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November  2017, 22(9): 3341-3367. doi: 10.3934/dcdsb.2017140

Computing stable hierarchies of fiber bundles

Department of Mathematics, Bielefeld University, POB 100131,33501 Bielefeld, Germany

Received  September 2016 Revised  December 2016 Published  April 2017

Fund Project: Supported by CRC 701 'Spectral Structures and Topological Methods in Mathematics'

Stable fiber bundles are important structures for understanding nonautonomous dynamics. These sets have a hierarchical structure ranging from stable to strong stable fibers. First, we compute corresponding structures for linear systems and prove an error estimate. The spectral concept of choice is the Sacker-Sell spectrum that is based on exponential dichotomies. Secondly, we tackle the nonlinear case and propose an algorithm for the numerical approximation of stable hierarchies in nonautonomous difference equations. This method generalizes the contour algorithm for computing stable fibers from [38,39]. It is based on Hadamard's graph transform and approximates fibers of the hierarchy by zero-contours of specific operators. We calculate fiber bundles and illustrate errors involved for several examples, including a nonautonomous Lorenz model.

Citation: Thorsten Hüls. Computing stable hierarchies of fiber bundles. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3341-3367. doi: 10.3934/dcdsb.2017140
References:
[1]

J. -P. Aubin and H. Frankowska, Set-valued Analysis, Modern Birkhäuser Classics, Birkhäuser Boston, Inc. , Boston, MA, 2009. doi: 10.1007/978-0-8176-4848-0. Google Scholar

[2]

B. Aulbach and J. Kalkbrenner, Exponential forward splitting for noninvertible difference equations Comput. Math. Appl. , 42 (2001), 743--754, doi: 10.1016/S0898-1221(01)00194-8. Google Scholar

[3]

B. Aulbach, The fundamental existence theorem on invariant fiber bundles, J. Differ. Equations Appl., 3 (1998), 501-537. doi: 10.1080/10236199708808118. Google Scholar

[4]

B. AulbachC. Pötzsche and S. Siegmund, A smoothness theorem for invariant fiber bundles, J. Dynam. Differential Equations, 14 (2002), 519-547. doi: 10.1023/A:1016383031231. Google Scholar

[5]

B. Aulbach and S. Siegmund, The dichotomy spectrum for noninvertible systems of linear difference equations, J. Differ. Equations Appl., 7 (2001), 895-913. doi: 10.1080/10236190108808310. Google Scholar

[6]

A. BergerT. S. Doan and S. Siegmund, A definition of spectrum for differential equations on finite time, J. Differential Equations, 246 (2009), 1098-1118. doi: 10.1016/j.jde.2008.06.036. Google Scholar

[7]

A. BergerD. T. Son and S. Siegmund, Nonautonomous finite-time dynamics, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 463-492. doi: 10.3934/dcdsb.2008.9.463. Google Scholar

[8]

W.-J. Beyn and W. Kleß, Numerical Taylor expansions of invariant manifolds in large dynamical systems, Numer. Math., 80 (1998), 1-38. doi: 10.1007/s002110050357. Google Scholar

[9]

D. Blazevski and J. Franklin, Using scattering theory to compute invariant manifolds and numerical results for the laser-driven Hénon-Heiles system Chaos, 22 (2012), 043138, 9pp. doi: 10.1063/1.4767656. Google Scholar

[10]

X. CabréE. Fontich and R. de~la Llave, The parameterization method for invariant manifolds. Ⅲ. Overview and applications, J. Differential Equations, 218 (2005), 444-515. doi: 10.1016/j.jde.2004.12.003. Google Scholar

[11]

M. Canadell and R. de la Llave, KAM tori and whiskered invariant tori for non-autonomous systems, Phys. D, 310 (2015), 104-113. doi: 10.1016/j.physd.2015.08.004. Google Scholar

[12]

M. J. Capiński and C. Simó, Computer assisted proof for normally hyperbolic invariant manifolds, Nonlinearity, 25 (2012), 1997-2026. doi: 10.1088/0951-7715/25/7/1997. Google Scholar

