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Parameterization of slow-stable manifolds and their invariant vector bundles: Theory and numerical implementation
1. | VU University Amsterdam, Department of Mathematics, de Boelelaan 1081, 1081 HV Amsterdam, Netherlands |
2. | Florida Atlantic University, Department of Mathematical Sciences, 777 Glades Road, Boca Raton, FL 33431, United States |
References:
[1] |
W.-J. Beyn, The numerical computation of connecting orbits in dynamical systems, IMA J. Numer. Anal., 10 (1990), 379-405.
doi: 10.1093/imanum/10.3.379. |
[2] |
M. Breden, J.-P. Lessard and J. D. Mireles James, Computation of maximal local (un)stable manifold patches by the parameterization method, Indagationes Mathematicae, 27 (2016), 340-367.
doi: 10.1016/j.indag.2015.11.001. |
[3] |
H. W. Broer, H. M. Osinga and G. Vegter, On the computation of normally hyperbolic invariant manifolds, In Nonlinear dynamical systems and chaos (Groningen, 1995), volume 19 of Progr. Nonlinear Differential Equations Appl., pages 423-447. Birkhäuser, Basel, 1996. |
[4] |
H. W. Broer, H. M. Osinga and G. Vegter, Algorithms for computing normally hyperbolic invariant manifolds, Z. Angew. Math. Phys., 48 (1997), 480-524.
doi: 10.1007/s000330050044. |
[5] |
X. Cabré, E. Fontich and R. de la Llave, The parameterization method for invariant manifolds. I. Manifolds associated to non-resonant subspaces, Indiana Univ. Math. J., 52 (2003), 283-328.
doi: 10.1512/iumj.2003.52.2245. |
[6] |
X. Cabré, E. Fontich and R. de la Llave, The parameterization method for invariant manifolds. {II}. Regularity with respect to parameters, Indiana Univ. Math. J., 52 (2003), 329-360.
doi: 10.1512/iumj.2003.52.2407. |
[7] |
X. Cabré, E. 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. |
[8] |
R. C. Calleja and J.-L. Figueras, Collision of invariant bundles of quasi-periodic attractors in the dissipative standard map, Chaos, 22 (2012), 033114, 10 pp.
doi: 10.1063/1.4737205. |
[9] |
M. J. Capiński, Covering relations and the existence of topologically normally hyperbolic invariant sets, Discrete Contin. Dyn. Syst., 23 (2009), 705-725.
doi: 10.3934/dcds.2009.23.705. |
[10] |
R. Castelli and J.-P. Lessard, Rigorous Numerics in Floquet Theory: Computing Stable and Unstable Bundles of Periodic Orbits, SIAM J. Appl. Dyn. Syst., 12 (2013), 204-245.
doi: 10.1137/120873960. |
[11] |
R. Castelli, J.-P. Lessard and J. D. Mireles James, Parameterization of invariant manifolds for periodic orbits i: Efficient numerics via the floquet normal form, SIAM Journal on Applied Dynamical Systems, 14 (2015), 132-167.
doi: 10.1137/140960207. |
[12] |
A. R. Champneys, Yu. A. Kuznetsov and B. Sandstede, A numerical toolbox for homoclinic bifurcation analysis, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 6 (1996), 867-887.
doi: 10.1142/S0218127496000485. |
[13] |
R. de la Llave, Invariant manifolds associated to nonresonant spectral subspaces, J. Statist. Phys., 87 (1997), 211-249.
doi: 10.1007/BF02181486. |
[14] |
R. de la Llave and C. Eugene Wayne, On Irwin's proof of the pseudostable manifold theorem, Math. Z., 219 (1995), 301-321.
doi: 10.1007/BF02572367. |
[15] |
M. Desroches, J. Guckenheimer, B. Krauskopf, C. Kuehn, H. M. Osinga and M. Wechselberger, Mixed-mode oscillations with multiple time scales, SIAM Rev., 54 (2012), 211-288.
doi: 10.1137/100791233. |
[16] |
E. J. Doedel and M. J. Friedman, Numerical computation of heteroclinic orbits, J. Comput. Appl. Math., 26 (1989), 155-170. Continuation techniques and bifurcation problems.
doi: 10.1016/0377-0427(89)90153-2. |
[17] |
N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J., 21 (1971/1972), 193-226.
