Skip to main content
SpringerLink
Log in

Computational Challenges of Nonlinear Systems

  • Chapter
  • First Online:
Emerging Frontiers in Nonlinear Science

Part of the book series: Nonlinear Systems and Complexity ((NSCH,volume 32))

Abstract

We survey some of the major types of dynamical-systems computations that can be carried out for two or three-dimensional systems of partial differential equations. In order of increasing complexity, we describe methods for calculating steady states and bifurcation diagrams, linear stability and Floquet analysis, and heteroclinic orbits. These are illustrated by computations for Rayleigh–Bénard convection in a cylindrical geometry, the Faraday instability of a fluid layer, the flow past a cylinder and over a square cavity, flow in a cylindrical container with counter-rotating lids, and Bose–Einstein condensation. We discuss some mathematical questions raised by these computations and the need for improved numerical tools.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. B. Hof, P.G. Lucas, T. Mullin, Phys. Fluids 11, 2815 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  2. K. Borońska, L.S. Tuckerman, Phys. Rev. E 81, 036320 (2010)

    Article  ADS  Google Scholar 

  3. K. Borońska, L.S. Tuckerman, Phys. Rev. E 81, 036321 (2010)

    Article  ADS  Google Scholar 

  4. Y. Saad, M.H. Schultz, SIAM J. Sci. Stat. Comput. 7, 856 (1986)

    Google Scholar 

  5. H.A. Van der Vorst, SIAM J. Sci. Stat. Comput. 13, 631 (1992)

    Google Scholar 

  6. P. Sonneveld, M.B. Van Gijzen, SIAM J. Sci. Comput. 31, 1035 (2008)

    Google Scholar 

  7. L.S. Tuckerman, in 11th International Conference on Numerical Methods in Fluid Dynamics, ed. by D.L. Dwoyer, M.Y. Hussaini, R.G. Voight. Lecture Notes in Physics, vol. 323 (Springer, Berlin, 1989), p. 573

    Google Scholar 

  8. C.K. Mamun, L.S. Tuckerman, Phys. Fluids 7, 80 (1995)

    Article  ADS  MathSciNet  Google Scholar 

  9. S. Xin, P. Le Quéré, L.S. Tuckerman, Phys. Fluids 10, 850 (1998)

    Article  ADS  Google Scholar 

  10. A. Bergeon, D. Henry, H. Benhadid, L.S. Tuckerman, J. Fluid Mech. 375, 143 (1998)

    Article  ADS  Google Scholar 

  11. C. Huepe, S. Metens, G. Dewel, P. Borckmans, M.E. Brachet, Phys. Rev. Lett. 82, 1616 (1999)

    Article  ADS  Google Scholar 

  12. L.S. Tuckerman, D. Barkley, in Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems, ed. by E. Doedel, L.S. Tuckerman. The IMA Volumes in Mathematics and its Applications, vol. 119 (Springer, New York, 2000), p. 453

    Google Scholar 

  13. C. Nore, L.S. Tuckerman, O. Daube, S. Xin, J. Fluid Mech. 477, 51 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  14. I. Mercader, A. Alonso, O. Batiste, Eur. Phys. J. E 15, 311 (2004)

    Article  Google Scholar 

  15. H.A. Dijkstra, F.W. Wubs, A.K. Cliffe, E. Doedel, I.F. Dragomirescu, B. Eckhardt, A.Y. Gelfgat, A.L. Hazel, V. Lucarini, A.G. Salinger, E.T. Phipps, J. Sanchez-Umbria, H. Schuttelaars, L.S. Tuckerman, U. Thiele, Commun. Comput. Phys. 15, 1 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. J.F. Gibson, J. Halcrow, P. Cvitanović, J. Fluid Mech. 638, 243 (2009)

    Article  ADS  Google Scholar 

  17. C.C. Pringle, Y. Duguet, R.R. Kerswell, Philos. Trans. R. Soc. A 367, 457 (2009)

    Article  ADS  Google Scholar 

  18. L.S. Tuckerman, J. Langham, A. Willis, in Computational Modelling of Bifurcations and Instabilities in Fluid Dynamics, ed. by A. Gelfgat (Springer, Cham, 2019), p. 3

