The European Physical Journal Special Topics

, Volume 225, Issue 13–14, pp 2465–2486 | Cite as

Numerical continuation methods for large-scale dissipative dynamical systems

  • Juan Sánchez Umbría
  • Marta Net
Review Review articles
Part of the following topical collections:
  1. Temporal and Spatio-Temporal Dynamic Instabilities: Novel Computational and Experimental Approaches


A tutorial on continuation and bifurcation methods for the analysis of truncated dissipative partial differential equations is presented. It focuses on the computation of equilibria, periodic orbits, their loci of codimension-one bifurcations, and invariant tori. To make it more self-contained, it includes some definitions of basic concepts of dynamical systems, and some preliminaries on the general underlying techniques used to solve non-linear systems of equations by inexact Newton methods, and eigenvalue problems by means of subspace or Arnoldi iterations.


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  1. 1.
    E. Doedel, AUTO: Software for Continuation and Bifurcation Problems in Ordinary Differential Equations, Tech. report, Applied Mathematics, California Institute of Technology, Pasadena CA (1986)Google Scholar
  2. 2.
    M. Kubíček, M. Marek, Computational Methods in Bifurcation Theory and Dissipative Structures (Springer-Verlag, 1983)Google Scholar
  3. 3.
    W.C. Rheinboldt, Numerical Analysis of Parametrized Nonlinear Equations (J. Wiley, 1986)Google Scholar
  4. 4.
    E.L. Allgower, K. Georg, Numerical Continuation Methods: An Introduction, Vol. 13 of Computational Mathematics (Springer, 1990)Google Scholar
  5. 5.
    R. Seydel, Practical bifurcation and stability analysis, in From equilibrium to chaos (Springer, New York, 1994)Google Scholar
  6. 6.
    Y.A. Kuznetsov, Elements of Applied Bifurcation Theory (Springer, Berlin, 1998)Google Scholar
  7. 7.
    K.A. Cliffe, A. Spence, S. Taverner, Acta Numer. 9, 39 (2000)CrossRefGoogle Scholar
  8. 8.
    H. Dankowicz, F. Schilder, Recipes for Continuation, Computational Science and Engineering (SIAM, 2013)Google Scholar
  9. 9.
    E. Riks, ASME J. Appl. Mech. 39, 1060 (1971)CrossRefGoogle Scholar
  10. 10.
    R. Meyer-Spasche, H.B. Keller, J. Comput. Phys. 35, 100 (1980)ADSCrossRefGoogle Scholar
  11. 11.
    Y. Saad, M.H. Schultz, SIAM J. Sci. Stat. Comput. 7, 856 (1986)MathSciNetCrossRefGoogle Scholar
  12. 12.
    R.B. Lehoucq, D.C. Sorensen, C. Yang, ARPACK User’s Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods, Software, Environments, Tools (SIAM, 1998)Google Scholar
  13. 13.
    E. Doedel, L.S. Tuckerman (Eds.), Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems, Vol. 119 of IMA Volumes in Mathematics and its Applications (Springer-Verlag, 2000)Google Scholar
  14. 14.
    H.A. Dijkstra, F.W. Wubs, A.K. Cliffe, E. Doedel, I.F. Dragomirescu, B. Eckhardt, A. Gelfgat, A. Hazel, V. Lucarini, A. Salinger, J. Sánchez, H. Schuttelaars, L. Tuckerman, U. Thiele, Commun. Comput. Phys. 15, 1 (2014)MathSciNetCrossRefGoogle Scholar
  15. 15.
    A.G. Salinger, N.M. Bou-Rabee, R.P. Pawlowsky, E.D. Wilkes, E.A. Burroughs, R.B. Lehoucq, L.A. Romero, LOCA 1.1. Library of Continuation Algorithms: Theory and Implementation Manual (Sandia National Laboratories, Albuquerque, NM, 2002)Google Scholar
  16. 16.
