Homoclinic trajectories of non-autonomous maps

Homoclinic trajectories of non-autonomous maps Thorsten Hüls Department of Mathematics Bielefeld University Germany huels@math.uni-bielefeld.de www.math.uni-bielefeld.de:~/huels Non-autonomous dynamical systems Spectral theory Invariant fiber bundles Homoclinic trajectories Hyperbolicity Non-autonomous bifurcations Skew product flow Asymptotic behavior Pullback attractors

Non-autonomous dynamical systems P. E. Kloeden and M. Rasmussen. Nonautonomous dynamical systems, volume176of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 211. C. Pötzsche. Geometric theory of discrete nonautonomous dynamical systems, volume22oflecture Notes in Mathematics. Springer, Berlin, 21. M. Rasmussen. Attractivity and bifurcation for nonautonomous dynamical systems, volume197oflecture Notes in Mathematics. Springer, Berlin, 27. Outline Homoclinic orbits in autonomous systems

Homoclinic orbits in autonomous systems Let f : k k be a smooth diffeomorphism, ξ be a hyperbolic fixed point, i.e. σ(df (ξ)) {x : x = 1} =. Assume that stable and unstable manifold of ξ intersect transversally. 1 x 1.5 x ξ W u (ξ).5 W s (ξ) x2 x1 1 1.5 1.5.5 1 1.5 Definition Homoclinic orbit: x = (xn )n : xn+1 = f (xn ), n Thorsten Hu ls (Bielefeld University), lim n ± xn = ξ. ICDEA 212 Homoclinic trajectories of non-autonomous maps Homoclinic orbits in autonomous systems Remark The dynamics near transversal homoclinic orbits is chaotic, cf. the celebrated Smale-S il nikov-birkhoff Theorem. L. P. S il nikov. Existence of a countable set of periodic motions in a neighborhood of a homoclinic curve. Dokl. Akad. Nauk SSSR, 172:298 31, 1967. Soviet Math. Dokl. 8 (1967), 12 16. S. Smale. Differentiable dynamical systems. Bull. Amer. Math. Soc., 73:747 817, 1967. K. J. Palmer. Shadowing in dynamical systems, volume 51 of Mathematics and its Applications. Kluwer Academic Publishers, Dordrecht, 2. Thorsten Hu ls (Bielefeld University) ICDEA 212 Homoclinic trajectories of non-autonomous maps

Approximation of homoclinic orbits in autonomous systems Compute a finite orbit segment by solving a boundary value problem. x n,...,x n+ Simplest case: periodic boundary conditions: x = x n.. x n+, Γ(x) = ( xn+1 f(x n ), n = n,...,n + 1 x n x n+ Often successful: Rough initial guess for Newton s method: ( ) u =(ξ,...,ξ, g, ξ,...,ξ) T 1, e.g. g = in the Hénon example. 1 ). Approximation of homoclinic orbits in autonomous systems Compute a finite orbit segment by solving a boundary value problem. x n,...,x n+ Simplest case: periodic boundary conditions: x = x n.. x n+, Γ(x) = ( xn+1 f(x n ), n = n,...,n + 1 x n x n+ Alternative: Compute initial guess via approximations of stable and unstable manifolds. R. K. Ghaziani, W. Govaerts, Y. A. Kuznetsov, and H. G. E. Meijer. Numerical continuation of connecting orbits of maps in MATLAB. J. Difference Equ. Appl., 15(8-9):849 875,29. ).

Approximation of homoclinic orbits in autonomous systems Example: Hénon s map: ( ) x H y ( 1 + y ax 2 = bx ), typical parameters: a = 1.4, b =.3. n = 6, n + = 6.3.2.1.1.2 1.5.5 1 1.5 Approximation of homoclinic orbits in autonomous systems W.-J. Beyn, The numerical computation of connecting orbits in dynamical systems. IMA J. Numer. Anal., 1,379 45, 199. W.-J. Beyn and J.-M. Kleinkauf, The numerical computation of homoclinic orbits for maps. SIAM J. Numer. Anal., 34, 127 1236, 1997. J.-M. Kleinkauf, Numerische Analyse tangentialer homokliner Orbits. PhD thesis, Universität Bielefeld, ShakerVerlag,Aachen,1998. Y. Zou and W.-J. Beyn, On manifolds of connecting orbits in discretizations of dynamical systems. Nonlinear Anal. TMA, 52(5),1499 152, 23. W.-J. Beyn and Th. Hüls. Error estimates for approximating non-hyperbolic heteroclinic orbits of maps. Numer. Math., 99(2):289 323, 24. Th. Hüls. Bifurcation of connecting orbits with one nonhyperbolic fixed point for maps. SIAM J. Appl. Dyn. Syst., 4(4):985 17,25. W.-J. Beyn, Th. Hüls, J.-M. Kleinkauf, and Y. Zou, Numerical analysis of degenerate connecting orbits for maps. Internat. J. Bifur. Chaos Appl. Sci. Engrg., 14,3385 347,24.

