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A unified approach to compute foliations, inertial manifolds, and tracking solutions

Authors: Y.-M. Chung and M. S. Jolly
Journal: Math. Comp. 84 (2015), 1729-1751
MSC (2010): Primary 34C40, 34C45, 37L25
Published electronically: December 9, 2014
MathSciNet review: 3335889
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Abstract | References | Similar Articles | Additional Information

Abstract: Several algorithms are presented for the accurate computation of the leaves in the foliation of an ODE near a hyperbolic fixed point. They are variations of a contraction mapping method used by Ricardo Rosa in 1995 to compute inertial manifolds, which represents a particular leaf in the unstable foliation. Such a mapping is combined with one for the leaf in the stable foliation to compute tracking solutions. The algorithms are demonstrated on the Kuramoto-Sivashinsky equation.

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  • [1] Peter W. Bates and Kening Lu, A Hartman-Grobman theorem for the Cahn-Hilliard and phase-field equations, J. Dynam. Differential Equations 6 (1994), no. 1, 101-145. MR 1262725 (94m:35280),
  • [2] Peter W. Bates, Kening Lu, and Chongchun Zeng, Existence and persistence of invariant manifolds for semiflows in Banach space, Mem. Amer. Math. Soc. 135 (1998), no. 645, viii+129. MR 1445489 (99b:58210),
  • [3] Peter W. Bates, Kening Lu, and Chongchun Zeng, Invariant foliations near normally hyperbolic invariant manifolds for semiflows, Trans. Amer. Math. Soc. 352 (2000), no. 10, 4641-4676. MR 1675237 (2001b:37031),
  • [4] Xavier Cabré, Ernest Fontich, and Rafael de la Llave, The parameterization method for invariant manifolds. III. Overview and applications, J. Differential Equations 218 (2005), no. 2, 444-515. MR 2177465 (2007b:37057),
  • [5] Nelson Castañeda and Ricardo Rosa, Optimal estimates for the uncoupling of differential equations, J. Dynam. Differential Equations 8 (1996), no. 1, 103-139. MR 1388166 (97d:34032),
  • [6] Shui-Nee Chow, Xiao-Biao Lin, and Kening Lu, Smooth invariant foliations in infinite-dimensional spaces, J. Differential Equations 94 (1991), no. 2, 266-291. MR 1137616 (92k:58210),
  • [7] Y.-M. Chung.
    FOLI8PAK software package.
    The Dynamical Systems Web Software List (
  • [8] Stephen M. Cox and A. J. Roberts, Initial conditions for models of dynamical systems, Phys. D 85 (1995), no. 1-2, 126-141. MR 1339235 (96e:34072),
  • [9] C. Foias, O. Manley, and R. Temam, Modelling of the interaction of small and large eddies in two-dimensional turbulent flows, RAIRO Modél. Math. Anal. Numér. 22 (1988), no. 1, 93-118 (English, with French summary). MR 934703 (89h:76022)
  • [10] C. Foias, B. Nicolaenko, G. R. Sell, and R. Temam, Inertial manifolds for the Kuramoto-Sivashinsky equation and an estimate of their lowest dimension, J. Math. Pures Appl. (9) 67 (1988), no. 3, 197-226. MR 964170 (90e:35137)
  • [11] Ciprian Foias, George R. Sell, and Roger Temam, Inertial manifolds for nonlinear evolutionary equations, J. Differential Equations 73 (1988), no. 2, 309-353. MR 943945 (89e:58020),
  • [12] Ciprian Foias, George R. Sell, and Edriss S. Titi, Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations, J. Dynam. Differential Equations 1 (1989), no. 2, 199-244. MR 1010966 (90k:35031),
  • [13] C. Foiaş and R. Temam, Some analytic and geometric properties of the solutions of the evolution Navier-Stokes equations, J. Math. Pures Appl. (9) 58 (1979), no. 3, 339-368. MR 544257 (81k:35130)
