A Mechanism for Spurious Solutions of Nonlinear Boundary Value Problems

  • H.-O. Peitgen
Part of the Springer Series in Synergetics book series (SSSYN, volume 21)


Typically numerical approximation schemes for many nonlinear boundary value problems generate spurious solutions. By developing a dynamical system approach a mechanism is presented which is able to explain a certain class of such solutions. In essence the spurious solutions here are a consequence of structural changes such as bifurcations in the homoclinic structure of the associated dynamical system.


Nonlinear Boundary Homoclinic Bifurcation Homoclinic Point Spurious Solution Symmetric Periodic Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Al]
    ALLGOWER, E. L.: On a discretization of y″ + λyk = o. Proc. Conf. Roy. Irish Acad., J. J. H. Miller (ed.), New York, Academic Press 1975, 1–15.Google Scholar
  2. [Be]
    BERRY, M. V.: Regular and irregular motion, in: Topics in Nonlinear Dynamics, S. Jorna (ed.), Amer. Inst. Phys. Conf. Proc., New York, 46, 16–120 (1978).Google Scholar
  3. [BD]
    BEYN, W.-J. and DOEDEL, E.: Stability and multiplicity of solutions to discretizations of ordinary differential equations, SIAM J. Sci. Stat. Comput. 2, 107–120 (1981).MATHCrossRefGoogle Scholar
  4. [BL]
    BEYN, W.-J. and LORENZ, J.: Spurious solutions for discrete superlinear boundary value problems, Computing 28, 43–51 (1982).MathSciNetMATHCrossRefGoogle Scholar
  5. [Bi]
    BIRKHOFF, G.D.: The restricted problem of three bodies, Rend. Circ. Mat. Palermo 39, 265–334 (1915).CrossRefGoogle Scholar
  6. [Bo]
    BOHL, E.: On the bifurcation diagram of discrete analogues for ordinary bifurcation problems, Math. Meth. Appl. in the Sci. 1 566–571 (1979).MathSciNetADSMATHCrossRefGoogle Scholar
  7. [GNN]
    GIDAS, B., NI, W. M. and NIRENBERG, L.: Symmetry and related properties via the maximum principle, Commun. Math. Phys. 68, 209–243 (1979).MathSciNetADSMATHCrossRefGoogle Scholar
  8. [G]
    GREENE, J. M.: A method for determining a stochastic transition, J. Math. Phys. 20, 1183–1201 (1979).ADSCrossRefGoogle Scholar
  9. [M]
    MOSER, J.: Stable and Random Motions in Dynamical Systems, Ann. of Math. Studies 77, Princeton Univ. Press, 1973.MATHGoogle Scholar
  10. NP] NUSSBAUM, R. D. and PEITGEN, H. O.: Special and spurious solution of x(t) = - αf(x(t-1)), to appearGoogle Scholar
  11. [Pa]
    PALIS, J.: On Morse - Smale dynamical systems, Topology 8, 365–404 (1968).MathSciNetGoogle Scholar
  12. [P1]
    PEITGEN, H. O.: Topologische Perturbationen beim globalen numerischen Studium nichtlinearer Eigenwert- und Verzweigungsprobleme, Jber. d. Dt. Math.-Verein. 84, 107–162 (1982).MathSciNetMATHGoogle Scholar
  13. [P2]
    PEITGEN, H. O.: Phase transitions in the homoclinic regime of area preserving diffeomorphisms, Proc. Intern. Symp. on Synergetics, H. Haken (ed.), Springer Series in Synergetics. 17 197–214 (1982).Google Scholar
  14. PR] PEITGEN, H. O. and RICHTER, P. H.: Homoclinic bifurcation and the fate of periodic points, to appear.Google Scholar
  15. [PSS]
    PEITGEN, H. O., SAUPE, D. and SCHMITT, K.: Nonlinear elliptic boundary value problems versus their finite difference approximations: numerically irrele¬vant solutions, J. reine angew. Math. 322, 74–117 (1981).MathSciNetMATHGoogle Scholar
  16. [PS]
    PEITGEN, H. O. and SCHMITT, K.: Positive and spurious solutions of nonlinear eigenvalue problems, in: Numerical solution of Nonlinear Equations, Allgower, E. L., Glashoff, K. and Peitgen, H. O. (eds.), Berlin-Heidelberg-New York, Springer Lecture Notes in Mathematics 878, 275–324 (1981).CrossRefGoogle Scholar
  17. [SA]
    SPREUER, H. and ADAMS, E.: Pathologische Beispiele von Differenzenverfahren bei nichtlinearen gewöhnlichen Randwertaufgaben, ZAMM 57, T 304–T305 (1977).MathSciNetGoogle Scholar
  18. SS] STEPHENS, A. B. and SHUBIN, G. R.: Multiple solutions and bifurcation of finite difference approximations to some steady state problems of fluid dynamics, to appear.Google Scholar
  19. [U]
    USHIKI, S.: Unstable manifolds of analytic dynamical systems, J. Math. Kyoto Univ. 2U 763–785 (1981).MathSciNetGoogle Scholar
  20. [V]
    VOGELAERE de, R.: On the structure of symmetric periodic solutions of conservative systems, with applications, in Contributions to the Theory of Nonlinear Oscillations, vol. 4, S. Lefschetz (ed.), Princeton University Press, 1968.Google Scholar
  21. [YU]
    YAMAGUTI, M. and USHIKI, S.: Chaos in numerical analysis of ordinary differential equations, Physica 3D, 618–626 (1981).MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  • H.-O. Peitgen
    • 1
  1. 1.Forschungsschwerpunkt “Dynamische Systeme”Universität BremenBremen 33Fed Rep. of Germany

Personalised recommendations