Numerical computation of dichotomy rates and projectors in discrete time

We introduce a characterization of exponential dichotomies for linear difference equations that can be tested numerically and enables the approximation of dichotomy rates and projectors with high accuracy. The test is based on computing the bounded solutions of a specific inhomogeneous difference equation. For this task a boundary value and a least squares approach is applied. The results are illustrated using Hénon’s map. We compute approximations of dichotomy rates and projectors of the variational equation, along a homoclinic orbit and an orbit on the attractor as well as for an almost periodic example. For the boundary value and the least squares approach, we analyze in detail errors that occur, when restricting the infinite dimensional problem to a finite interval.


Introduction
Hyperbolicity of the linear difference equation can be expressed in terms of an exponential dichotomy, see [8,12,16].At each time n, an exponential dichotomy defines a splitting into two subspaces, in which the solution decays exponentially fast in forward-and backward-time, respectively.
Denote by Φ the solution operator of (1), defined as for n < m.

Definition 1
The linear difference equation ( 1) possesses an exponential dichotomy with data (K, α s , α u , P s n , P u n ) on J ⊂ , if there exist two families of projectors P s n and P u n = I − P s n and constants K, α s , α u > 0, such that the following statements hold: Φ(m, n)P u n ≤ Ke −αu(n−m) ∀n ≥ m, n, m ∈ J.
A typical example, where exponential dichotomies have a geometric interpretation, is a diffeomorphism f : Ê k → Ê k having a fixed point ξ and a homoclinic orbit x = (x n ) n∈ with respect to this fixed point, i.e. lim n→±∞ xn = ξ.This orbit is called transversal, if stable and unstable manifolds of ξ intersect transversally.
Equivalence of this transversality assumption to an exponential dichotomy on of the variational equation is well known, see [16].In a non-autonomous setup, in which f n depends on n, the same connection between transversality of fiber bundles and an exponential dichotomy of the variational equation holds true, when considering homoclinic trajectories, cf.[18,13].Therefore, the approximation of dichotomy rates and dichotomy projectors is an important task, which we face in this paper from a numerical point of view.First we note, that it is not our attention to find a computer assisted proof of an exponential dichotomy, like, for example, the rigorous computational shadowing results, introduced in [7].For an arbitrary sequence of matrices A = (A n ) n∈ , a computer assisted proof of an exponential dichotomy on is impossible without having further structural information.Numerically, one can only compute on finite intervals J = [n − , n + ]∩ .But if (1) possesses an exponential dichotomy on J, it does not necessarily has one on .A counterexample is A n = I for n / ∈ J. On the other hand, an exponential dichotomy on follows for almost periodic systems from an exponential dichotomy on a sufficiently large interval, see for example [1,17].Further note that for linear upper triangular ODEs, techniques for computing Sacker-Sell spectral intervals are introduced in [9], that are based on the relationships between different spectra.
In Section 2, we first prove that the dichotomy rates are related to the exponential rates of decay of the solution of the inhomogeneous equation where δ n,m is the Kronecker symbol.This result is applied in Section 3, to develop an algorithm for approximating dichotomy rates and the corresponding dichotomy projectors.Here it is crucial to solve system (4) on finite intervals with high accuracy.For this task, two different approaches are introduced.The first one is based on solving linear boundary value problems while the second one computes least squares solutions, see also [5] where similar techniques apply for a global error analysis of initial value ODEs.
As a toy model, Hénon's map is used.We approximate dichotomy rates of the variational equation ( 3), where x is a transversal or tangential homoclinic orbit, a sequence on the Hénon attractor or an almost periodic bounded trajectory.
Approximation errors of the solution of (4) on finite intervals are discussed in Section 4. For the boundary value approach, we analyze the influence of the chosen boundary operator.Furthermore, we show that effects, caused by matrices outside the finite interval J, decay exponentially fast.For the least squares approach, we prove an exponential estimate for the approximation error.

