# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - A, 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. 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. 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), 423. [4] H. W. Broer, H. M. Osinga and G. Vegter, Algorithms for computing normally hyperbolic invariant manifolds,, Z. Angew. Math. Phys., 48 (1997), 480. 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. 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. 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. 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). 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. 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. 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. 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. doi: 10.1142/S0218127496000485. [13] R. de la Llave, Invariant manifolds associated to nonresonant spectral subspaces,, J. Statist. Phys., 87 (1997), 211. 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. 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. doi: 10.1137/100791233. [16] E. J. Doedel and M. J. Friedman, Numerical computation of heteroclinic orbits,, J. Comput. Appl. Math., 26 (1989), 155. doi: 10.1016/0377-0427(89)90153-2. [17] N. Fenichel, Persistence and smoothness of invariant manifolds for flows,, Indiana Univ. Math. J., 21 (): 193. 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. 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. 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. doi: 10.1142/S0218127411029604. [21] J. Guckenheimer and C. Kuehn, Computing slow manifolds of saddle type,, SIAM J. Appl. Dyn. Syst., 8 (2009), 854. 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. 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. doi: 10.1080/10586458.2005.10128904. [25] D. E. Knuth, The Art of Computer Programming. Vol. 2,, Addison-Wesley Publishing Co., (1981). [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. doi: 10.1007/s10884-014-9367-0. [27] E. N. Lorenz, The slow manifold-what is it?,, J. Atmospheric Sci., 49 (1992), 2449. 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. 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. 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. 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'',, , (). [32] C. Pötzsche and M. Rasmussen, Local approximation of invariant fiber bundles: an algorithmic approach,, In Difference equations and discrete dynamical systems, (2005), 155. doi: 10.1142/9789812701572_0011. [33] W. Tucker, Validated Numerics,, Princeton University Press, (2011). [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. 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. 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. 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. 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), 423. [4] H. W. Broer, H. M. Osinga and G. Vegter, Algorithms for computing normally hyperbolic invariant manifolds,, Z. Angew. Math. Phys., 48 (1997), 480. 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. 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. 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. 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). 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. 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. 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. 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. doi: 10.1142/S0218127496000485. [13] R. de la Llave, Invariant manifolds associated to nonresonant spectral subspaces,, J. Statist. Phys., 87 (1997), 211. 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. 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. doi: 10.1137/100791233. [16] E. J. Doedel and M. J. Friedman, Numerical computation of heteroclinic orbits,, J. Comput. Appl. Math., 26 (1989), 155. doi: 10.1016/0377-0427(89)90153-2. [17] N. Fenichel, Persistence and smoothness of invariant manifolds for flows,, Indiana Univ. Math. J., 21 (): 193. 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. 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. 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. doi: 10.1142/S0218127411029604. [21] J. Guckenheimer and C. Kuehn, Computing slow manifolds of saddle type,, SIAM J. Appl. Dyn. Syst., 8 (2009), 854. 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. 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. doi: 10.1080/10586458.2005.10128904. [25] D. E. Knuth, The Art of Computer Programming. Vol. 2,, Addison-Wesley Publishing Co., (1981). [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. doi: 10.1007/s10884-014-9367-0. [27] E. N. Lorenz, The slow manifold-what is it?,, J. Atmospheric Sci., 49 (1992), 2449. 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. 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. 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. 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'',, , (). [32] C. Pötzsche and M. Rasmussen, Local approximation of invariant fiber bundles: an algorithmic approach,, In Difference equations and discrete dynamical systems, (2005), 155. doi: 10.1142/9789812701572_0011. [33] W. Tucker, Validated Numerics,, Princeton University Press, (2011). [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. 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. doi: 10.1016/j.cnsns.2012.10.017.