[13]

M. J. Capiński and P. Zgliczyński, Geometric proof for normally hyperbolic invariant manifolds, J. Differential Equations, 259 (2015), 6215-6286. doi: 10.1016/j.jde.2015.07.020. Google Scholar

[14]

F. Colonius and W. Kliemann, The Dynamics of Control, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc. , Boston, MA, 2000. doi: 10.1007/978-1-4612-1350-5. Google Scholar

[15]

W. A. Coppel, Dichotomies in Stability Theory, Springer, Berlin, 1978, Lecture Notes in Mathematics, Vol. 629. doi: 10.1007/BFb0067780. Google Scholar

[16]

J. L. Daleckiĭ and M. G. Kreĭn, Stability of Solutions of Differential Equations in Banach Space, American Mathematical Society, Providence, R. I. , 1974. Google Scholar

[17]

M. Dellnitz and A. Hohmann, A subdivision algorithm for the computation of unstable manifolds and global attractors, Numer. Math., 75 (1997), 293-317. doi: 10.1007/s002110050240. Google Scholar

[18]

L. DieciC. Elia and E. Van Vleck, Exponential dichotomy on the real line: SVD and QR methods, J. Differential Equations, 248 (2010), 287-308. doi: 10.1016/j.jde.2009.07.004. Google Scholar

[19]

L. Dieci and E. S. Van Vleck, Lyapunov and Sacker-Sell spectral intervals, J. Dynam. Differential Equations, 19 (2007), 265-293. doi: 10.1007/s10884-006-9030-5. Google Scholar

[20]

E. J. DoedelB. Krauskopf and H. M. Osinga, Global organization of phase space in the transition to chaos in the Lorenz system, Nonlinearity, 28 (2015), R113-R139. doi: 10.1088/0951-7715/28/11/R113. Google Scholar

[21]

L. H. Duc and S. Siegmund, Hyperbolicity and invariant manifolds for planar nonautonomous systems on finite time intervals, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 18 (2008), 641-674. doi: 10.1142/S0218127408020562. Google Scholar

[22]

T. Eirola and J. von Pfaler, Numerical Taylor expansions for invariant manifolds, Numer. Math., 99 (2004), 25-46. doi: 10.1007/s00211-004-0537-6. Google Scholar

[23]

J. P. EnglandB. Krauskopf and H. M. Osinga, Computing one-dimensional stable manifolds and stable sets of planar maps without the inverse, SIAM J. Appl. Dyn. Syst., 3 (2004), 161-190. doi: 10.1137/030600131. Google Scholar

[24]

J.-L. Figueras and À. Haro, Reliable computation of robust response tori on the verge of breakdown, SIAM J. Appl. Dyn. Syst., 11 (2012), 597-628. doi: 10.1137/100809222. Google Scholar

[25]

G. FroylandT. HülsG. P. Morriss and T. M. Watson, Computing covariant Lyapunov vectors, Oseledets vectors, and dichotomy projectors: A comparative numerical study, Phys. D, 247 (2013), 18-39. doi: 10.1016/j.physd.2012.12.005. Google Scholar

[26]

G. FroylandS. Lloyd and N. Santitissadeekorn, Coherent sets for nonautonomous dynamical systems, Phys. D, 239 (2010), 1527-1541. doi: 10.1016/j.physd.2010.03.009. Google Scholar

[27]

A. Girod and T. Hüls, Nonautonomous systems with transversal homoclinic structures under discretization, BIT, 56 (2016), 605-631. doi: 10.1007/s10543-015-0567-8. Google Scholar

[28]

J. Hadamard, Sur l'itératio et les solutions asymptotiques des équations différentielles, Bull. Soc. Math. France, 29 (1901), 224-228. Google Scholar

[29]

G. Haller, Finding finite-time invariant manifolds in two-dimensional velocity fields, Chaos, 10 (2000), 99-108. doi: 10.1063/1.166479. Google Scholar

[30]