Computing dichotomy rates
In [12,Theorem 7.6.5],Henry proved that (1) possesses an exponential dichotomy on , if and only if the inhomogeneous equation has for each bounded sequence r := (r n ) n∈ a unique bounded solution, see also [6].
In this section, we give a modified result that allows the numerical computation of the corresponding dichotomy rates.
Theorem 2 Assume that A n is invertible for all n ∈ .Then the following assertions are equivalent.
(i) The inhomogeneous equation possesses for all N ∈ and r ∈ Ê k a unique bounded solution, fulfilling (ii) The difference equation ( 1) possesses an exponential dichotomy on with dichotomy constants K, α s , α u .
Proof: Assume (i) and denote by Φ the solution operator of (1).Since (5) has for each r ∈ Ê k and N ∈ a unique bounded solution, the homogeneous equation has no non-trivial bounded solution.We define the subspaces and it follows that S + ∩ S − = {0} and S + ⊕ S − = Ê k .Let P s 0 be the projector with R(P s 0 ) = S + and N (P s 0 ) = S − .The complementary projector is given as P u 0 := I − P s 0 and P s,u n := Φ(n, 0)P s,u 0 Φ(0, n), n ∈ .A solution of ( 5) can be constructed, using Green's function, see [16] where G is defined as By construction and as a consequence, u is the unique bounded solution that converges by assumption (6) exponentially fast to 0. It holds for n ≥ N + 1 and therefore This proves the first dichotomy estimate, since (9) holds for all N ∈ .The second dichotomy estimate follows from Thus, (1) possesses an exponential dichotomy on with data (K, α s , α u , P s n , P u n ).Using the explicit solution (7), it immediately follows, that the converse also holds true.
Note that due to linearity, it suffices to consider statement (i) for vectors r from a basis of Ê k .However, for the numerical tests in Section 3 we only take one generic vector r, which we choose at random.
Approximations of dichotomy projectors can be computed, by solving inhomogeneous linear systems.Corollary 3 Let condition (i) from Theorem 2 be fulfilled.Fix N ∈ and let u i be the solution of (5) for r = e i , i = 1, . . ., k, where e i is the i-th unit vector.
Then (1) possesses an exponential dichotomy on with stable projector Proof: For each i = 1, . . ., k, the solution u i of has the explicit form, see ( 7) and using (8), we get Finally, we analyze, whether two half-sided dichotomies on + and − with projectors of equal rank, lead to an exponential dichotomy on .Note that the variational equation (3) along a tangential homoclinic orbit possesses half sided dichotomies on + and − , but no exponential dichotomy on .
Proposition 4 Assume that the homogeneous difference equation ( 1) possesses an exponential dichotomy on + and − with projectors of equal rank.Then (1) has an exponential dichotomy on if and only if the inhomogeneous equation ( 5) has a bounded solution for N = 0 and all r ∈ Ê k .
Proof: Let u be a bounded solution of (5) on .To obtain a contradiction, suppose that the homogeneous equation (1) has non-trivial bounded solutions on which is equivalent to, see [16 where P +s,u n and P −s,u n denote the stable and unstable dichotomy projectors of (1) on + and − , respectively.
Since R(P +s 0 ) + R(P −u 0 ) ⊂ N (P +s 0 ) it follows that in general, an u 0 , fulfilling (10), does not exist, and as a consequence, the inhomogeneous equation has not for all r ∈ Ê k a bounded solution.
Thus, R(P +s 0 ) ∩ R(P −u 0 ) = {0} and we obtain an exponential dichotomy on .The converse is a direct consequence of Theorem 2.
The dichotomy test in Section 3 is based on Theorem 2. In order to prove an exponential dichotomy, we first have to check, if condition (6) holds true.This is realized in Section 3, by solving (5) on a finite interval.The logarithmic slope of the solution then gives approximations of the dichotomy constants α u and α s .For an approximation of dichotomy projectors with high accuracy, we use the idea, introduced in Corollary 3.

Numerical computations
Given a sequence of matrices A J = (A n ) n∈J , A n ∈ Ê k,k , we compute approximations of dichotomy constants and projectors of (1) as follows: (i) Solve (5) on a sufficiently large interval J.
(ii) Determine the exponential slope of the solution.
For the main task (i), we introduce two different approaches.We compute finite approximations applying a boundary value and a least squares ansatz, respectively.

A boundary value approach
The boundary value approach requires to solve the following linear equations on the finite interval J = [n − , n + ]: In (11), an extra variable c is introduced, to determine whether the homogeneous equation possesses a non-trivial bounded solution, see Proposition 4. The normalizing condition ( 13) is needed to get a square system.Finally, b : Ê 2k → Ê k is an appropriately chosen boundary operator.The concrete choice of boundary conditions and corresponding approximation results are discussed in Section 4. If the boundary operator is linear, system ( 11)-( 13) may equivalently be written as For the infinite dimensional problem ( 5), the dichotomy rates can be computed, by Theorem 2, from the logarithmic slope of the solution of ( 14).We use this idea also for our finite dimensional approximations and present its formal justification in Section 4.