Outline Nonautonomous analog of homoclinic orbits Non-autonomous analog of a fixed point Non-autonomous discrete time dynamical system x n+1 = f n (x n ), n, f n family of smooth diffeomorphisms autonomous vs. non-autonomous fixed point bounded trajectory ξ ξ : ξ n+1 = f n (ξ n ), n J. A. Langa, J. C. Robinson, and A. Suárez. ξ n C n Stability, instability, and bifurcation phenomena in non-autonomous differential equations. Nonlinearity, 15(3):887 93,22.

Non-autonomous analog of hyperbolicity Non-autonomous discrete time dynamical system x n+1 = f n (x n ), n, f n family of smooth diffeomorphisms Hyperbolicity of afixedpointξ aboundedtrajectoryξ Df(ξ)has The variational equation no center eigenvalue u n+1 = Df n (ξ n )u n, n has an exponential dichotomy on. Acounterexample,showingthateigenvaluesaremeaninglessforanalyzing stability in non-autonomous systems was given by Vinograd (1952), cf. F. Colonius and W. Kliemann. The dynamics of control. Systems & Control: Foundations & Applications. Birkhäuser Boston Inc., Boston, MA, 2. Exponential dichotomy Consider the linear difference equation u n+1 = A n u n, n, u n k, A n GL(k; ). (1) A n 1...A m for n > m Solution operator of (1): Φ(n, m) := I for n = m Definition A 1 n...a 1 m 1 for n < m The linear difference equation (1) has an exponential dichotomy with data (K, α, Pn s, Pu) n on J =[n, n + ] if there exist 2familiesofprojectorsPn s, Pu n, n J, withps + n Pu n = I for all n J and constants K, α >, such that P κ n Φ(n, m) =Φ(n, m)pκ m n, m J, κ {s, u}, Φ(n, m)p s m K e α(n m) Φ(m, n)p u n K e α(n m) n m, n, m J.

Exponential dichotomy Some references W. A. Coppel. Dichotomies in Stability Theory. Springer-Verlag, Berlin, 1978. Lecture Notes in Mathematics, Vol. 629. D. Henry. Geometric Theory of Semilinear Parabolic Equations. Springer-Verlag, Berlin, 1981. K. J. Palmer. Exponential dichotomies, the shadowing lemma and transversal homoclinic points. In Dynamics reported, Vol. 1, pages265 36.Teubner,Stuttgart,1988. Non-autonomous analog of a homoclinic orbit Non-autonomous discrete time dynamical system x n+1 = f n (x n ), n, f n family of smooth diffeomorphisms autonomous vs. non-autonomous homoclinic orbit homoclinic trajectories: An orbit x that satisfies Two bounded trajectories x, ξ satisfying lim n ± x n = ξ lim n ± x n ξ n = Note that: x is homoclinic to ξ ξ is homoclinic to x

Outline Step 1 Step 2 Approximation of aboundedtrajectoryξ. Approximation of asecondtrajectoryx that is homoclinic to ξ. Setup x n+1 = f n (x n ), n f n is generated by a parameter-dependent map f n = f(, λ n ), λ sequence of parameters. Assumptions ❶ Smoothness: f ( k, k ), f(, λ) diffeomorphism for all λ. ❷ There exists λ such that ξ n+1 = f( ξ n, λ n ), n has a bounded solution ξ. ❸ The variational equation u n+1 = D x f( ξ n, λ n )u n, n has an exponential dichotomy on.