doi: 10.1512/iumj.1972.21.21017. |
[18] |
J.-L. Figueras and À. Haro, Triple collisions of invariant bundles, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2069-2082.
doi: 10.3934/dcdsb.2013.18.2069. |
[19] |
M. J. Friedman and E. J. Doedel, Computational methods for global analysis of homoclinic and heteroclinic orbits: A case study, J. Dynam. Differential Equations, 5 (1993), 37-57.
doi: 10.1007/BF01063734. |
[20] |
R. H. Goodman and J. K. Wróbel, High-order bisection method for computing invariant manifolds of two-dimensional maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 21 (2011), 2017-2042.
doi: 10.1142/S0218127411029604. |
[21] |
J. Guckenheimer and C. Kuehn, Computing slow manifolds of saddle type, SIAM J. Appl. Dyn. Syst., 8 (2009), 854-879.
doi: 10.1137/080741999. |
[22] |
A. Haro, Automatic differentiation methods in computational dynamical systems: Invariant manifolds and normal forms of vector fields at fixed points, Manuscript. |
[23] |
M. W. Hirsch and C. C. Pugh, Stable manifolds for hyperbolic sets, Bull. Amer. Math. Soc., 75 (1969), 149-152.
doi: 10.1090/S0002-9904-1969-12184-1. |
[24] |
À. Jorba and M. Zou, A software package for the numerical integration of {ODE}s by means of high-order Taylor methods, Experiment. Math., 14 (2005), 99-117.
doi: 10.1080/10586458.2005.10128904. |
[25] |
D. E. Knuth, The Art of Computer Programming. Vol. 2, Addison-Wesley Publishing Co., Reading, Mass., second edition, 1981. Seminumerical algorithms, Addison-Wesley Series in Computer Science and Information Processing. |
[26] |
J.-P. Lessard, J. D. Mireles James and C. Reinhardt, Computer assisted proof of transverse saddle-to-saddle connecting orbits for first order vector fields, J. Dynam. Differential Equations, 26 (2014), 267-313.
doi: 10.1007/s10884-014-9367-0. |
[27] |
E. N. Lorenz, The slow manifold-what is it?, J. Atmospheric Sci., 49 (1992), 2449-2451.
doi: 10.1175/1520-0469(1992)049<2449:TSMII>2.0.CO;2. |
[28] |
J. D. Mireles James, Quadratic volume-preserving maps: (Un)stable manifolds, hyperbolic dynamics, and vortex-bubble bifurcations, J. Nonlinear Sci., 23 (2013), 585-615.
doi: 10.1007/s00332-012-9162-1. |
[29] |
J. D. Mireles James and H. Lomelí, Computation of heteroclinic arcs with application to the volume preserving Hénon family, SIAM J. Appl. Dyn. Syst., 9 (2010), 919-953.
doi: 10.1137/090776329. |
[30] |
J. D. Mireles James and K. Mischaikow, Rigorous a posteriori computation of (un)stable manifolds and connecting orbits for analytic maps, SIAM J. Appl. Dyn. Syst., 12 (2013), 957-1006.
doi: 10.1137/12088224X. |
[31] |
J. D. Mireles James and J. B. Van den Berg, Matlab codes for "parameterization of slow-stable manifold and their invariant vector bundles: Theory and numerical implementation'', http://cosweb1.fau.edu/~jmirelesjames/fastSlowPage.html. |
[32] |
C. Pötzsche and M. Rasmussen, Local approximation of invariant fiber bundles: an algorithmic approach, In Difference equations and discrete dynamical systems, pages 155-170. World Sci. Publ., Hackensack, NJ, 2005.
doi: 10.1142/9789812701572_0011. |
[33] |
W. Tucker, Validated Numerics, Princeton University Press, Princeton, NJ, 2011. A short introduction to rigorous computations. |
[34] |
J. B. Van den Berg, J. D. Mireles James and C. Reinhardt, Computing (un)stable manifolds with validated error bounds: Non-resonant and resonant spectra, To appear in Journal of Nonlinear Science, 2016. |
[35] |
J. B. Van den Berg, J. D. Mireles-James, J.-P. Lessard and K. Mischaikow, Rigorous numerics for symmetric connecting orbits: Even homoclinics of the Gray-Scott equation, SIAM J. Math. Anal., 43 (2011), 1557-1594.
