# American Institute of Mathematical Sciences

September  2016, 36(9): 4637-4664. doi: 10.3934/dcds.2016002

## 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

Received  May 2015 Revised  February 2016 Published  May 2016

The present work deals with numerical methods for computing slow stable invariant manifolds as well as their invariant stable and unstable normal bundles. The slow manifolds studied here are sub-manifolds of the stable manifold of a hyperbolic equilibrium point. Our approach is based on studying certain partial differential equations equations whose solutions parameterize the invariant manifolds/bundles. Formal solutions of the partial differential equations are obtained via power series arguments, and truncating the formal series provides an explicit polynomial representation for the desired chart maps. The coefficients of the formal series are given by recursion relations which are amenable to computer calculations. The parameterizations conjugate the dynamics on the invariant manifolds and bundles to a prescribed linear dynamical systems. Hence in addition to providing accurate representation of the invariant manifolds and bundles our methods describe the dynamics on these objects as well. Example computations are given for vector fields which arise as Galerkin projections of a partial differential equation. As an application we illustrate the use of the parameterized slow manifolds and their linear bundles in the computation of heteroclinic orbits.
Citation: J. B. van den Berg, J. D. Mireles James. Parameterization of slow-stable manifolds and their invariant vector bundles: Theory and numerical implementation. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 4637-4664. doi: 10.3934/dcds.2016002
##### 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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##### 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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