Approximation of bounded trajectories Bounded trajectory ξ Zero of the operator Γ : X X,definedas Γ(ξ, λ ):= ( ξ n+1 f(ξ n, λ n ) ) n. Space of bounded sequences on the discrete interval J { X J := u J =(u n ) n J ( } k ) J : sup u n < n J Approximation of bounded trajectories Lemma Assume ❶ ❸. Then there exist two neighborhoods U( λ ) and V( ξ ), suchthat Γ(ξ, λ )= has for all λ U( λ ) auniquesolutionξ V( ξ ). Lemma Assume ❶ ❸. Then there exist two neighborhoods U( λ ) and V( ξ ), suchthat u n+1 = D x f(x n, λ n )u n, n has an exponential dichotomy on for any sequence x V( ξ ), λ U( λ ). Thedichotomyconstantsareindependentofthespecific sequence x.

Approximation of bounded trajectories Aim: Numerical approximations of a bounded solution of Problem ξ n+1 = f(ξ n, λ n ) on the finite interval J =[n, n + ]. ξ J =(ξ n ) n J depends on λ n, n J, but also on λ n, n / J. J n n + Fortunately The influence of λ n, n / J decays exponentially fast towards the middle of the interval J. Thus, numerical approximations with high accuracy can be achieved. Similar observation in autonomous systems P. Diamond, P. Kloeden, V. Kozyakin, and A. Pokrovskii. Expansivity of semi-hyperbolic Lipschitz mappings. Bull. Austral. Math. Soc., 51(2):31 38,1995. K. J. Palmer. Shadowing in dynamical systems, volume51ofmathematics and its Applications. Kluwer Academic Publishers, Dordrecht, 2.

Approximation of bounded trajectories Theorem Assume ❶ ❸. Let J =[n, n + ] be a finite interval and U( λ ), V( ξ ) as stated above. Choose λ, µ U( λ ) such that λ n = µ n for n J. Denote by ξ, η V( ξ ) the bounded solutions w.r.t. λ and µ. Then there exist constants C, α > that do not depend on λ ξ n η n C λ µ J and µ,suchthat ( e α(n n ) + e α(n + n) n n+ holds for all n J. Proof ( ξ n η n C ( e α(n n ) + e α(n + n) ),whereλ n = µ n, n J =[n, n + ]) ξ n+1 = f(ξ n, λ n ), η n+1 = f(η n,µ n ), d := η ξ, h := µ λ. d n+1 = f(ξ n + d n, λ n + h n ) f(ξ n, λ n ) = f(ξ n + d n, λ n )+ D λ f(ξ n + d n, λ n + τ h n )dτ h n f(ξ n, λ n ) = f(ξ n, λ n )+ D x f(ξ n + τ d n, λ n )dτ d n + D λ f(ξ n + d n, λ n + τ h n )dτ h n f(ξ n, λ n ) = = A n d n + r n D x f(ξ n + τ d n, λ n )dτ d n + D λ f(ξ n + d n, λ n + τ h n )dτ h n where A n = D x f(ξ n + τ d n, λ n )dτ r n = D λ f(ξ n + d n, λ n + τ h n )dτ h n.

Proof ( ξ n η n C ( e α(n n ) + e α(n + n) ),whereλ n = µ n, n J =[n, n + ]) A n = D x f(ξ n + τ d n, λ n )dτ r n = D λ f(ξ n + d n, λ n + τ h n )dτ h n By assumption ❸ u n+1 = D x f( ξ n, λ n )u n, n has an exponential dichotomy on. Due to the Roughness-Theorem and our construction of neighborhoods, u n+1 = A n u n, n has an exponential dichotomy on with data (K, α, P s n, Pu n ). Solution operator: Φ(n, m), i.e.u n = Φ(n, m)u m Proof ( ξ n η n C ( e α(n n ) + e α(n + n) ),whereλ n = µ n, n J =[n, n + ]) A n = D x f(ξ n + τ d n, λ n )dτ r n = D λ f(ξ n + d n, λ n + τ h n )dτ h n Unique bounded solution of u n+1 = A n u n + r n on : u n = m G(n, m + 1)r m, where G is Green s function, defined as G(n, m) = { Φ(n, m)p s m, n m, Φ(n, m)pm u, n < m. Estimates: G(n, m) = Φ(n, m)p s m K e α(n m), for n m, G(n, m) = Φ(n, m)p u m K e α(m n), for n < m.