G. Haller, Lagrangian structures and the rate of strain in a partition of two-dimensional turbulence, Phys. Fluids, 13 (2001), 3365-3385. doi: 10.1063/1.1403336. Google Scholar

[31]

G. Haller, Lagrangian coherent structures, Annual Review of Fluid Mechanics, 47 (2015), 137-162. doi: 10.1146/annurev-fluid-010313-141322. Google Scholar

[32]

À. Haro, M. Canadell, J. -L. Figueras, A. Luque and J. -M. Mondelo, The Parameterization Method for Invariant Manifolds, -From Rigorous Results to Effective Computations, vol. 195 of Applied Mathematical Sciences, Springer, 2016. doi: 10.1007/978-3-319-29662-3. Google Scholar

[33]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, Berlin, 1981. doi: 10.1007/BFb0089647. Google Scholar

[34]

M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, Vol. 583, Springer, Berlin, 1977. doi: 10.1007/BFb0092042. Google Scholar

[35]

T. Hüls, Numerical computation of dichotomy rates and projectors in discrete time, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 109-131. doi: 10.3934/dcdsb.2009.12.109. Google Scholar

[36]

T. Hüls, Computing Sacker-Sell spectra in discrete time dynamical systems, SIAM J. Numer. Anal., 48 (2010), 2043-2064. doi: 10.1137/090754509. Google Scholar

[37]

T. Hüls, Homoclinic trajectories of non-autonomous maps, J. Difference Equ. Appl., 17 (2011), 9-31. doi: 10.1080/10236190902932742. Google Scholar

[38]

T. Hüls, A contour algorithm for computing stable fiber bundles of nonautonomous, noninvertible maps, SIAM J. Appl. Dyn. Syst., 15 (2016), 923-951. doi: 10.1137/140999815. Google Scholar

[39]

T. Hüls, On the approximation of stable and unstable fiber bundles of (non)autonomous ODEs -A contour algorithm, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 26 (2016), 1650118, 10pp. doi: 10.1142/S0218127416501182. Google Scholar

[40]

T. Hüls and C. Pötzsche, Qualitative analysis of a nonautonomous Beverton-Holt Ricker model, SIAM J. Appl. Dyn. Syst., 13 (2014), 1442-1488. doi: 10.1137/140955434. Google Scholar

[41]

J. Kalkbrenner, Exponentielle Dichotomie und chaotische Dynamik nichtinvertierbarer Differenzengleichungen, vol. 1 of Augsburger Mathematisch-Naturwissenschaftliche Schriften, Dr. Bernd Wiß ner, Augsburg, 1994. Google Scholar

[42]

B. KrauskopfH. M. OsingaE. J. DoedelM. E. HendersonJ. GuckenheimerA. VladimirskyM. Dellnitz and O. Junge, A survey of methods for computing (un)stable manifolds of vector fields, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 763-791. doi: 10.1142/S0218127405012533. Google Scholar

[43]

E. N. Lorenz, Deterministic nonperiodic flow, The Theory of Chaotic Attractors, (2004), 25-36. doi: 10.1007/978-0-387-21830-4_2. Google Scholar

[44]

O. Perron, Die Stabilitätsfrage bei Differentialgleichungen, Math. Z., 32 (1930), 703-728. doi: 10.1007/BF01194662. Google Scholar

[45]

H. Poincaré, Mémoire sur les courbes définies par une équation différentielle (ⅳ), Journal de Mathématiques Pures et Appliquées, 2 (1886), 151-217. Google Scholar

[46]

C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems, vol. 2002 of Lecture Notes in Mathematics, Springer, Berlin, 2010. doi: 10.1007/978-3-642-14258-1. Google Scholar

[47]

C. Pötzsche, Fine structure of the dichotomy spectrum, Integral Equations Operator Theory, 73 (2012), 107-151. doi: 10.1007/s00020-012-1959-7. Google Scholar

[48]

C. Pötzsche and M. Rasmussen, Taylor approximation of invariant fiber bundles for nonautonomous difference equations, Nonlinear Anal., 60 (2005), 1303-1330. doi: 10.1016/j.na.2004.10.019. Google Scholar

