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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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[16]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[28]

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

[29]

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

[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.

[31]

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

[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.

[33]

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

[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.

[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.

[36]

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

[37]

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

[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.

[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.

[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.

[41]

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

[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.

[43]

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

[44]

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

[45]

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

[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.

[47]

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

[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.

[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.

[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.

[51]

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

[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.

[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.

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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[16]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[28]

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

[29]

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

[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.

[31]

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

[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.

[33]

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

[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.

[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.

[36]

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

[37]

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

[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.

[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.

[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.

[41]

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

[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.

[43]

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

[44]

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

[45]

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

[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.

[47]

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

[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.

[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.

[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.

[51]

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

[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.

[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.

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 $t = 0$
[1]

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