Advertisement

Scaling and Delay

  • Christian Kuehn
Chapter
Part of the Applied Mathematical Sciences book series (AMS, volume 191)

Abstract

This chapter has two major goals. The first is to analyze delayed loss of stability near fast subsystem bifurcation points with a focus on Hopf bifurcation. The second goal is to introduce several algebraic-combinatorial flavored tools, which turn out to be very helpful for multiscale systems.

Bibliography

  1. [Abe85b]
    E.H. Abed. Singularly perturbed Hopf bifurcation. IEEE Transactions on Circuits and Systems, 32(12):1270–1280, 1985.CrossRefzbMATHMathSciNetGoogle Scholar
  2. [AFR13]
    A.H. Abbasian, H. Fallah, and M.R. Razvan. Symmetric bursting behaviors in the generalized FitzHugh–Nagumo model. Biol. Cybern., pages 1–12, 2013. to appear.Google Scholar
  3. [AG13]
    G.G. Avalos and N.B. Gallegos. Quasi-steady state model determination for systems with singular perturbations modelled by bond graphs. Math. Computer Mod. Dyn. Syst., pages 1–21, 2013. to appear.Google Scholar
  4. [AGMR89]
    F.T. Arecchi, W. Gadomski, R. Meucci, and J.A. Roversi. Delayed bifurcation at the threshold of a swept gain CO2 laser. Optics Commun., 70(2):155–160, 1989.CrossRefGoogle Scholar
  5. [Arn94]
    V.I. Arnold. Encyclopedia of Mathematical Sciences: Dynamical Systems V. Springer, 1994.Google Scholar
  6. [Bae91a]
    C. Baesens. Noise effect on dynamic bifurcations: the case of a period-doubling cascade. In E. Benoît, editor, Dynamic Bifurcations, volume 1493 of Lecture Notes in Mathematics, pages 107–130. Springer, 1991.Google Scholar
  7. [Bae91b]
    C. Baesens. Slow sweep through a period-doubling cascade: delayed bifurcations and renormalisation. Physica D, 53(2):319–375, 1991.CrossRefzbMATHMathSciNetGoogle Scholar
  8. [Bae95]
    C. Baesens. Gevrey series and dynamic bifurcations for analytic slow–fast mappings. Nonlinearity, 8(2):179, 1995.Google Scholar
  9. [BBRS91]
    W. Balser and J. Mozo-Fernández. Multisummability of formal power series solutions of linear ordinary differential equations. Asymp. Anal., 5(1):27–45, 1991.zbMATHGoogle Scholar
  10. [Ben91a]
    E. Benoît, editor. Dynamic Bifurcations, volume 1493 of Lecture Notes in Mathematics. Springer, 1991.Google Scholar
  11. [Ben91b]
    E. Benoît. Linear dynamic bifurcation with noise. In E. Benoît, editor, Dynamic Bifurcations, volume 1493 of Lecture Notes in Mathematics, pages 131–150. Springer, 1991.Google Scholar
  12. [Ben09]
    E. Benoît. Bifurcation delay - the case of the sequence: stable focus - unstable focus - unstable node. Discr. Cont. Dyn. Sys. - Series S, 2(4):911–929, 2009.CrossRefzbMATHGoogle Scholar
  13. [BER89]
    S.M. Baer, T. Erneux, and J. Rinzel. The slow passage through a Hopf bifurcation: delay, memory effects, and resonance. SIAM J. Appl. Math., 49(1):55–71, 1989.CrossRefzbMATHMathSciNetGoogle Scholar
  14. [Ber98b]
    N. Berglund. Adiabatic Dynamical Systems and Hysteresis. PhD thesis, EPFL, 1998.Google Scholar
  15. [Ber00]
    N. Berglund. Control of dynamic Hopf bifurcations. Nonlinearity, 13:225–248, 2000.CrossRefzbMATHMathSciNetGoogle Scholar
  16. [BG08a]