A least squares approach
Assume that the homogeneous equation possesses an exponential dichotomy on .
As an alternative to the boundary value approach, we compute the least squares solution of and approximate dichotomy rates in a second step.System (5) can equivalently be written as Bu = R, where and we seek for the solution, that is minimal w.r.t.• 2 .The least squares solution ūJ of ( 15) is ūJ = B + R, and since B has full rank, the Moore-Penrose inverse B + can be computed as see [19].Note that BB T is a sparse matrix with a block-tridiagonal structure, which one can invert efficiently.Errors between the unique bounded solution on and the finite least squares approximation, are discussed in Section 4.2.

Dichotomy rates along a homoclinic orbit
For numerical computations, we consider the well known Hénon map [11] and compute a homoclinic orbit xn+1 = f (x n ), n ∈ with respect to the fixed point i.e. lim n→±∞ xn = ξ, cf.[3].Then our method applies to the variational equation Note that the matrix Df (ξ) has one stable and one unstable eigenvalue λ s and λ u , respectively.Therefore, the difference equation possesses an exponential dichotomy on with dichotomy rates Furthermore, Df (x n ) converges exponentially fast towards Df (ξ) as n → ±∞, see Figure 2.An L 1 roughness-theorem applies, see [8,2] and we obtain exponential dichotomies on − and + of ( 17) with the same exponential rates ᾱs , ᾱu .Since stable and unstable manifolds of ξ intersect transversally, [16, Proposition 2.6] proves an exponential dichotomy of (17) on with rates ᾱs , ᾱu .
We compare these exact values with numerical approximations.On the interval J = [−100, 100] a homoclinic orbit is computed, using the techniques introduced in [4], see Figure 1 (left).In a second step, we solve (14) for N = 0 and plot its solution u J in Figure 1 (right) in a logarithmic scale.
Due to Theorem 2, the dichotomy rates α s and α u are the slopes the solution u J .To obtain an approximation of α s , we take points (n, log( u n )) such that n ≥ 3 and u n > 10 −14 .By a least squares approach, we fit these points and get α s as the slope of the resulting line.For α u , the same approach is applied for n < 2. The resulting rates are α s = 1.85803258 and α u = 0.65401 when computing u J via the boundary value approach, while α s = 1.85803259 and α u = 0.65402 in case of minimizing.On the other hand, the exact dichotomy rates from (18) are ᾱs = 1.858243418746994 and ᾱu = 0.6542706144210578, thus the difference between theoretical and numerical results is of order 10 −4 .
The constant c from (11) indicates transversality, and its value is c = −0.971.Now we vary N and discuss the dependence of errors for α u , α s .For N = −20 the solution of ( 14) looks similar to Figure 1 (right), where the peak is shifted to n = −20.16) (left) and the solution u J of ( 14), plotted over n in a logarithmic scale, where  Since Df (ξ) − Df (x n ) converges exponentially fast to 0, see Figure 2, the equation is for n ≤ −20 nearly a constant coefficient problem.Thus the computation of α u yields better results for N = −20 than for N = 0. On the other hand, the influence of the non-constant matrices Df (x n ), |n| small, results in a worse approximation of α s .Numerically, we obtain A computation for N = 20 gives an opposite result.Note that one should not solve (14) for N close to n ± , since extra points on both sides of N are needed for an approximation of the dichotomy rates.

Approximation of dichotomy projectors
For the example from Section 3.3, the corresponding dichotomy projectors P s 0 are approximated numerically.Following the approach, introduced in Corollary 3, we fix N = −1 and solve ( 11)-( 13) for r = e 1 and r = e 2 on the interval J = [−100, 100].Denote by (u 1 J , c 1 ) and (u 2 J , c 2 ) the corresponding solutions.We compensate the scaling factors c 1 , c 2 and define Note that if one neglects the test for transversality, one can directly solve the inhomogeneous system (11), (12) with fixed c = 1, and right hand sides r = e 1 and r = e 2 , respectively.Alternatively, the least squares solution of ( 15) can be computed for r = e 1 and r = e 2 .
Numerically, we get in all cases For testing whether P s 0 is a precise approximation of the stable dichotomy projector, we first note that P s 0 • P s 0 − P s 0 2 = 2.8 • 10 −17 .Secondly, take a random vector v, define u 0 = P s 0 v, and compute the iterates In a hyperbolic system, the iteration of points from the stable subspace is highly sensitive to rounding errors.Thus the results in Table 1 suggest, that the approximation of the dichotomy projector is exact up to machine accuracy.This result is formalized in Section 4.1.3.