doi: 10.1137/100812008. |
[36] |
J. K. Wróbel and R. H. Goodman, High-order adaptive method for computing two-dimensional invariant manifolds of three-dimensional maps, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 1734-1745.
doi: 10.1016/j.cnsns.2012.10.017. |
show all references
References:
[1] |
W.-J. Beyn, The numerical computation of connecting orbits in dynamical systems, IMA J. Numer. Anal., 10 (1990), 379-405.
doi: 10.1093/imanum/10.3.379. |
[2] |
M. Breden, J.-P. Lessard and J. D. Mireles James, Computation of maximal local (un)stable manifold patches by the parameterization method, Indagationes Mathematicae, 27 (2016), 340-367.
doi: 10.1016/j.indag.2015.11.001. |
[3] |
H. W. Broer, H. M. Osinga and G. Vegter, On the computation of normally hyperbolic invariant manifolds, In Nonlinear dynamical systems and chaos (Groningen, 1995), volume 19 of Progr. Nonlinear Differential Equations Appl., pages 423-447. Birkhäuser, Basel, 1996. |
[4] |
H. W. Broer, H. M. Osinga and G. Vegter, Algorithms for computing normally hyperbolic invariant manifolds, Z. Angew. Math. Phys., 48 (1997), 480-524.
doi: 10.1007/s000330050044. |
[5] |
X. Cabré, E. Fontich and R. de la Llave, The parameterization method for invariant manifolds. I. Manifolds associated to non-resonant subspaces, Indiana Univ. Math. J., 52 (2003), 283-328.
doi: 10.1512/iumj.2003.52.2245. |
[6] |
X. Cabré, E. Fontich and R. de la Llave, The parameterization method for invariant manifolds. {II}. Regularity with respect to parameters, Indiana Univ. Math. J., 52 (2003), 329-360.
doi: 10.1512/iumj.2003.52.2407. |
[7] |
X. Cabré, E. 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. |
[8] |
R. C. Calleja and J.-L. Figueras, Collision of invariant bundles of quasi-periodic attractors in the dissipative standard map, Chaos, 22 (2012), 033114, 10 pp.
doi: 10.1063/1.4737205. |
[9] |
M. J. Capiński, Covering relations and the existence of topologically normally hyperbolic invariant sets, Discrete Contin. Dyn. Syst., 23 (2009), 705-725.
doi: 10.3934/dcds.2009.23.705. |
[10] |
R. Castelli and J.-P. Lessard, Rigorous Numerics in Floquet Theory: Computing Stable and Unstable Bundles of Periodic Orbits, SIAM J. Appl. Dyn. Syst., 12 (2013), 204-245.
doi: 10.1137/120873960. |
[11] |
R. Castelli, J.-P. Lessard and J. D. Mireles James, Parameterization of invariant manifolds for periodic orbits i: Efficient numerics via the floquet normal form, SIAM Journal on Applied Dynamical Systems, 14 (2015), 132-167.
doi: 10.1137/140960207. |
[12] |
A. R. Champneys, Yu. A. Kuznetsov and B. Sandstede, A numerical toolbox for homoclinic bifurcation analysis, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 6 (1996), 867-887.
doi: 10.1142/S0218127496000485. |
[13] |
R. de la Llave, Invariant manifolds associated to nonresonant spectral subspaces, J. Statist. Phys., 87 (1997), 211-249.
doi: 10.1007/BF02181486. |
[14] |
R. de la Llave and C. Eugene Wayne, On Irwin's proof of the pseudostable manifold theorem, Math. Z., 219 (1995), 301-321.
doi: 10.1007/BF02572367. |
[15] |
M. Desroches, J. Guckenheimer, B. Krauskopf, C. Kuehn, H. M. Osinga and M. Wechselberger, Mixed-mode oscillations with multiple time scales, SIAM Rev., 54 (2012), 211-288.
doi: 10.1137/100791233. |
[16] |
E. J. Doedel and M. J. Friedman, Numerical computation of heteroclinic orbits, J. Comput. Appl. Math., 26 (1989), 155-170. Continuation techniques and bifurcation problems.
doi: 10.1016/0377-0427(89)90153-2. |
[17] |
N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J., 21 (1971/1972), 193-226.
doi: 10.1512/iumj.1972.21.21017. |
[18] |
J.-L. Figueras and À. Haro, Triple collisions of invariant bundles, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2069-2082.
