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Numerical Deterkination of Bifurcation Points in Steady State and Periodic Solutions — Numerical Algorithms and Examples

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Numerical Methods for Bifurcation Problems

Abstract

Numerical algorithms for determination of bifurcation points in steady State and periodic solutions are presented. Results of an evaluation of the limit points in a distributed parameter system where shooting method cannot be used are shown in the form of bifurcation diagram. Four direct iteration algorithms for an evaluation of complex (Hopf) bifurcation points in lumped parameter systems (ordinary differential equations) are described and applied to an example taken from the chemical reactor theory. An algorithm for determination of complex bifurcation points in distributed parameter systems (parabolic par-tial differential equations) is developed and results for a model of tubulär reactor with axial dispersion are presented.Two algorithms for evaluation of period doubling bifurcation points in periodic solutions of ordinary differential equations are suggested. Results of the application to a model of two inter-connected reaction cells are presented.

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Literature

  1. V.I. Arnold: Additional chapters of the theory of ordinary differential equations, Nauka, Moscow, 1978 (in Russianj

    Google Scholar 

  2. D.W.Decker, H.B.Keller: Solution branching - A constructive technique, in New Approaches to Nonlinear Problems in Dyna-mics ( P.J.Holmes, Ed.). SIAM publ. Philadelphia 1980, p. 53

    Google Scholar 

  3. D.Dellwo, H.B.Keller, B.J.Matkowsky, E.L.Reiss: On the birth of isolas, SIAM, J. Appl. Math, in press

    Google Scholar 

  4. V.Hlavacek, H.Hofmann, M.Kubicek: Modelling of chemical reactors XXIV., Chem. Engng. Sei. 26 (1971), 1629

    Google Scholar 

  5. V. Hlavacek, M.Kubicek, M.Marek: Analysis of non stationary heat and mass tranfer in a porous catalyst particle I-II, J. Catal. 15, (1969), 17, 31

    Google Scholar 

  6. M.Holodniok, M.Kubicek: New algorithms for evaluation of complex bifurcation points in ordinary differential equa-tions. A comparative numerical study, Appl. Math. Comput., to be published

    Google Scholar 

  7. M.Holodniok, M.Kubxcek: Continuation of periodic solutions - Algorithm and applications to the Lorenz model, this pro-ceedings

    Google Scholar 

  8. M.Holodniok, M.Kubicek, V.Hlavacek: Computation of the flow between two rotating coaxial disks, J. Pluid Mech. 81 (1977) 689

    Google Scholar 

  9. M.Holodniok, M.Kubicek, V.Hlavacek: Computation of the flow between two rotating coaxial disks. Multiplicity of steady State solutions, J. Pluid Mech. 108, (1981), 227

    Google Scholar 

  10. G.Iooss, D.D.Joseph: Elementary stability and bifurcation theory, Springer Verlag, New York, 1980

    Google Scholar 

  11. K»F.Jensen, W.H.Ray: The bifurcation behavior of tubulär reactors, Chem. Engng, Sei. 37, (1982), 199

    Article  Google Scholar 

  12. A.D.Jepson: Numerical Hopf bifurcation, PhD Thesis, Dept. of Math* Calif. Inst, of Technology, 1981

    Google Scholar 

  13. H.B.Keller: Numerical Solution of bifurcation and nonlinear eigenvalue problems, in Applications of Bifurcation Theory (P.H.Rabinowitz, Ed.), Academic Press, New York 1977,p#359

    Google Scholar 

  14. H.B.Keller, R.K.-H.Szeto: Calculations of flows between ro-tating disks, preprint.

    Google Scholar 

  15. M.Kubicek: Evaluation of branching points for nonlinear boundary value problems based on the GPM technique, Appl. Math* Comput. 1 (1975), 341

    Google Scholar 

  16. M.Kubicek: Linear systems with almost band matrix, Sei. papers of the Inst, of Chem. Technol. Prague, X 12 (1977), 5

    Google Scholar 

  17. M.Kubicek: Algorithm for evaluation of complex bifurcation points in ordinary differential equations. SIAM J.Appl. Math. 38 (1980), 103

    Google Scholar 

  18. M.Kubicek: Occurrence of oscillatory regimes in lumped parameter systems. Determination of Hopf’s bifurcation points. Chem. Engng. Sei. 34 (1979), 1078

    Google Scholar 

  19. BH.Kubxcek, V.Hlaväcek: Solution of boundary value problems IX. Evaluation of branching points based on the differenti-ation with respect to boundary condition. Chem. Engng. Sei. 30 (1975), 1439

    Google Scholar 

  20. M.Kubicek, M.Holodniok: Evaluation of Hopf bifurcation points in parabolic equations describing heat and mass transfer in chemical reactors. Chem, Engng. Sei. in press