    Chapter  Google Scholar 

  19. P. Cvitanović, B. Eckhardt, J. Phys. A 24, L237 (1991)

    Article  ADS  Google Scholar 

  20. J.F. Gibson, Channelflow: a spectral Navier-Stokes simulator in C++ (2014), http://channelflow.org. Accessed 27 Jan 2020

  21. A.P. Willis, SoftwareX 6, 124 (2017)

    Article  ADS  Google Scholar 

  22. L.S. Tuckerman, Commun. Comput. Phys. 18, 1336 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. T.B. Benjamin, F.J. Ursell, Proc. R. Soc. A 225, 505 (1954)

    ADS  Google Scholar 

  24. K. Kumar, L.S. Tuckerman, J. Fluid Mech. 279, 49 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  25. N. Perinet, D. Juric, L.S. Tuckerman, J. Fluid Mech. 635, 1 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  26. Y. Bengana, J.C. Loiseau, J.C. Robinet, L.S. Tuckerman, J. Fluid Mech. 875, 725 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  27. D. Barkley, R.D. Henderson, J. Fluid Mech. 322, 215 (1996)

    Article  ADS  Google Scholar 

  28. L.S. Tuckerman, D. Barkley, Phys. Rev. Lett. 61, 408 (1988)

    Article  ADS  Google Scholar 

  29. A. Andronov, E. Leontovich, Mat. Sb. 48, 335 (1959)

    MathSciNet  Google Scholar 

  30. Y.A. Kuznetsov, Elements of Applied Bifurcation Theory (Springer, New York, 2004)

    Book  MATH  Google Scholar 

  31. L. Bordja, L.S. Tuckerman, L.M. Witkowski, M.C. Navarro, D. Barkley, R. Bessaih, Phys. Rev. E 81, 036322 (2010)

    Article  ADS  Google Scholar 

  32. J.M. Lopez, A. Rubio, F. Marques, J. Fluid Mech. 569, 331 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  33. J.H. Siggers, J. Fluid Mech. 475, 357 (2003)

    Article  ADS  Google Scholar 

  34. F. Pétrélis, S. Fauve, E. Dormy, J.-P. Valet, Phys. Rev. Lett. 102, 144503 (2009)

    Article  ADS  Google Scholar 

  35. D. Armbruster, J. Guckenheimer, P. Holmes, Physica D 29, 257 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  36. Y. Bengana, L.S. Tuckerman, Phys. Rev. Fluids 4, 044402 (2019)

    Article  ADS  Google Scholar 

  37. I. Mercader, J. Prat, E. Knobloch, Int. J. Bifurc. Chaos 12, 2501 (2002)

    Article  Google Scholar 

  38. J. Halcrow, J.F. Gibson, P. Cvitanović, D. Viswanath, J. Fluid Mech. 621, 365 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  39. L. Van Veen, G. Kawahara, M. Atsushi, SIAM J. Sci. Comput. 33, 25 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  40. M. Farano, S. Cherubini, J.C. Robinet, P. De Palma, T.M. Schneider, J. Fluid Mech. 858, (2019)

    Google Scholar 

  41. C. Huepe, L.S. Tuckerman, S. Métens, M.E. Brachet, Phys. Rev. A 68, 023609 (2003)

    Article  ADS  Google Scholar 

  42. K.J. Law, P.G. Kevrekidis, L.S. Tuckerman, Phys. Rev. Lett. 105, 160405 (2010)

    Article  ADS  Google Scholar 

  43. N. Périnet, D. Juric, L.S. Tuckerman, Rev. Cuba. Fís. 29, 1E6 (2012)

    Google Scholar 

  44. N. Périnet, D. Juric, L.S. Tuckerman, Phys. Rev. Lett. 109, 164501 (2012)

    Article  ADS  Google Scholar 

  45. L. Kahouadji, N. Périnet, L.S. Tuckerman, S. Shin, J. Chergui, D. Juric, J. Fluid Mech. 772, (2015)

    Google Scholar 

  46. A.H. Ebo-Adou, L.S. Tuckerman, S. Shin, J. Chergui, D. Juric, J. Fluid Mech. 870, 433 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  47. P. Chossat, R. Lauterbach, I. Melbourne, Arch. Ration. Mech. Anal. 113, 313 (1991)