[49]

C. Pötzsche and M. Rasmussen, Computation of nonautonomous invariant and inertial manifolds, Numer. Math., 112 (2009), 449-483. doi: 10.1007/s00211-009-0215-9. Google Scholar

[50]

R. J. Sacker and G. R. Sell, A spectral theory for linear differential systems, J. Differential Equations, 27 (1978), 320-358. doi: 10.1016/0022-0396(78)90057-8. Google Scholar

[51]

M. Shub, Global Stability of Dynamical Systems, Springer, New York, 1987. doi: 10.1007/978-1-4757-1947-5. Google Scholar

[52]

C. Simó, On the analytical and numerical approximation of invariant manifolds, in Les méthodes modernes de la mécanique céleste (eds. D. Benest and C. Froeschlé), 1989,285-329.Google Scholar

[53]

S. Wiggins, Normally Hyperbolic Invariant Manifolds in Dynamical Systems, vol. 105 of Applied Mathematical Sciences, Springer, New York, 1994. doi: 10.1007/978-1-4612-4312-0. Google Scholar

show all references

References:
[1]

J. -P. Aubin and H. Frankowska, Set-valued Analysis, Modern Birkhäuser Classics, Birkhäuser Boston, Inc. , Boston, MA, 2009. doi: 10.1007/978-0-8176-4848-0. Google Scholar

[2]

B. Aulbach and J. Kalkbrenner, Exponential forward splitting for noninvertible difference equations Comput. Math. Appl. , 42 (2001), 743--754, doi: 10.1016/S0898-1221(01)00194-8. Google Scholar

[3]

B. Aulbach, The fundamental existence theorem on invariant fiber bundles, J. Differ. Equations Appl., 3 (1998), 501-537. doi: 10.1080/10236199708808118. Google Scholar

[4]

B. AulbachC. Pötzsche and S. Siegmund, A smoothness theorem for invariant fiber bundles, J. Dynam. Differential Equations, 14 (2002), 519-547. doi: 10.1023/A:1016383031231. Google Scholar

[5]

B. Aulbach and S. Siegmund, The dichotomy spectrum for noninvertible systems of linear difference equations, J. Differ. Equations Appl., 7 (2001), 895-913. doi: 10.1080/10236190108808310. Google Scholar

[6]

A. BergerT. S. Doan and S. Siegmund, A definition of spectrum for differential equations on finite time, J. Differential Equations, 246 (2009), 1098-1118. doi: 10.1016/j.jde.2008.06.036. Google Scholar

[7]

A. BergerD. T. Son and S. Siegmund, Nonautonomous finite-time dynamics, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 463-492. doi: 10.3934/dcdsb.2008.9.463. Google Scholar

[8]

W.-J. Beyn and W. Kleß, Numerical Taylor expansions of invariant manifolds in large dynamical systems, Numer. Math., 80 (1998), 1-38. doi: 10.1007/s002110050357. Google Scholar

[9]

D. Blazevski and J. Franklin, Using scattering theory to compute invariant manifolds and numerical results for the laser-driven Hénon-Heiles system Chaos, 22 (2012), 043138, 9pp. doi: 10.1063/1.4767656. Google Scholar

[10]

X. CabréE. Fontich and R. de~la Llave, The parameterization method for invariant manifolds. Ⅲ. Overview and applications, J. Differential Equations, 218 (2005), 444-515. doi: 10.1016/j.jde.2004.12.003. Google Scholar

[11]

M. Canadell and R. de la Llave, KAM tori and whiskered invariant tori for non-autonomous systems, Phys. D, 310 (2015), 104-113. doi: 10.1016/j.physd.2015.08.004. Google Scholar

[12]

M. J. Capiński and C. Simó, Computer assisted proof for normally hyperbolic invariant manifolds, Nonlinearity, 25 (2012), 1997-2026. doi: 10.1088/0951-7715/25/7/1997. Google Scholar

[13]

M. J. Capiński and P. Zgliczyński, Geometric proof for normally hyperbolic invariant manifolds, J. Differential Equations, 259 (2015), 6215-6286. doi: 10.1016/j.jde.2015.07.020. Google Scholar