    H. Uecker, D. Wetzel, J. Rademacher, Th. Meth. Appl. 7, 58 (2014)MathSciNetGoogle Scholar
  17. 17.
    K. Lust, D. Roose, A. Spence, A. Champneys, SIAM J. Sci. Comput. 19, 1188 (1998)MathSciNetCrossRefGoogle Scholar
  18. 18.
    T.L. van Noorden, S.M. Verduyn Lunel, A. Bliek, SIAM J. Sci. Comput. 25, 1921 (2004)MathSciNetCrossRefGoogle Scholar
  19. 19.
    J. Sánchez, M. Net, B. García-Archilla, C. Simó, J. Comput. Phys. 201, 13 (2004)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    J. Sánchez, M. Net, Int. J. Bifur. Chaos 20, 1 (2010)CrossRefGoogle Scholar
  21. 21.
    L. van Veen, G. Kawahara, M. Atsushi, SIAM J. Sci. Comput. 33, 25 (2011)MathSciNetCrossRefGoogle Scholar
  22. 22.
    J. Sánchez, M. Net, C. Simó, Physica D 239, 123 (2010)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    J. Sánchez, M. Net, Physica D 252, 22 (2013)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    M. Net, J. Sánchez, SIAM J. Appl. Dynam. Systems 14, 674 (2015)CrossRefGoogle Scholar
  25. 25.
    Y. Saad, Iterative methods for sparse linear systems (PWS pub. company, New York, 1996)Google Scholar
  26. 26.
    T.L. van Noorden, S.M. Verduyn Lunel, A. Bliek, IMA J. Appl. Math. 68, 149 (2003)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    K.I. Dickson, C.T. Kelley, I.C.F. Ipsen, I.G. Kevrekidis, SIAM J. Numer. Anal. 45, 263 (2007)MathSciNetCrossRefGoogle Scholar
  28. 28.
    C.T. Kelley, Iterative methods for linear and nonlinear equations (Frontiers in applied mathematics, SIAM, 1995)Google Scholar
  29. 29.
    R.S. Dembo, S.C. Eisenstat, T. Steihaug, SIAM J. Numer. Anal. 19, 400 (1982)ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    J.E. Dennis, Jr., R.B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Classics in Applied Mathematics, 16, SIAM, 1996)Google Scholar
  31. 31.
    S.C. Eisenstat, H.F. Walker, SIAM J. Sci. Comput. 17, 16 (1996)MathSciNetCrossRefGoogle Scholar
  32. 32.
    M. Pernice, H.F. Walker, SIAM J. Sci. Comput. 19, 302 (1998)MathSciNetCrossRefGoogle Scholar
  33. 33.
    S.L. Campbell, I.C.F. Ipsen, C.T. Kelley, C.D. Meyer, BIT Numerical Mathematics 36, 664 (1996)MathSciNetCrossRefGoogle Scholar
  34. 34.
    R. Barrett, M. Berry, T.F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, H.V. der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (SIAM, Philadelphia, 1994)Google Scholar
  35. 35.
    V. Frayssé, L. Giraud, S. Gratton, J. Langou, ACM Trans. Math. Softw. 31, 228 (2005)CrossRefGoogle Scholar
  36. 36.
    Y. Saad, Numerical Methods for Large Eigenvalue Problems (Manchester University Press, Manchester, 1992)Google Scholar
  37. 37.
    W. Stewart, A. Jennings, ACM Trans. Math. Soft. 7, 230 (1981)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Z. Bai, G.W. Stewart, ACM Trans. Math. Softw. 23, 494 (1997)MathSciNetCrossRefGoogle Scholar
  39. 39.
    E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, D. Sorensen, LAPACK Users’ Guide, 3rd edn. (SIAM, 1999)Google Scholar
  40. 40.
    D.C. Sorensen, SIAM J. Matrix Anal. Appl. 13, 357 (1992)MathSciNetCrossRefGoogle Scholar
  41. 41.
    B. García-Archilla, J. Sánchez, C. Simó, BIT 46, 731 (2006)MathSciNetCrossRefGoogle Scholar
  42. 42.
    K. Meerbergen, D. Roose, IMA J. Numer. Anal. 16, 297 (1996)MathSciNetCrossRefGoogle Scholar
  43. 43.
    A. Fortin, M. Jardak, J.J. Gervais, R. Pierre, Int. J. Numer. Meth. Fluids 24, 1185 (1997)MathSciNetCrossRefGoogle Scholar