    Article  Google Scholar 

  48. S. Shin, J. Chergui, D. Juric, J. Mech. Sci. Technol. 31, 1739 (2017)

    Article  Google Scholar 

  49. P. Beltrame, U. Thiele, SIAM J. Appl. Dyn. Syst. 9, 484 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  50. M. Kubicek, M. Marek, Computational Methods in Bifurcation Theory and Dissipative Structures (Springer, Berlin, 1983)

    Book  MATH  Google Scholar 

  51. H.B. Keller, Lectures on Numerical Methods in Bifurcation Problems. Lectures on Mathematics and Physics Mathematics, vol. 79 (Springer, Berlin, 1988)

    Google Scholar 

  52. R. Seydel, From Equilibrium to Chaos: Practical Bifurcation and Stability Analysis (Elsevier, Amsterdam, 1988)

    MATH  Google Scholar 

  53. W.J.F. Govaerts, Numerical Methods for Bifurcations of Dynamical Equilibria (SIAM, Philadelphia, 2000)

    Book  MATH  Google Scholar 

  54. E. Doedel, L.S. Tuckerman (eds.), Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems. The IMA Volumes in Mathematics and Its Applications, vol. 119 (Springer, New York, 2000)

    Google Scholar 

  55. B. Krauskopf, H.M. Osinga, J. Galán-Vioque (eds.), Numerical Continuation Methods for Dynamical Systems (Springer Netherlands, Dordrecht, 2007)

    Google Scholar 

  56. A.S.D. Roose, B. De Dier, A. Spence (eds.), Continuation and Bifurcations: Numerical Techniques and Applications. NATO Science Series C, vol. 313 (Springer Netherlands, Dordrecht, 1990)

    Google Scholar 

  57. K.A. Cliffe, A. Spence, S.J. Tavener, Acta Numer. 9, 39 (2000)

    Article  Google Scholar 

  58. E.J. Doedel, in Numerical Continuation Methods for Dynamical Systems, ed. by B. Krauskopf, H.M. Osinga, J. Galán-Vioque (Springer Netherlands, Dordrecht, 2007), p. 1

    Google Scholar 

  59. E.J. Doedel, Congr. Numer. 30, 265 (1981)

    MathSciNet  Google Scholar 

  60. A. Back, J. Guckenheimer, M. Myers, F. Wicklin, P. Worfolk, Not. Am. Math. Soc. 39, 303 (1992)

    Google Scholar 

  61. K. Lust, D. Roose, SIAM J. Sci. Comput. 19, 1188 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  62. A. Dhooge, W. Govaerts, Y.A. Kuznetsov, A.C.M. Trans, ACM Trans. Math. Softw. 29, 141 (2003)

    Article  Google Scholar 

  63. A.G. Salinger, N.M. Bou-Rabee, R.P. Pawlowski, E.D. Wilkes, E.A. Burroughs, R.B. Lehoucq, L.A. Romero, LOCA 1.0 library of continuation algorithms: theory and implementation manual. Tech. Rep. SAND2002-0396 (Sandia National Laboratories, 2002)

    Google Scholar 

  64. G. Datseris, J. Open Source Softw. 3, 598 (2018)

    Article  ADS  Google Scholar 

  65. H. Uecker, D. Wetzel, J.D.M. Rademacher, Numer. Math.: Theory Methods Appl. 7, 58 (2014)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Physique et Mécanique des Milieux Hétérogènes (PMMH), CNRS, ESPCI Paris, PSL Research University, Sorbonne Université, Université Paris Diderot, 75005, Paris, France

    Laurette S. Tuckerman

Authors
  1. Laurette S. Tuckerman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Laurette S. Tuckerman .

Editor information

Editors and Affiliations

  1. Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA, USA

    Panayotis G. Kevrekidis

  2. Grupo de Fisica No Lineal, Departamento de Fisica Aplicada I, Escuela Politécnica Superior, Universidad de Sevilla, Seville, Spain

    Jesús Cuevas-Maraver

  3. Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA

    Avadh Saxena

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Tuckerman, L.S. (2020). Computational Challenges of Nonlinear Systems. In: Kevrekidis, P., Cuevas-Maraver, J., Saxena, A. (eds) Emerging Frontiers in Nonlinear Science. Nonlinear Systems and Complexity, vol 32. Springer, Cham. https://doi.org/10.1007/978-3-030-44992-6_11

Download citation

Publish with us

Policies and ethics

Search

Navigation