Proof ( ξ n η n C ( e α(n n ) + e α(n + n) ),whereλ n = µ n, n J =[n, n + ]) A n = D x f(ξ n + τ d n, λ n )dτ r n = D λ f(ξ n + d n, λ n + τ h n )dτ h n u n = m G(n, m+1)r m, G(n, m) = { Φ(n, m)p s m K e α(n m), n m, Φ(n, m)pm K u e α(m n), n < m. u n = n 1 m= n 1 m= G(n, m + 1)r m + RK e α(n m 1) + m=n + +1 m=n + +1 G(n, m + 1)r m RK e α(m+1 n) ( RK ) 1 e α e α(n n ) + e α(n + n+2), n J. Proof ( ξ n η n C ( e α(n n ) + e α(n + n) ),whereλ n = µ n, n J =[n, n + ]) A n = D x f(ξ n + τ d n, λ n )dτ r n = D λ f(ξ n + d n, λ n + τ h n )dτ h n u n u n+1 = A n u n + r n, n (2) RK ( ) 1 e α e α(n n ) + e α(n + n+2), n J. d = ξ η is the unique bounded solution of (2), thus ( ) d n = ξ n η n C e α(n n ) + e α(n + n), n J.

Approximation of bounded trajectories ξ :zerooftheoperatorγ : X X Γ(ξ, λ ):= ( ξ n+1 f(ξ n, λ n ) ) n. Finite approximation z J on J =[n, n + ] (zn+1 Γ J (z J, λ J ):=( f(z n, λ n ) ) ) n [n,n + 1], b(z n, z n + ) = with periodic boundary operator b(z n, z n+ ):=z n z n+. Approximation of bounded trajectories For all parameter sequences that coincide on J, λ we get the same numerical approximation. µ µ Thus, we choose µ n = const for all n / J. J J

Approximation of bounded trajectories with constant tails (zn+1 Γ J (z J, λ J ):=( f(z n, λ n ) ) ) n [n,n + 1], b(z n, z n + ) = Assumption ❹ There exist sequence µ U( λ ) with solution η V( ξ ) and µ, η k such that lim µ n = n + Theorem Assume ❶ ❹. lim µ n =: µ and lim η n = n n + lim η n =: η. n Then constants δ, N,C> exist, such that Γ J (z J, µ J )=, withperiodic boundary conditions, has a unique solution z J B δ ( η J ) for J =[n, n + ], n, n + N. Approximation error: η J z J C η n η n+. Approximation of bounded trajectories with tolerance λ J 1 : ξ J η J 2 if λ n = µ n, n J 1 µ choose µ n n / J 1 = µ for ξ I : η I z I 2 where Γ I (z I )= For n J we get η ξ n z n ξ n η n + η n z n I J 1 J 2 + 2 =

Outline Numerical experiments: Computation of bounded trajectories Example: Computation of bounded trajectories Hénon s map x h(x, λ, b) =( 1 + x2 λx 2 1 bx 1 ) Fix b =.3 andchooseλ [1, 2] at random. Non-autonomous difference equation x n+1 = h(x n, λ n, b), n. M. Hénon. Atwo-dimensionalmappingwithastrangeattractor. Comm. Math. Phys., 5(1):69 77,1976.

Example: Computation of bounded trajectories Solutions of Γ J = onj =[ 4, 4] for two sequences λ J that coincide on [-2,2]. 2 λ n 1 4 3 2 1 1 2 3 4.8 n.7 x n.6.5 4 3 2 1 1 2 3 4 n Example: Computation of bounded trajectories on J =[ 15, 15] Given a sequence λ J [1, 2] J with corresponding solution ξ J. Let µ J [1, 2] J such that λ n = µ n for n [ 1, 1] and solution η J. 1 d n = ξ n η n, n J for 1 different sequences µ J. 1 2 d n 1 4 1 6 1 8 1 1 1 12 1 14 1 16 15 1 5 5 1 15 n n n + n + Interval where d n n = n + = n + n + log α, log α +, α ± dichotomy constants, = 1 16.

Example: Computation of bounded trajectories on J =[ 2, 2] Choose a buffer interval [ n, n + ] such that we get an accurate approximation on [ 2, 2]: n = n log α = 4 and n + = n + + log = 74. α +.8.8.75.75.7.7.65.65 x 1.6 x 1.6.55.55.5.5.45 4 2 2 4 6.45 2 15 1 5 5 1 15 2 n n n + n + n n + Outline Step 1 (done) Step 2 Approximation of aboundedtrajectoryξ. Approximation of asecondtrajectoryx that is homoclinic to ξ.