  • [14] M. S. Jolly, I. G. Kevrekidis, and E. S. Titi, Approximate inertial manifolds for the Kuramoto-Sivashinsky equation: analysis and computations, Phys. D 44 (1990), no. 1-2, 38-60. MR 1069671 (91f:35217),
  • [15] M. S. Jolly and R. Rosa, Computation of non-smooth local centre manifolds, IMA J. Numer. Anal. 25 (2005), no. 4, 698-725. MR 2170520 (2006g:37135),
  • [16] M. S. Jolly, R. Rosa, and R. Temam, Accurate computations on inertial manifolds, SIAM J. Sci. Comput. 22 (2000), no. 6, 2216-2238 (electronic). MR 1856310 (2002k:35044),
  • [17] M. S. Jolly, R. Rosa, and R. Temam, Evaluating the dimension of an inertial manifold for the Kuramoto-Sivashinsky equation, Adv. Differential Equations 5 (2000), no. 1-3, 31-66. MR 1734536 (2001a:37127)
  • [18] U. Kirchgraber and K. J. Palmer, Geometry in the Neighborhood of Invariant Manifolds of Maps and Flows and Linearization, Pitman Research Notes in Mathematics Series, vol. 233, Longman Scientific & Technical, Harlow, 1990. MR 1068954 (91k:58115)
  • [19] B. Krauskopf, H. M. Osinga, E. J. Doedel, M. E. Henderson, J. Guckenheimer, A. Vladimirsky, M. 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), no. 3, 763-791. MR 2136745 (2006h:37116),
  • [20] Kening Lu, A Hartman-Grobman theorem for scalar reaction-diffusion equations, J. Differential Equations 93 (1991), no. 2, 364-394. MR 1125224 (92k:35147),
  • [21] Yves Nievergelt, Aitken's and Steffensen's accelerations in several variables, Numer. Math. 59 (1991), no. 3, 295-310. MR 1106386 (92a:65172),
  • [22] Christian Pötzsche and Martin Rasmussen, Computation of nonautonomous invariant and inertial manifolds, Numer. Math. 112 (2009), no. 3, 449-483. MR 2501313 (2010a:65135),
  • [23] A. J. Roberts, Appropriate initial conditions for asymptotic descriptions of the long term evolution of dynamical systems, J. Austral. Math. Soc. Ser. B 31 (1989), no. 1, 48-75. MR 1002091 (90h:58075),
  • [24] A. J. Roberts, Computer algebra derives correct initial conditions for low-dimensional dynamical models, Computer Physics Communication 126 (2000), 187-206.
  • [25] James C. Robinson, Computing inertial manifolds, Discrete Contin. Dyn. Syst. 8 (2002), no. 4, 815-833. MR 1920645 (2003g:37142),
  • [26] Ricardo Rosa, Approximate inertial manifolds of exponential order, Discrete Contin. Dynam. Systems 1 (1995), no. 3, 421-448. MR 1355883 (96j:34114),
  • [27] Roger Temam and Xiao Ming Wang, Estimates on the lowest dimension of inertial manifolds for the Kuramoto-Sivashinsky equation in the general case, Differential Integral Equations 7 (1994), no. 3-4, 1095-1108. MR 1270121 (95b:35219)

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Additional Information

Y.-M. Chung
Affiliation: Department of Mathematics, University of Kansas, Lawrence, Kansas 66045

M. S. Jolly
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405

Keywords: Foliations, inertial manifolds, tracking solution, spectral gap condition
Received by editor(s): September 25, 2012
Received by editor(s) in revised form: September 12, 2013, and October 27, 2013
Published electronically: December 9, 2014
Additional Notes: This work was supported in part by NSF grant numbers DMS-1008661 and DMS-1109638. The authors thank Ricardo Rosa for several stimulating discussions, and the referees for their helpful comments.
Article copyright: © Copyright 2014 American Mathematical Society

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