doi: 10.3934/dcdsb.2013.18.2069. |
[19] |
M. J. Friedman and E. J. Doedel, Computational methods for global analysis of homoclinic and heteroclinic orbits: A case study, J. Dynam. Differential Equations, 5 (1993), 37-57.
doi: 10.1007/BF01063734. |
[20] |
R. H. Goodman and J. K. Wróbel, High-order bisection method for computing invariant manifolds of two-dimensional maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 21 (2011), 2017-2042.
doi: 10.1142/S0218127411029604. |
[21] |
J. Guckenheimer and C. Kuehn, Computing slow manifolds of saddle type, SIAM J. Appl. Dyn. Syst., 8 (2009), 854-879.
doi: 10.1137/080741999. |
[22] |
A. Haro, Automatic differentiation methods in computational dynamical systems: Invariant manifolds and normal forms of vector fields at fixed points, Manuscript. |
[23] |
M. W. Hirsch and C. C. Pugh, Stable manifolds for hyperbolic sets, Bull. Amer. Math. Soc., 75 (1969), 149-152.
doi: 10.1090/S0002-9904-1969-12184-1. |
[24] |
À. Jorba and M. Zou, A software package for the numerical integration of {ODE}s by means of high-order Taylor methods, Experiment. Math., 14 (2005), 99-117.
doi: 10.1080/10586458.2005.10128904. |
[25] |
D. E. Knuth, The Art of Computer Programming. Vol. 2, Addison-Wesley Publishing Co., Reading, Mass., second edition, 1981. Seminumerical algorithms, Addison-Wesley Series in Computer Science and Information Processing. |
[26] |
J.-P. Lessard, J. D. Mireles James and C. Reinhardt, Computer assisted proof of transverse saddle-to-saddle connecting orbits for first order vector fields, J. Dynam. Differential Equations, 26 (2014), 267-313.
doi: 10.1007/s10884-014-9367-0. |
[27] |
E. N. Lorenz, The slow manifold-what is it?, J. Atmospheric Sci., 49 (1992), 2449-2451.
doi: 10.1175/1520-0469(1992)049<2449:TSMII>2.0.CO;2. |
[28] |
J. D. Mireles James, Quadratic volume-preserving maps: (Un)stable manifolds, hyperbolic dynamics, and vortex-bubble bifurcations, J. Nonlinear Sci., 23 (2013), 585-615.
doi: 10.1007/s00332-012-9162-1. |
[29] |
J. D. Mireles James and H. Lomelí, Computation of heteroclinic arcs with application to the volume preserving Hénon family, SIAM J. Appl. Dyn. Syst., 9 (2010), 919-953.
doi: 10.1137/090776329. |
[30] |
J. D. Mireles James and K. Mischaikow, Rigorous a posteriori computation of (un)stable manifolds and connecting orbits for analytic maps, SIAM J. Appl. Dyn. Syst., 12 (2013), 957-1006.
doi: 10.1137/12088224X. |
[31] |
J. D. Mireles James and J. B. Van den Berg, Matlab codes for "parameterization of slow-stable manifold and their invariant vector bundles: Theory and numerical implementation'', http://cosweb1.fau.edu/~jmirelesjames/fastSlowPage.html. |
[32] |
C. Pötzsche and M. Rasmussen, Local approximation of invariant fiber bundles: an algorithmic approach, In Difference equations and discrete dynamical systems, pages 155-170. World Sci. Publ., Hackensack, NJ, 2005.
doi: 10.1142/9789812701572_0011. |
[33] |
W. Tucker, Validated Numerics, Princeton University Press, Princeton, NJ, 2011. A short introduction to rigorous computations. |
[34] |
J. B. Van den Berg, J. D. Mireles James and C. Reinhardt, Computing (un)stable manifolds with validated error bounds: Non-resonant and resonant spectra, To appear in Journal of Nonlinear Science, 2016. |
[35] |
J. B. Van den Berg, J. D. Mireles-James, J.-P. Lessard and K. Mischaikow, Rigorous numerics for symmetric connecting orbits: Even homoclinics of the Gray-Scott equation, SIAM J. Math. Anal., 43 (2011), 1557-1594.
doi: 10.1137/100812008. |
[36] |
J. K. Wróbel and R. H. Goodman, High-order adaptive method for computing two-dimensional invariant manifolds of three-dimensional maps, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 1734-1745.
doi: 10.1016/j.cnsns.2012.10.017. |
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