    S.M. Baer and E.M. Gaekel. Slow acceleration and deacceleration through a Hopf bifurcation: power ramps, target nucleation, and elliptic bursting. Phys. Rev. E, 78:036205, 2008.CrossRefMathSciNetGoogle Scholar
  17. [BGK12]
    N. Berglund, B. Gentz, and C. Kuehn. Hunting French ducks in a noisy environment. J. Differential Equat., 252(9):4786–4841, 2012.CrossRefzbMATHMathSciNetGoogle Scholar
  18. [BH98a]
    J.D. Brothers and R. Haberman. Accurate phase after slow passage through subharmonic resonance. SIAM J. Appl. Math., 59(1):347–364, 1998.CrossRefMathSciNetGoogle Scholar
  19. [BH98b]
    J.D. Brothers and R. Haberman. Slow passage through a homoclinic orbit with subharmonic resonances. Stud. Appl. Math., 101(2):211–232, 1998.CrossRefzbMATHMathSciNetGoogle Scholar
  20. [BHS87]
    C. De Boor, K. Höllig, and M. Sabin. High accuracy geometric Hermite interpolation. Computer Aided Geometric Design, 4(4):269–278, 1987.CrossRefzbMATHMathSciNetGoogle Scholar
  21. [BK99]
    N. Berglund and H. Kunz. Memory effects and scaling laws in slowly driven systems. J. Phys. A: Math. Gen., 32:15–39, 1999.CrossRefzbMATHMathSciNetGoogle Scholar
  22. [BO99]
    C.M. Bender and S.A. Orszag. Asymptotic Methods and Perturbation Theory. Springer, 1999.Google Scholar
  23. [Bom78]
    E. Bombieri. On exponential sums in finite fields, II. Invent. Math., 47(1):29–39, 1978.CrossRefzbMATHMathSciNetGoogle Scholar
  24. [Bru89]
    Alexander D. Bruno. Local Methods in Nonlinear Differential Equations. Springer, 1989.Google Scholar
  25. [BT97a]
    D. Bertsimas and J.N. Tsitsiklis. Introduction to Linear Optimization. Athena Scientific, 1997.Google Scholar
  26. [CKN13]
    K. Cho, J. Kamimoto, and T. Nose. Asymptotic analysis of oscillatory integrals via the Newton polyhedra of the phase and the amplitude. J. Math. Soc. Japan, 65(2):521–562, 2013.CrossRefzbMATHMathSciNetGoogle Scholar
  27. [CMA90]
    M. Ciofini, R. Meucci, and F.T. Arecchi. Transient statistics in a CO 2 laser with a slowly swept pump. Phys. Rev. A, 42:482–486, 1990.CrossRefGoogle Scholar
  28. [DH00a]
    D.C. Diminnie and R. Haberman. Slow passage through a saddle-center bifurcation. J. Nonlinear Sci., 10:197–221, 2000.CrossRefzbMATHMathSciNetGoogle Scholar
  29. [DH02b]
    D.C. Diminnie and R. Haberman. Slow passage through homoclinic orbits for the unfolding of a saddle-center bifurcation and the change in the adiabatic invariant. Physica D, 162:34–52, 2002.CrossRefzbMATHMathSciNetGoogle Scholar
  30. [DKO09]
    M. Desroches, B. Krauskopf, and H.M. Osinga. The geometry of mixed-mode oscillations in the Olsen model for the perioxidase-oxidase reaction. DCDS-S, 2(4):807–827, 2009.CrossRefzbMATHMathSciNetGoogle Scholar
  31. [DMV98]
    R. Denk, R. Mennicken, and L. Volevich. The newton polygon and elliptic problems with parameter. Math. Nachr., 192(1):125–157, 1998.CrossRefzbMATHMathSciNetGoogle Scholar
  32. [DNS05]
    J. Denef, J. Nicaise, and P. Sargos. Oscillating integrals and Newton polyhedra. J. Anal. Math., 95:147–172, 2005.CrossRefzbMATHMathSciNetGoogle Scholar
  33. [DSS08]