[14]

F. Colonius and W. Kliemann, The Dynamics of Control, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc. , Boston, MA, 2000. doi: 10.1007/978-1-4612-1350-5. Google Scholar

[15]

W. A. Coppel, Dichotomies in Stability Theory, Springer, Berlin, 1978, Lecture Notes in Mathematics, Vol. 629. doi: 10.1007/BFb0067780. Google Scholar

[16]

J. L. Daleckiĭ and M. G. Kreĭn, Stability of Solutions of Differential Equations in Banach Space, American Mathematical Society, Providence, R. I. , 1974. Google Scholar

[17]

M. Dellnitz and A. Hohmann, A subdivision algorithm for the computation of unstable manifolds and global attractors, Numer. Math., 75 (1997), 293-317. doi: 10.1007/s002110050240. Google Scholar

[18]

L. DieciC. Elia and E. Van Vleck, Exponential dichotomy on the real line: SVD and QR methods, J. Differential Equations, 248 (2010), 287-308. doi: 10.1016/j.jde.2009.07.004. Google Scholar

[19]

L. Dieci and E. S. Van Vleck, Lyapunov and Sacker-Sell spectral intervals, J. Dynam. Differential Equations, 19 (2007), 265-293. doi: 10.1007/s10884-006-9030-5. Google Scholar

[20]

E. J. DoedelB. Krauskopf and H. M. Osinga, Global organization of phase space in the transition to chaos in the Lorenz system, Nonlinearity, 28 (2015), R113-R139. doi: 10.1088/0951-7715/28/11/R113. Google Scholar

[21]

L. H. Duc and S. Siegmund, Hyperbolicity and invariant manifolds for planar nonautonomous systems on finite time intervals, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 18 (2008), 641-674. doi: 10.1142/S0218127408020562. Google Scholar

[22]

T. Eirola and J. von Pfaler, Numerical Taylor expansions for invariant manifolds, Numer. Math., 99 (2004), 25-46. doi: 10.1007/s00211-004-0537-6. Google Scholar

[23]

J. P. EnglandB. Krauskopf and H. M. Osinga, Computing one-dimensional stable manifolds and stable sets of planar maps without the inverse, SIAM J. Appl. Dyn. Syst., 3 (2004), 161-190. doi: 10.1137/030600131. Google Scholar

[24]

J.-L. Figueras and À. Haro, Reliable computation of robust response tori on the verge of breakdown, SIAM J. Appl. Dyn. Syst., 11 (2012), 597-628. doi: 10.1137/100809222. Google Scholar

[25]

G. FroylandT. HülsG. P. Morriss and T. M. Watson, Computing covariant Lyapunov vectors, Oseledets vectors, and dichotomy projectors: A comparative numerical study, Phys. D, 247 (2013), 18-39. doi: 10.1016/j.physd.2012.12.005. Google Scholar

[26]

G. FroylandS. Lloyd and N. Santitissadeekorn, Coherent sets for nonautonomous dynamical systems, Phys. D, 239 (2010), 1527-1541. doi: 10.1016/j.physd.2010.03.009. Google Scholar

[27]

A. Girod and T. Hüls, Nonautonomous systems with transversal homoclinic structures under discretization, BIT, 56 (2016), 605-631. doi: 10.1007/s10543-015-0567-8. Google Scholar

[28]

J. Hadamard, Sur l'itératio et les solutions asymptotiques des équations différentielles, Bull. Soc. Math. France, 29 (1901), 224-228. Google Scholar

[29]

G. Haller, Finding finite-time invariant manifolds in two-dimensional velocity fields, Chaos, 10 (2000), 99-108. doi: 10.1063/1.166479. Google Scholar

[30]

G. Haller, Lagrangian structures and the rate of strain in a partition of two-dimensional turbulence, Phys. Fluids, 13 (2001), 3365-3385. doi: 10.1063/1.1403336. Google Scholar

[31]