Hyperbolic almost periodic bounded sequences
Consider the non-autonomous difference equation where a n = 1.4 + 1 2 sin( 1 5 n) is an almost periodic sequence of parameters.We compute on the interval J = [−500, 500] a finite solution xJ of (20), see Figure 3 (left), using the algorithm that is introduced in [14].Consequently, the variational equation along this orbit is almost periodic.The solution of the inhomogeneous problem Note that an exponential dichotomy on a sufficiently large interval J guarantees for almost periodic systems the existence of an exponential dichotomy on , see [1,17].

Hyperbolic sequences on the Hénon attractor
We analyze, if a sequence on the Hénon attractor for the classical parameters a = 1.4,b = 0.3 is hyperbolic, i.e. the variational equation along this sequence possesses an exponential dichotomy.In the first step, a sequence on the attractor is constructed by iteration.We iterate 100 steps for an appropriate initial vector, to reach the attractor, and save the next 1001 steps in xn , n = −500, . . .500.With A n = Df (x n ), we then solve (5) for N = 0, see Figure 6, and the resulting dichotomy rates are α s = 1.66 and α u = 0.39.The dependence of α s , α u on the chosen N is shown in Figure 7. Here, boundary value and least squares approach lead to nearly identical results.
Since r is chosen at random, and therefore has a generic position, our computations suggest that the variational equation possesses an exponential dichotomy on the finite interval [−400, 400].Note that we cannot conclude from this result that Hénon's attractor is hyperbolic.Although the chosen trajectory lies dense on the attractor as J → , it does not contain, all bounded trajectories, as periodic orbits, for example.

A tangential homoclinic orbit
Consider parameters for which tangential homoclinic orbits of Hénon's map exist.In this case, the variational equation ( 17) possesses exponential dichotomies on − and + but no dichotomy on , since (17) has non-trivial bounded solutions, cf.[15,3].In this case, the value of the constant c in (11) should be 0, cf.Proposition 4.
For the parameters a = 1.4 and b = 0.1014563589 in Hénon's map, we are close to a homoclinic tangency.When solving (14) on the interval J = [−200, 200] with N = 0, we obtain a transversality constant c of magnitude 10 −4 , and an error in the dichotomy rates |α s,u − ᾱs,u | of order 10 −3 .

Justification of the numerical process
In Sections 3.1, 3.2 two approaches are introduced for getting finite approximations of the solution of (5).In both cases, we will present a detailed error analysis under the assumption that the homogeneous difference equation possesses an exponential dichotomy on .

A1 The linear difference equation
with uniformly bounded invertible matrices A n , possesses an exponential dichotomy on with data (K, α s , α u , P s n , P u n ).

The boundary value approach
By assuming an exponential dichotomy on , the free variable c in (11) that tests for transversality, is not needed and thus, the normalizing condition (13) also vanishes.When choosing a linear boundary operator, ( 14) reduces to We analyze errors that occur, when solving this finite dimensional problem (23) instead of the infinite dimensional system (22).

Effects of matrices with large |n|
Let J be a finite interval.Note that the system (23) does not depend on the matrices A n , for which n lies outside the interval J. Thus no matter how A n is defined for n / ∈ J, we always obtain the same finite approximation.Fortunately, the influence from the outer matrices decays exponentially fast towards the middle of the interval J.This is proven for non-autonomous systems, generated by a parameter dependent non-linear map, in [14, Theorem 4].Here, we state the corresponding linear result.
Proposition 5 Assume A1.Let J be a finite interval and let B J be a family of matrix sequences, fulfilling B J n = A n , for n ∈ J. Assume that the difference equation possesses for each J an exponential dichotomy on with J-independent constants L, β s , β u .
For N ∈ J and r ∈ Ê k denote by ū and v the unique bounded solutions of Then there exists a J-independent positive constant C, such that the following estimate holds for n Proof: For n ∈ define d n := ūn − vn .Then where q n = 0, n ∈ J, A n vn − vn+1 , n / ∈ J. Since ( 22) possesses an exponential dichotomy on , the unique bounded solution of where Green's function G is defined in (8).By Theorem 2 and since A n is uniformly bounded, there exists a constant C > 0, such that Using the dichotomy estimates, we get for n ∈ J By construction, d is the unique bounded solution of (24) and therefore 1−e −αu −βs .