Homoclinic trajectories Assumptions ❺ Let λ as in ❷. Asolution x of x n+1 = f(x n, λ n ), n exists, that is homoclinic to ξ and non-trivial, i.e. x ξ. ❻ The trajectory x is transversal, i.e. u n+1 = D x f( x n, λ n )u n, n for u X u =. Lemma Assume ❶ ❻. Thenthedifferenceequation has an exponential dichotomy on. u n+1 = D x f( x n, λ n )u n, n. Homoclinic trajectories x is homoclinic to ξ if and only if ȳ defined as ȳ n = x n ξ n is a homoclinic orbit of y n+1 = g(y n, λ n ):=f(y n + ξ n, λ n ) ξ n+1 n w.r.t. the fixed point. The f and g-system are topologically equivalent due to the kinematic transformation, cf. B. Aulbach and T. Wanner. Invariant foliations and decoupling of non-autonomous difference equations. J. Difference Equ. Appl., 9(5):459 472,23.

Approximation of homoclinic trajectories g n (y) :=f(y + ξ n, λ n ) ξ n+1, y n+1 = g n (y n ), g n () =, n. Approximation Γ J (y J ):=( (yn+1 g n (y n ) ) n [n,n + 1], y n y n + ) = with periodic boundary conditions. Assumptions ❼ Denote by P s n, P u n the dichotomy projectors of u n+1 = Df( ξ n, λ n )u n, n. Assume for all sufficiently large n, n + : (R(P s n ), R(P u n + )) > σ, for a < σ < π 2. Approximation of homoclinic trajectories g n (y) :=f(y + ξ n, λ n ) ξ n+1, y n+1 = g n (y n ), g n () =, n. Theorem Assume ❶ ❽. Then there exist constants δ, N,C >, such that Γ J (y J )=, withprojectionboundaryconditions,hasauniquesolution y J B δ (ȳ J ) for all J =[n, n + ], where n,n + N. Approximation error: ȳ J y J C ȳ n ȳ n+. Th. Hüls. Homoclinic orbits of non-autonomous maps and their approximation. J. Difference Equ. Appl., 12(11):113 1126,26.

Outline Numerical experiments: Computation of homoclinic trajectories (a) for the Hénon system, (b) for a predator-prey model. Hénon system: Computation of homoclinic trajectories Homoclinic orbit of the transformed system y n+1 = h(y n + ξ n, λ n, b) ξ n+1, n J, h : Hénon s map.5.5 1 y 2 y 1.5.5 2 1 n 1 2 1 1.5 Homoclinic orbit w.r.t. the fixed point.

Hénon system: Computation of homoclinic trajectories Homoclinic trajectories Let x n = y n + ξ n, n J. Then x J and ξ J are two homoclinic trajectories. 1.5 1.5 1.5.5 1 x 1 x 2.5.5 2 1 n 1 2.5 1 x 1.5 1 2 15 1 5 5 1 15 2 n Predator-prey model: Computation of homoclinic trajectories Predator-prey model ( xn+1 y n+1 ) = ( x n exp ( ( ) ) a 1 x n K n by n ( cx n 1 exp( byn ) ) ), n. x n prey at time n, y n predator at time n, K n carrying capacity, a = 7, b =.2, c = 2. J. R. Beddington, C. A. Free, and J. H. Lawton. Dynamic complexity in predator-prey models framed in difference equations. Nature, 255(553):58 6,1975. J. D. Murray. Mathematical biology. I, volume17ofinterdisciplinary Applied Mathematics. Springer-Verlag, New York, third edition, 22.

Predator-prey model: Computation of homoclinic trajectories ( xn+1 y n+1 ) = ( ( ( ) )) x n exp a 1 x n K n by n ( cx n 1 exp( byn ) ), n. K n = 1 + µ r n, r n [ 1, 1 ] uniformly distributed, µ [, 1]. 2 2 15 x 1 5 1 µ.5 2 1 n 1 2 Outline Some remarks and extended results

