ1(x), φ2(x),· · · )Tis an invariant manifold for the map T(z) = Lz +F(z), such that Φ(0) = 0, ∂xΦ(0) = 0. Then φ1, φ2, φ3,··· , φn,··· should satisfy the following system of equations: φ1(ax) = aφ1(x) + εx2 φ2(ax) = δφ1(x) + aφ2(x) φ3(ax) = δφ2(x) + aφ2(x) ··· ··· ··· ··· ··· ··· φn+1(ax) = δφn(x) + ayn+1(x) ··· ··· ··· ··· ··· ··· ··· Using Lemma 1 recursively, we obtain φ1(x) = 1 a−a2ε x2, φ2(x) = δ (a−a2)2ε x2,··· , φn+1(x) = δn (a−a2)n+1 ε x2,··· . Lemma 2. The spectrum σ(J)of the operator Jdefined above is the closed disk of radius 1 centered at zero. When a < 1and 0< δ < min{a, 1−a}the operator L=  a0 0δJ +aI  is an invertible contraction and its spectrum is the disk of center aand radius δin the complex plane. INVERTIBLE CONTRACTIONS AND ODE’S THAT ARE NOT C1-LINEARIZABLE. 7 Proof: It is clear that kJk= 1. Suppose |λ|>1. Then kλ−1Jk=|λ|−1<1, I−λ−1Jis invertible and kI−λ−1Jk ≤ |λ|(|λ|−1)−1. So, λbelongs to the resolvent set of J. Suppose now |λ| ≤ 1. Let ~e1= (1,0,0,···)>, and let us write the infinite system (J−λI)~y =~e1: −λ y1= 1 y1−λ y2= 0 y2−λ y3= 0 ············ yn−λ yn+1 = 0 ············ If λ= 0, the system is clearly incompatible. For 0 <|λ| ≤ 1 the components of the solution should be yn=−λ−nand so ~y does not belong to `2. Thus λ∈σ(L). This completes the proof of the first part of our lemma. The second part of our lemma follows from the fact that L=  a0 0δJ +aI =δ  0 0 0J +aI, and so σ(L) = δσ(J) + a. Since 0 /∈σ(L) and kLk ≤ a+δ < 1 we conclude that Lis an invertible contraction. Proof of Theorem 2. For T=L+Fwe suppose that a local linearization map Rexists such that RTR−1= L. If both Rand R−1are differentiable at zero then from RTR−1=Lone obtains that ∂xR(0)L=L∂xR(0) and so (∂xR(0))−1RT((∂xR(0))−1R)−1=L. So we can suppose that ∂xR(0) = I. Now, the linear subspace {(x, 0)T} ⊂ `2is invariant by L, so R−1{(x, 0)T}is invariant by T. Let us write R−1(x, 0)T= (x+ψ(x),Φ(x))T. We have ψ(0) = 0, ∂xψ(0) = 0, Φ(0) = 0, ∂xΦ(0) = 0. Let us see that ψ≡0. From TR−1(x, 0)T=R−1L(x, 0)Twe take the first component and we see that aψ(x) = ψ(ax) for all xnear zero. Then, by applying Lemma 1 with r= 0 we see that ψ(x) = 0 in a neighborhood of zero. So, the set R−1{(x, 0)T}can be expressed in a neighborhood of zero as {(x, Φ(x))T} ⊂ `2. 8 HILDEBRANDO M. RODRIGUES AND J. SOL ` A-MORALES From Proposition 1 it follows that for any integer n≥1, kΦ(x)k ≥ δn (a−a2)n+1 εx2. From the assumption (1.6), since a−a2≤δ, it follows that Φ(x) does not belong to `2. This completes the first part of the proof of our main theorem. Let us consider now the differential equation: ˙z=Az +G(z) (2.2) where A:= log L, and G:`2→`2is defined by G(z) :=   0 x2~ β  for some ~ β∈`2that will be found later. It is easy to see that the operator A= log Lis a bounded operator, but next we will obtain an estimate for its norm. Since L=  a0 0δJ +aI =a I−µ−δ a¶  0 0 0J  , if we let D:= µ−δ a¶  0 0 0J  we obtain log L= (log a)I+ log(I−D) = (log a)I−(D+D2 2+···Dn n+···). Therefore, klog Lk ≤ − log a+δ a+ (δ a)2 2+··· (δ a)n n+···=−log a−log(1 −δ a) = −log(a−δ). Using the variation of constants formula in the time t= 1, we obtain z(1, z) = eAz+Z1 0 eA(1−s)G(z(s, z))ds (2.3) where z(t, z) indicates the solution such that z(0, z) = z. INVERTIBLE CONTRACTIONS AND ODE’S THAT ARE NOT C1-LINEARIZABLE. 9 Using the special form of G, we can show that the above equation is equivalent to the system: x(1, x) = e−αx ~y(1, ~y) = eA1~y +R1 0eA1(1−s)~ β(x(s, x))2ds =eA1~y +R1 0eA1(1−s)e−2αs ds ~ β x2(2.4) where −α= log aand A1= log(δJ +aI). Since F(z) = (0, ε~ f(x))Tand ~ f(x) = (x2,0,0,···)Twe must show that there exists ~ β∈`2 such that: Z1 0 eA1(1−s)e−2αsds~ β= (ε, 0,0,···)T. Now we let τ= 1−s. Then dτ =−ds and 2ατ −2α=−2αs and so the previous equation is equivalent to: Z1 0 e(A1+2αI)τdτ ~ β= (ε e2α,0,0,···)T. Let B:= A1+ 2αI and h(B) := R1 0eBτ dτ. By taking the series expansion of eBτ and integrating term by term, we obtain: h(B) = Z1 0 eBτ dτ =I+B 2! +B2 3! +··· Bn (n+ 1)! ··· Next we are going to prove that (h(B))−1exists and is bounded, or in other words, that 0 does not belong to σ(h(B)). Let h:C→Cthe analytic function: h(ξ) := Z1 0 eξτ dτ =   1,if ξ= 0 ξ−1(eξ−1),if ξ6= 0 From the Spectral Mapping Theorem it follows that h(σ(B)) = σ(h(B)).But h(ξ) = 0 if and only if ξ= 2nπi for ninteger, n6= 0. Let us now estimate the spectrum of B= log(δJ +aI) + 2αI. Here we are considering log as the principal branch of the logarithm function. If ξ∈log(σ(δJ +aI)) then from Lemma 2 it follows that ξ= log(reiθ) = log r+iθ, where r∈[a−δ, a +δ] and θ∈(−γ,γ) , for some γ∈(0,π 2).Since α=−log a, we can conclude