    R. Denk, J. Saal, and J. Seiler. Inhomogeneous symbols, the Newton polygon, and maximal L p-regularity. Russ. J. Math. Phys., 15(2):171–191, 2008.CrossRefzbMATHMathSciNetGoogle Scholar
  34. [Dum93]
    F. Dumortier. Techniques in the theory of local bifurcations: Blow-up, normal forms, nilpotent bifurcations, singular perturbations. In D. Schlomiuk, editor, Bifurcations and Periodic Orbits of Vector Fields, pages 19–73. Kluwer, Dortrecht, The Netherlands, 1993.Google Scholar
  35. [EM86]
    T. Erneux and P. Mandel. Imperfect bifurcation with a slowly-varying control parameter. SIAM J. Appl. Math., 46(1):1–15, 1986.CrossRefzbMATHMathSciNetGoogle Scholar
  36. [EM91]
    T. Erneux and P. Mandel. Slow passage through the laser first threshold: influence of the initial conditions. Optics Commun., 85(1):43–46, 1991.CrossRefGoogle Scholar
  37. [ERHG91]
    T. Erneux, E.L. Reiss, L.J. Holden, and M. Georgiou. Slow passage through bifurcations and limit points. Asymptotic theory and applications. In E. Benoît, editor, Dynamic Bifurcations, volume 1493 of Lecture Notes in Mathematics, pages 14–28. Springer, 1991.Google Scholar
  38. [EW05]
    G. Everest and T. Ward. An Introduction to Number Theory. Springer, 2005.Google Scholar
  39. [Fru91]
    A. Fruchard. Existence of bifurcation delay: the discrete case. In E. Benoît, editor, Dynamic Bifurcations, volume 1493 of Lecture Notes in Mathematics, pages 87–106. Springer, 1991.Google Scholar
  40. [FS03b]
    A. Fruchard and R. Schäfke. Bifurcation delay and difference equations. Nonlinearity, 16:2199–2220, 2003.CrossRefzbMATHMathSciNetGoogle Scholar
  41. [FS09]
    A. Fruchard and R. Schäfke. A survey of some results on overstability and bifurcation delay. Discr. Cont. Dyn. Sys. - Series S, 2(4):941–965, 2009.Google Scholar
  42. [Gam01]
    T.W. Gamelin. Complex Analysis. Springer, 2001.Google Scholar
  43. [GO12]
    J. Guckenheimer and H.M. Osinga. The singular limit of a Hopf bifurcation. Discr. Cont. Dyn. Syst. A, 32:2805–2823, 2012.CrossRefzbMATHMathSciNetGoogle Scholar
  44. [Gre10]
    M. Greenblatt. Oscillatory integral decay, sublevel set growth, and the Newton polyhedron. Math. Ann., 346:857–895, 2010.CrossRefzbMATHMathSciNetGoogle Scholar
  45. [Hab79]
    R. Haberman. Slowly varying jump and transition phenomena associated with algebraic bifurcation problems. SIAM J. Appl. Math., 37(1):69–106, 1979.CrossRefzbMATHMathSciNetGoogle Scholar
  46. [HE93a]
    L. Holden and T. Erneux. Slow passage through a Hopf bifurcation: from oscillatory to steady state solutions. SIAM J. Appl. Math., 53(4):1045–1058, 1993.CrossRefzbMATHMathSciNetGoogle Scholar
  47. [HE93b]
    L. Holden and T. Erneux. Understanding bursting oscillations as periodic slow passages through bifurcation and limit points. J. Math. Biol., 31(4):351–365, 1993.CrossRefzbMATHMathSciNetGoogle Scholar
  48. [Hir67]
    H. Hironaka. Characteristic polyhedra of singularities. Kyoto J. Math., 7(3):251–293, 1967.zbMATHMathSciNetGoogle Scholar
  49. [How01]
    J. Howald. Multiplier ideals of monomial ideals. Trans. Amer. Math. Soc., 353(7):2665–2671, 2001.CrossRefzbMATHMathSciNetGoogle Scholar
  50. [IS63]