    Google Scholar 

  21. M.Kubicek, A.KLic: Direction of branches bifurcating at a bifurcation point. Determination of starting points for a continuation algorithm, Appl. Math. Comput. 13 (1983), 125

    Google Scholar 

  22. M.Kubicek, M.Marek: Computational methods in bifurcation theory and dissipative struetures. Springer Verlag, New York, 1983

    Book  Google Scholar 

  23. M.Kubicek, M.Marek: Evaluation of limit and bifurcation points for algebraic and nonlinear boundary value problems. Appl. Math. Comput. 5, (1979), 253

    Article  Google Scholar 

  24. M.Kubicek, I.Stuchl, M.Marek: “Isolas11 in Solution diagrams, J.Comput. Phys. 48 (1982) 106

    Google Scholar 

  25. G.N.Lance, M.H.Rogers: The axially Symmetrie flow of a vis«* cous fluid between two infinite rotating disks, Proc. Roy, Soc. A 266 (1962), 109

    Article  Google Scholar 

  26. J.E.Marsden, M.McCracken, The Hopf bifurcation and its app«* lications. Springer Verlag, Berlin 1976

    Google Scholar 

  27. G.L.Mellor, P.J.Chapple, V.K.Stokes: Qn the flow between a rotating and a stationary disk, J. Fluid. Mech. 31 (1968), 95

    Article  Google Scholar 

  28. H.D.Mittelmann, H.Weber, ed.: Bifurcation problems and their numerical Solution, Birkhauser, Basel 1980

    Google Scholar 

  29. H.D.Mittelmann, H.Weber: Numerical methods for bifurcation Problems - A survey and Classification, in [28], p. 1

    Google Scholar 

  30. N.D’.Nguyen, J.P.Ribault, P.Florent: Multiple solutions for flow between coaxial disks, J. Pluid. Mech. 68 (1975), 369

    Google Scholar 

  31. G.Nicolis, I.Prigogine: Self-organization in nonequilibrium systems, J. Wiley, New York 1977

    Google Scholar 

  32. J.Peterson, K.A.Overholser, R.P.Heinemann: Hopf bifurcation in a radiating laminar flame, Chem. Engng. Sei. 36 (1981), 628

    Article  Google Scholar 

  33. S.M.Roberts, J.S.Shipman: Computation of the flow between a rotating and a stationary disk, J. Pluid. Mech. 73 (1976), 53

    Google Scholar 

  34. D.H.Sattinger: Topics in stability and bifurcation theory. Lecture Notes in Math. 309, Springer Verlag, Berlin 1973

    Google Scholar 

  35. I.Schreiber, M.Kubicek, M.Holodniok, M.Marek: Periodic phe-nomena in coupled cells, in preparation

    Google Scholar 

  36. I.Schreiber, M.Kubicek, M.Marek: On ooupled cells, In New Approaches to Nonlinear Problems in Dynamics, ed. by P.J. Holmes, SIAM Publ., Philadelphia 1980, p. 496

    Google Scholar 

  37. R« Seydel: Numerical computation of branch points in non-linear equations, Numer. Math. 33 (1979), 339

    Google Scholar 

  38. R. Seydel: Numerical computation of branch points in ordina-ry differential equations, Numer. Math. 32 (1979), 51

    Google Scholar 

  39. A.Varma, N.R.Amundson: Some problems concerning the non- -adiabatic tubulär reactor: A-priori bounds, qualitative behavior, preliminary uniqueness and stability considera-tions, Canad. J. Chem. Engng. 50 (1972), 470

    Article  Google Scholar 

  40. J.H.Wilkinson, C.Reinsch: Handbook for automatic computa-tion II, Linear algebra, Springer Verlag, Berlin, 1971

    Google Scholar 

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Authors and Affiliations

  1. Department of Chemical Engineering and Computer, Center Prague Institute of Chemical Technology Suchbätarova, 5 166 28, Praha 6, Czechoslovakia

    M. Kubíček & M. Holodniok

Authors
  1. M. Kubíček
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  2. M. Holodniok
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Editor information

Editors and Affiliations

  1. Abt. Mathematik, Universität Dortmund, Postfach 50 05 00, D-4600, Dortmund 50, Germany

    T. Küpper  & H. D. Mittelmann  & 

  2. Rechenzentrum, Johannes Gutenberg-Universität, Postfach 3980, D-6500, Mainz 1, Germany

    H. Weber

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Kubíček, M., Holodniok, M. (1984). Numerical Deterkination of Bifurcation Points in Steady State and Periodic Solutions — Numerical Algorithms and Examples. In: Küpper, T., Mittelmann, H.D., Weber, H. (eds) Numerical Methods for Bifurcation Problems. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique, vol 70. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-6256-1_17

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  • DOI: https://doi.org/10.1007/978-3-0348-6256-1_17

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-6257-8

  • Online ISBN: 978-3-0348-6256-1

  • eBook Packages: Springer Book Archive

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