  44. 44.
    M. Rieutord, L. Valdettaro, J. Fluid Mech. 341, 77 (1997)ADSMathSciNetCrossRefGoogle Scholar
  45. 45.
    J. Sánchez, F. Garcia, M. Net, J. Comput. Phys. 308, 273 (2016)ADSMathSciNetCrossRefGoogle Scholar
  46. 46.
    R.B. Lehoucq, A.G. Salinger, Int. J. Numer. Meth. Fluids 36, 309 (2001)CrossRefGoogle Scholar
  47. 47.
    C. Wales, A.L. Gaitonde, D.P. Jones, D. Avitabile, A.R. Champneys, Int. J. Numer. Meth. Fluids 68, 135 (2012)MathSciNetCrossRefGoogle Scholar
  48. 48.
    M. Net, F. Garcia, J. Sánchez, J. Fluid Mech. 601, 317 (2008)ADSMathSciNetCrossRefGoogle Scholar
  49. 49.
    D. Barkley, R.D. Henderson, J. Fluid Mech. 322, 215 (1996)ADSCrossRefGoogle Scholar
  50. 50.
    W.-J. Beyn, A. Champneys, E. Doedel, W. Govaerts, Y.A. Kuznetsov, B. Sandstede, Chapter 4, Numerical continuation, and computation of normal forms Vol. 2 of Handbook of Dynamical Systems (Elsevier Science, 2002), p. 149Google Scholar
  51. 51.
    A.G. Salinger, E.A. Burroughs, R.P. Pawlowski, E.T. Phipps, L.A. Romero, Int. J. Bifur. Chaos 15, 1015 (2005)MathSciNetCrossRefGoogle Scholar
  52. 52.
    L.O. Chua, A. Ushida, IEEE Tran. Circuits Sys. 28, 953 (1981)MathSciNetCrossRefGoogle Scholar
  53. 53.
    C. Kaas-Petersen, Physica D 25, 288 (1987)ADSMathSciNetCrossRefGoogle Scholar
  54. 54.
    C. Simó, Effective Computations in Hamiltonian Dynamics, in: Cent ans après les Méthodes Nouvelles de H. Poincaré (Société Mathématique de France, 1996), pp. 1–23Google Scholar
  55. 55.
    C. Simó, Effective computations in celestial mechanics and astrodynamics, in: V. Rumyantsev, A. Karapetyan (Eds.), Modern Methods of Analytical Mechanics and their Applications (CISM Courses and Lectures 387, Springer, 1998), pp. 55–102Google Scholar
  56. 56.
    À. Jorba, Nonlinearity 14, 943 (2001)ADSMathSciNetCrossRefGoogle Scholar
  57. 57.
    F. Schilder, H.M. Osinga, W. Vogt, SIAM J. Appl. Dynam. Syst. 4, 459 (2005)ADSMathSciNetCrossRefGoogle Scholar
  58. 58.
    G. Kawahara, M. Uhlmann, L. van Veen, Ann. Rev. Fluid Mech. 44, 203 (2012)ADSMathSciNetCrossRefGoogle Scholar
  59. 59.
    T. Watanabe, M. Iima, Y.Y. Nishiura, J. Fluid Mech. 712, 219 (2012)ADSMathSciNetCrossRefGoogle Scholar
  60. 60.
    J. Sánchez, F. Garcia, M. Net, Phys. Rev. E. 87, 033014 (2013)ADSCrossRefGoogle Scholar
  61. 61.
    F. Feudel, L.S. Tuckerman, M. Gellert, N. Seehafer, Phys. Rev. E 92, 053015 (2015)ADSCrossRefGoogle Scholar
  62. 62.
    D. Viswanath, J. Fluid Mech. 580, 339 (2007)ADSMathSciNetCrossRefGoogle Scholar
  63. 63.
    Y. Duguet, C.C.T. Pringle, R.R. Kerswell, Phys. Fluids 20, 114102 (2008)ADSCrossRefGoogle Scholar
  64. 64.
    D. Puigjaner, J. Herrero, C. Simó, F. Giralt, Physica D 240, 920 (2011)ADSMathSciNetCrossRefGoogle Scholar
  65. 65.
    I.C. Waugh, S.J. Illingworth, M.P. Juniper, J. Comput. Phys. 240, 225 (2013)ADSMathSciNetCrossRefGoogle Scholar
  66. 66.
    I.C. Waugh, K. Kashinath, M.P. Juniper, J. Fluid Mech. 759, 1 (2014)ADSMathSciNetCrossRefGoogle Scholar
  67. 67.
    F. Garcia, M. Net, J. Sánchez, Phys. Rev. E 93, 013119 (2016)ADSCrossRefGoogle Scholar

Copyright information

© EDP Sciences and Springer 2016

Authors and Affiliations

  1. 1.Departament de Física, Universitat Politècnica de CatalunyaBarcelonaSpain

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