Approximation results
We show that errors of the boundary value approach essentially depend on the chosen boundary operator.Assume A1, and define the operator Γ on a finite interval J = [n − , n + ]: .

Projection boundary conditions require the following assumptions:
A2 There exist two complementary projectors Q s and Q u having the same rank as the stable and unstable dichotomy projectors P s n and P u n , and an angle 0 The angle between two subspaces A and B is defined as (see [10]) , where cos θ = max As boundary operator, we either take periodic or projection boundary conditions, defined as b per (x, y) := x − y, (25) where the columns of Y s and Y u form a basis of R(Q u ) ⊥ and R(Q s ) ⊥ .We impose the following regularity assumption, cf.[13,Lemma 4.3]: is invertible for sufficiently large −n − , n + , and has a uniformly bounded inverse, i.e. there exist a C > 0 and an L ∈ AE such that Note that for projection boundary conditions (26), assumption A3 directly follows from A2, see Appendix A, Lemma 11.
Denote by ū the unique bounded solution of the infinite dimensional problem ( 5) and let ū|J be its restriction to the finite interval J. Then Γ J u J = r J possesses for sufficiently large intervals J a unique solution u J and the error can be estimated as Proof: By [13, Lemma 4.3] it follows that the operator Γ J has for sufficiently large −n − , n + a uniformly bounded inverse and consequently In case of periodic boundary conditions, (27) reads, The disadvantage of these boundary conditions obviously lies in the coupling of the end-points, which is not the case for projection boundary conditions.In numerical computations, this effect can clearly be observed, cf. Figure 8. Combining Propositions 5 and 6, we prove for projection boundary conditions that approximation errors (27) decrease exponentially fast towards the middle of the interval J.
Proposition 7 Assume A1 and A2.Let u J be the solution of Γ J u J = r J with projection boundary conditions (26), where J = [n − , n + ], r J := (0, . . ., 0, r, 0, . . ., 0) T , and r ∈ Ê k is placed at N-th position, n − ≤ N ≤ n + − 1.Then, a J-independent positive constant C exists, such that the following inequality for the error is satisfied: Proof: Applying Proposition 6 and the definition of the projection boundary operator (26), it follows that Note that we get the same finite approximation for all sequences B that coincide with A on the finite interval J. On the other hand, the numerical approximation and the exact solution coincide on J, if the exact solution is a zero of the right hand side of (29), which means that the boundary operator is exact.Thus, the idea is, to construct a sequence B , fulfilling Furthermore, we construct B , such that the homogeneous equation possesses an exponential dichotomy with J-independent rates α s and α u .The error is the difference between ū| Ĵ and v| Ĵ for which, due to Proposition 5, the estimate ūn − vn ≤ C e −αs(n−n − −1) e −αu(N −n − −1) + e −αu(n + −2−n) e −αs(n + −2−N ) , n ∈ Ĵ holds and by adjusting the constant C, we obtain (28).
In the remaining part of the proof, we construct the sequence B .Let J be a finite interval and let P s n , P u n be the dichotomy projectors of (22) on J. Let A be a matrix with stable subspace R(Q s ) and unstable subspace R(Q u ), having in absolute value the largest stable and smallest unstable eigenvalues λ s = e −αs , λ u = e αu , respectively.Assuming that these eigenvalues are simple, u n+1 = A n u n possesses an exponential dichotomy on with rates α s and α u .
We define the sequence B :