Remarks: Invariant fiber bundles Stable and unstable fiber bundles are the non-autonomous generalization of stable and unstable manifolds: Ψ : solution operator of x n+1 = f n (x n ), } S s (ξ ) = {x k : lim Ψ(m, )(x) ξ m =, m } S u (ξ ) = {x k : lim Ψ(m, )(x) ξ m =. m Approximation results: C. Pötzsche and M. Rasmussen. Taylor approximation of invariant fiber bundles for non-autonomous difference equations. Nonlinear Anal., 6(7):133 133,25. Let x be a homoclinic trajectory w.r.t. ξ.then x S s (ξ ) S u (ξ ). Heteroclinic trajectories Autonomous world: Heteroclinic orbit. Let ξ + and ξ two fixed points. Atrajectoryx is heteroclinic w.r.t. ξ ±,if lim n x n = ξ, lim n x n = ξ +. Non-autonomous analog: Heteroclinic trajectories. Let ξ be a trajectory that is bounded in backward time, and let ξ + be a trajectory that is bounded in forward time. Atrajectoryx is heteroclinic w.r.t. ξ ±,if lim n x n ξ n =, lim n x n ξ + n =. Th. Hüls and Y. Zou. On computing heteroclinic trajectories of non-autonomous maps. Discrete Contin. Dyn. Syst. Ser. B, 17(1):79 99,212.

Heteroclinic trajectories Σ u S s (x ) x ξ ξ + + S u (x ) Σ s Problem: Families of semi-bounded trajectories exist. Separate one semi-bounded trajectory by posing an initial condition. Heteroclinic trajectories One achieves accurate approximations of semi-bounded and heteroclinic trajectories, by solving appropriate boundary value problems. approximation of ξ approximation of ξ + n 1 n 2 n+ 1 n+ 2 n n approximation of x + Boundary operator: b(x n, x n+ )= ( Y T (x n ξ ) n ) Y+(x T n+ ξ n + ), Y :baseofr(p u n ), Y + :baseofr(p s n + ).

Computation of dichotomy projectors Fix N and compute P s N as follows: For each i = 1,...,k solve for n u i = n+1 A nu i + δ n n,n 1e i, n, A n = Df n (ξ ± ) n e i : i-th unit vector, δ : Kronecker symbol. Unique bounded solution for n : u i n = G(n, N)e i, G(n, N) = { Φ(n, N)P s N, n N, Φ(n, N)P u, N n < N. Thus Therefore u i N = G(N, N)e i = P s N e i. P s N = ( u 1 N u 2 N... u k N). Finite approximations can be achieved since errors decay exponentially fast towards the midpoint. Computation of dichotomy projectors Error estimates for approximate dichotomy projectors: Th. Hüls. Numerical computation of dichotomy rates and projectors in discrete time. Discrete Contin. Dyn. Syst. Ser. B, 12(1):19 131,29. Extended results: Th. Hüls. Computing Sacker-Sell spectra in discrete time dynamical systems. SIAM J. Numer. Anal., 48(6):243 264,21.

Example: Heteroclinic trajectories Discrete time model for two competing species x and y: ( xn+1 y n+1 ) ( = s n x n x n +y n y n e r (x n+y n ) ), r = 4, s n = 2 + 1 5 sin( 1 5 n) y ξ + n 4 3 z n 2 1 ξ n.5 1 1.5 2 2.5 x Y. Kang and H. Smith. Global Dynamics of a Discrete Two-species Lottery-Ricker Competition Model. Journal of Biological Dynamics 6(2):358-376, 212. Homoclinic and heteroclinic trajectories z ξ ξ + + S s (x ) S u (x ) x Heteroclinic trajectories: ξ +, ξ and x. If S s (x ) and S u (x ) intersect transversally, we find homoclinic trajectories x, z.

Conclusion We derive approximation results for homoclinic (and heteroclinic) trajectories via boundary value problems. Justification: Due to our hyperbolicity assumptions, errors on finite intervals decay exponentially fast toward the midpoint. One can verify these hyperbolicity assumption, using techniques that have been introduced, for example, in: L. Dieci, C. Elia, and E. Van Vleck. Exponential dichotomy on the real line: SVD and QR methods. J. Differential Equations, 248(2):287 38,21. Th. Hüls. Computing Sacker-Sell spectra in discrete time dynamical systems. SIAM J. Numer. Anal., 48(6):243 264,21. Conclusion The exponential decay of error enables accurate computation of covariant vectors: G. Froyland, Th. Hüls, G.P. Morriss and Th.M. Watson. Computing covariant vectors, Lyapunov vectors, Oseledets vectors, and dichotomy projectors: a comparative numerical study. arxiv:124.871, 212.