G. Haller, Lagrangian coherent structures, Annual Review of Fluid Mechanics, 47 (2015), 137-162. doi: 10.1146/annurev-fluid-010313-141322. Google Scholar

[32]

À. Haro, M. Canadell, J. -L. Figueras, A. Luque and J. -M. Mondelo, The Parameterization Method for Invariant Manifolds, -From Rigorous Results to Effective Computations, vol. 195 of Applied Mathematical Sciences, Springer, 2016. doi: 10.1007/978-3-319-29662-3. Google Scholar

[33]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, Berlin, 1981. doi: 10.1007/BFb0089647. Google Scholar

[34]

M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, Vol. 583, Springer, Berlin, 1977. doi: 10.1007/BFb0092042. Google Scholar

[35]

T. Hüls, Numerical computation of dichotomy rates and projectors in discrete time, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 109-131. doi: 10.3934/dcdsb.2009.12.109. Google Scholar

[36]

T. Hüls, Computing Sacker-Sell spectra in discrete time dynamical systems, SIAM J. Numer. Anal., 48 (2010), 2043-2064. doi: 10.1137/090754509. Google Scholar

[37]

T. Hüls, Homoclinic trajectories of non-autonomous maps, J. Difference Equ. Appl., 17 (2011), 9-31. doi: 10.1080/10236190902932742. Google Scholar

[38]

T. Hüls, A contour algorithm for computing stable fiber bundles of nonautonomous, noninvertible maps, SIAM J. Appl. Dyn. Syst., 15 (2016), 923-951. doi: 10.1137/140999815. Google Scholar

[39]

T. Hüls, On the approximation of stable and unstable fiber bundles of (non)autonomous ODEs -A contour algorithm, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 26 (2016), 1650118, 10pp. doi: 10.1142/S0218127416501182. Google Scholar

[40]

T. Hüls and C. Pötzsche, Qualitative analysis of a nonautonomous Beverton-Holt Ricker model, SIAM J. Appl. Dyn. Syst., 13 (2014), 1442-1488. doi: 10.1137/140955434. Google Scholar

[41]

J. Kalkbrenner, Exponentielle Dichotomie und chaotische Dynamik nichtinvertierbarer Differenzengleichungen, vol. 1 of Augsburger Mathematisch-Naturwissenschaftliche Schriften, Dr. Bernd Wiß ner, Augsburg, 1994. Google Scholar

[42]

B. KrauskopfH. M. OsingaE. J. DoedelM. E. HendersonJ. GuckenheimerA. VladimirskyM. Dellnitz and O. Junge, A survey of methods for computing (un)stable manifolds of vector fields, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 763-791. doi: 10.1142/S0218127405012533. Google Scholar

[43]

E. N. Lorenz, Deterministic nonperiodic flow, The Theory of Chaotic Attractors, (2004), 25-36. doi: 10.1007/978-0-387-21830-4_2. Google Scholar

[44]

O. Perron, Die Stabilitätsfrage bei Differentialgleichungen, Math. Z., 32 (1930), 703-728. doi: 10.1007/BF01194662. Google Scholar

[45]

H. Poincaré, Mémoire sur les courbes définies par une équation différentielle (ⅳ), Journal de Mathématiques Pures et Appliquées, 2 (1886), 151-217. Google Scholar

[46]

C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems, vol. 2002 of Lecture Notes in Mathematics, Springer, Berlin, 2010. doi: 10.1007/978-3-642-14258-1. Google Scholar

[47]

C. Pötzsche, Fine structure of the dichotomy spectrum, Integral Equations Operator Theory, 73 (2012), 107-151. doi: 10.1007/s00020-012-1959-7. Google Scholar

[48]

C. Pötzsche and M. Rasmussen, Taylor approximation of invariant fiber bundles for nonautonomous difference equations, Nonlinear Anal., 60 (2005), 1303-1330. doi: 10.1016/j.na.2004.10.019. Google Scholar

[49]

C. Pötzsche and M. Rasmussen, Computation of nonautonomous invariant and inertial manifolds, Numer. Math., 112 (2009), 449-483. doi: 10.1007/s00211-009-0215-9. Google Scholar