 [1] C. M. Groothedde, J. D. Mireles James. Parameterization method for unstable manifolds of delay differential equations. Journal of Computational Dynamics, 2017, 4 (1&2) : 21-70. doi: 10.3934/jcd.2017002 [2] Michihiro Hirayama, Naoya Sumi. Hyperbolic measures with transverse intersections of stable and unstable manifolds. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1451-1476. doi: 10.3934/dcds.2013.33.1451 [3] Ale Jan Homburg. Heteroclinic bifurcations of $\Omega$-stable vector fields on 3-manifolds. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 559-580. doi: 10.3934/dcds.1998.4.559 [4] Inmaculada Baldomá, Ernest Fontich, Rafael de la Llave, Pau Martín. The parameterization method for one- dimensional invariant manifolds of higher dimensional parabolic fixed points. Discrete & Continuous Dynamical Systems - A, 2007, 17 (4) : 835-865. doi: 10.3934/dcds.2007.17.835 [5] Rafael de la Llave, Jason D. Mireles James. Parameterization of invariant manifolds by reducibility for volume preserving and symplectic maps. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4321-4360. doi: 10.3934/dcds.2012.32.4321 [6] Nguyen Thieu Huy, Pham Van Bang. Invariant stable manifolds for partial neutral functional differential equations in admissible spaces on a half-line. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 2993-3011. doi: 10.3934/dcdsb.2015.20.2993 [7] Carlos Arnoldo Morales. Strong stable manifolds for sectional-hyperbolic sets. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 553-560. doi: 10.3934/dcds.2007.17.553 [8] Luis Barreira, Claudia Valls. Stable manifolds with optimal regularity for difference equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1537-1555. doi: 10.3934/dcds.2012.32.1537 [9] Mark Pollicott. Ergodicity of stable manifolds for nilpotent extensions of Anosov flows. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 599-604. doi: 10.3934/dcds.2002.8.599 [10] Luis Barreira, Claudia Valls. Characterization of stable manifolds for nonuniform exponential dichotomies. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1025-1046. doi: 10.3934/dcds.2008.21.1025 [11] Arne Ogrowsky, Björn Schmalfuss. Unstable invariant manifolds for a nonautonomous differential equation with nonautonomous unbounded delay. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1663-1681. doi: 10.3934/dcdsb.2013.18.1663 [12] Henk Broer, Aaron Hagen, Gert Vegter. Numerical approximation of normally hyperbolic invariant manifolds. Conference Publications, 2003, 2003 (Special) : 133-140. doi: 10.3934/proc.2003.2003.133 [13] Andrei Fursikov. Local existence theorems with unbounded set of input data and unboundedness of stable invariant manifolds for 3D Navier-Stokes equations. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 269-289. doi: 10.3934/dcdss.2010.3.269 [14] Thorsten Hüls. Computing stable hierarchies of fiber bundles. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3341-3367. doi: 10.3934/dcdsb.2017140 [15] Tao Jiang, Xianming Liu, Jinqiao Duan. Approximation for random stable manifolds under multiplicative correlated noises. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3163-3174. doi: 10.3934/dcdsb.2016091 [16] Redouane Qesmi, Hans-Otto Walther. Center-stable manifolds for differential equations with state-dependent delays. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 1009-1033. doi: 10.3934/dcds.2009.23.1009 [17] Luis Barreira, Claudia Valls. Existence of stable manifolds for nonuniformly hyperbolic $c^1$ dynamics. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 307-327. doi: 10.3934/dcds.2006.16.307 [18] Alin Pogan, Kevin Zumbrun. Stable manifolds for a class of singular evolution equations and exponential decay of kinetic shocks. Kinetic & Related Models, 2019, 12 (1) : 1-36. doi: 10.3934/krm.2019001 [19] Àlex Haro, Rafael de la Llave. A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Numerical algorithms. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1261-1300. doi: 10.3934/dcdsb.2006.6.1261 [20] Jaemin Shin, Yongho Choi, Junseok Kim. An unconditionally stable numerical method for the viscous Cahn--Hilliard equation. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1737-1747. doi: 10.3934/dcdsb.2014.19.1737

2017 Impact Factor: 1.179