Approximation with exact boundary conditions
By computing the exact boundary operator first, we obtain an algorithm that gives high accuracy approximations of the solution of ( 5) on a given finite interval J = [n − , n + ]: • Approximate the dichotomy projectors P s n − and P u n + with accuracy ∆. • Compute bases Y s , Y u of the orthogonal complement of R(P u n − ) and R(P s n + ), respectively, and define the boundary operator b opt as in (26).
Note that it requires solving 2k inhomogeneous equations on intervals around n − and n + , to get the two dichotomy projectors.
Denote by u J the solution of ( 23) with boundary operator b opt .By Proposition 6, the error can be estimated as These results are illustrated in Figure 9 for the example from Section 3.6.As reference orbit ū , we take a long orbit segment of length −n − , n + = 10000 which we compare with short segments u J of length n − = −30, n + = 10.In the left picture we take projection boundary conditions w.r.t. a fixed subspace.As one can see, the error decreases exponentially fast towards the middle, see Proposition 7. In the right of Figure 9, the error is plotted, when solving w.r.t. the exact boundary operator b opt .In this case, errors are of magnitude 10 −22 .
We prove that the difference between ū and the least squares solution of (31) on the finite interval J = [n − , n + ], decreases exponentially fast to 0 as −n − , n + → ∞.
Theorem 8 Assume A1.Denote by u J the least squares solution of (31) on J. Then a J independent constant C > 0 exists, such that where α = min{α s , α u }.
Proof: Let B s n , B u n be orthonormal bases of R(P s n ), R(P u n ), and denote by k s , k u the dimensions of these subspaces.The general solution of (31) on J is We find the minimal solution of (31) on J by computing the least squares solution of Note that A has rank k, therefore F is minimal for see [19], and the least squares solution of (31) on J, is given as We derive an estimate for the difference between ū and least squares solution u J : First it is shown that A T A has for all sufficiently large −n − , n + a uniformly bounded inverse.
The non-diagonal blocks tend exponentially fast to 0 as The matrices D 1 J and D 2 J are uniformly bounded from above: and similarly D 2 J 2 ≤ K 2 1 1−e −2αu .Furthermore D 1,2 J are symmetric, positive definite matrices and B s,u n ∓ are orthonormal bases, thus it follows that As a consequence A T A is invertible for sufficiently large intervals J and has a uniformly bounded inverse.Using the dichotomy estimates, one sees that where C does not depend on the interval J.
Denote by A n the n-th block component of A. We derive an estimate for the n-th block component of where In case ℓ ≤ 0 we get and similarly, it holds for ℓ ≥ 1 As a consequence, with some constant C > 0 that does not depend on n − and n + .
Finally, we show numerically that the derived estimate for the error is quite sharp.We take the example from Section 3.3 for which the dichotomy rates are given in (18), choose symmetric intervals of the form J = [−N, N] and compute a reference solution, by setting N = 10000.The differences of this solution to short least squares solutions for N ∈ {2, . . ., 50} are shown in Figure 10 (black line).Neglecting constant terms, the approximation error (32) has the form Ne − ᾱuN , see the grey line in Figure 10.

A Appendix
Some technical results, needed in Section 4.1.2are presented in this appendix.
Lemma 9 Fix σ ∈ (0, π 2 ].Let A, B be two projectors in Ê k with range A s , B s and nullspace A u , B u , respectively, where dim(A s ) = dim(B s ), such that ∡(A s , A u ) ≥ σ, ∡(B s , B u ) ≥ σ.
Then there exist a constant C > 0 that depends on σ but not on the specific choice of A, B, and an invertible matrix T ∈ Ê k,k , such that Proof: Let α s , α u , β s , β u be orthonormal bases of range and nullspace of A and B. We determine T uniquely as T (α s ) = β s , T (α u ) = β u .
Then each x ∈ Ê k can be written in the form x = α s λ s + α u λ u , λ s ∈ Ê dim(As) , λ u ∈ Ê dim(Au) and thus, we get Note that This finishes the proof, since the same estimate holds for T −1 .
Proof: Let α, β be orthonormal bases of R(P ), N (P ), respectively.It holds = sup Lemma 11 Assume A2 and let the columns of Y s form an orthonormal basis of R(Q u ) ⊥ .Then there exists for sufficiently large n, an n-independent constant C > 0 such that Y T s x ≥ C x for all x ∈ R(P s n ).Proof: Let y be a column of Y s , and let x ∈ R(P s n ).It holds that y T x = y 2 x 2 cos(∡(y, x)) ≥ x 2 cos(∡(R(Q u ) ⊥ , R(P s n ))) ≥ x 2 cos σ, since ∡(R(Q u ) ⊥ , R(P s n )) ≤ σ, see A2.

Figure 1 :
Figure1: A homoclinic orbit x J of (16) (left) and the solution u J of (14), plotted over n in a logarithmic scale, where N = 0, A n = Df (x n ), n ∈ J (right).

2 nFigure 5 :
Figure 5: An orbit of length 1001 on the Hénon attractor, plotted over its index n (left), and in phase space (right).

Figure 6 :
Figure 6: Solution u J of (14), plotted over n in a logarithmic scale.

Figure 9 :
Figure 9: Error d n = u n − ūn for projection boundary conditions w.r.t.fixed subspaces, and for the boundary operator b opt that uses accurate approximations of dichotomy projectors (right).

Figure 11
the error d n := ūn − u n between reference and least squares solution is plotted over n for N ∈ {5, 50, 200}. of the least squares solution is analyzed in Theorem 8: u J − ū|J ≤ C(n + − n − ) e αn − + e −αn + .