[50]

R. J. Sacker and G. R. Sell, A spectral theory for linear differential systems, J. Differential Equations, 27 (1978), 320-358. doi: 10.1016/0022-0396(78)90057-8. Google Scholar

[51]

M. Shub, Global Stability of Dynamical Systems, Springer, New York, 1987. doi: 10.1007/978-1-4757-1947-5. Google Scholar

[52]

C. Simó, On the analytical and numerical approximation of invariant manifolds, in Les méthodes modernes de la mécanique céleste (eds. D. Benest and C. Froeschlé), 1989,285-329.Google Scholar

[53]

S. Wiggins, Normally Hyperbolic Invariant Manifolds in Dynamical Systems, vol. 105 of Applied Mathematical Sciences, Springer, New York, 1994. doi: 10.1007/978-1-4612-4312-0. Google Scholar

Figure 1.  Illustration of spectral and resolvent intervals
Figure 2.  Optimal choices of $\gamma_i$ and $\gamma_{i+1}$
Figure 3.  Sacker-Sell spectrum and resolvent set of (26)
Figure 4.  Errors of approximate spectral bundles of (26) for $n_-=-400$, $n_+ =10,\dots,100$
Figure 5.  Errors of approximate spectral bundles of (26) for $n_+=400$, $n_- =-10,\dots,-100$
Figure 6.  Illustration of strong and weak stable fibers in nonlinear systems
Figure 7.  Approximation of the Lorenz manifold (left) and of its intersection with the $(x_2,x_3)$ plane (right)
Figure 8.  Intersection of the stable manifold of (36) with threedimensional cubes
Figure 9.  Intersection of the stable manifold of (36) with twodimensional coordinate planes
Figure 10.  Approximation of $\mathcal{H}^2$ for (39) w.r.t. $\xi = 0$ (red ball). Zero-contour $\mathcal{N}_2$ (left) and interpolated graph representation $\tilde g_2$ w.r.t. the tangent space $\text{span}\{v_1,v_2\}$ (right). The approximate strong stable manifold $\mathcal{N}_3$ is shown in white
Figure 11.  Approximation error of $\mathcal{N}_2$ for $m=10$ w.r.t. the parameterization (40) (left). Distance of $\mathcal{H}^3$ to its numerical approximation for $m=1,\dots,20$ (right)
Figure 12.  Approximation of $\mathcal{H}^2$ and $\mathcal{H}^3$ for (41) (left) and distance of $\mathcal{H}^3$ to its approximation w.r.t. $m=1,\dots,15$ (right)
Figure 13.  Approximation of $\mathcal{L}^2$ and $\mathcal{H}^3$ for (42) (left) and distance of $\mathcal{H}^3$ to its approximation w.r.t. $m=1,\dots,15$ (right)
Figure 14.  Parametrizations of $\mathcal{H}^2$, computed using the cutoff function (44) with $\mu = \tfrac 14$
Figure 15.  Zero-contours (38) w.r.t. the parametrizations from Figure 14
Figure 16.  Approximation of the stable manifold $\mathcal{H}^2$ of (43) w.r.t. the fixed point $0$ (red ball). The red lines are parts of the strong stable manifold $\mathcal{H}^3$
Figure 17.  Illustration of parts of the strong stable manifold of the fixed point 0
Figure 18.  Approximation of the stable and strong stable fiber (in red) of the fixed point $0$ (red ball) for the nonautonomous Lorenz system (45) at time $t = 0$
Figure 19.  Approximation of the stable and strong stable fiber of the fixed point $0$ (red ball) for the nonautonomous Lorenz system (45) at times $t\in \{\tfrac 12, 1,\tfrac 32, 2\}$. In the right diagrams, the strong stable manifold is hidden by parts of the stable manifold
Figure 20.  Approximation of the stable fiber (left: solid, right: transparent) and of the strong stable fiber (in red) of the fixed point $0$ for the nonautonomous Lorenz system (45) at time $t = 0$
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