    M. Iwano and Y. Sibuya. Reduction of the order of a linear ordinary differential equation containing a small parameter. Kodai Math. J., 15:1–28, 1963.CrossRefMathSciNetGoogle Scholar
  51. [Kam04]
    J. Kamimoto. Newton polyhedra and the Bergman kernel. Math. Z., 246:405–440, 2004.CrossRefzbMATHMathSciNetGoogle Scholar
  52. [Kev71]
    J. Kevorkian. Passage through resonance for a one-dimensional oscillator with slowly varying frequency. SIAM J. Appl. Math., 20:364–373, 1971.CrossRefzbMATHGoogle Scholar
  53. [KW10]
    M. Krupa and M. Wechselberger. Local analysis near a folded saddle-node singularity. J. Differential Equat., 248(12): 2841–2888, 2010.CrossRefzbMATHMathSciNetGoogle Scholar
  54. [Lob91]
    C. Lobry. Dynamic bifurcations. In E. Benoît, editor, Dynamic Bifurcations, volume 1493 of Lecture Notes in Mathematics, pages 1–13. Springer, 1991.Google Scholar
  55. [Man87]
    P. Mandel. Properties of a good cavity laser with swept losses. Optics Commun., 64(6):549–552, 1987.CrossRefGoogle Scholar
  56. [Maz73]
    B. Mazur. Frobenius and the Hodge filtration (estimates). Ann. Math., 98(1):58–95, 1973.CrossRefzbMATHMathSciNetGoogle Scholar
  57. [ME84]
    P. Mandel and T. Erneux. Laser Lorenz equations with a time-dependent parameter. Phys. Rev. Lett., 53(19):1818–1821, 1984.CrossRefGoogle Scholar
  58. [ME87]
    P. Mandel and T. Erneux. The slow passage through a steady bifurcation: delay and memory effects. J. Stat. Phys., 48(5):1059–1070, 1987.CrossRefMathSciNetGoogle Scholar
  59. [Mik08]
    O. Mikitchenko. Applications of the resolution of singularities to asymptotic analysis of differential equations. PhD thesis, Boston University, Boston, USA, 2008.Google Scholar
  60. [Nei87b]
    A.I. Neishtadt. Persistence of stability loss for dynamical bifurcations. I. Differential Equations Translations, 23:1385–1391, 1987.Google Scholar
  61. [Nei88]
    A.I. Neishtadt. Persistence of stability loss for dynamical bifurcations. II. Differential Equations Translations, 24:171–176, 1988.Google Scholar
  62. [Nei09]
    A.I. Neishtadt. On the stability loss delay for dynamical bifurcations. Discr. Cont. Dyn. Sys. S, 2(4):897–909, 2009.CrossRefzbMATHMathSciNetGoogle Scholar
  63. [Oor00]
    F. Oort. Newton polygons and formal groups: conjectures by Manin and Grothendieck. Ann. Math., 152(1):183–206, 2000.CrossRefzbMATHMathSciNetGoogle Scholar
  64. [PDL11]
    Y. Park, Y. Do, and J.M. Lopez. Slow passage through resonance. Phys. Rev. E, 84:056604, 2011.CrossRefGoogle Scholar
  65. [RB88]
    J. Rinzel and S.M. Baer. Threshold for repetitive activity for a slow stimulus ramp: a memory effect and its dependence on fluctuations. Biophys. J., 54(3):551–555, 1988.CrossRefGoogle Scholar
  66. [RS03]
    D. Rachinskii and K. Schneider. Delayed loss of stability in systems with degenerate linear parts. Zeitschr. Anal. Anwend., 22(2):433–454, 2003.CrossRefMathSciNetGoogle Scholar
  67. [Sam91]
    S.N. Samborski. Rivers from the point of view of the qualitative theory. In E. Benoît, editor, Dynamic Bifurcations, volume 1493 of Lecture Notes in Mathematics, pages 168–180. Springer, 1991.Google Scholar
  68. [Shi73]
    M.A. Shishkova. Analysis of a system of differential equations with a small parameter at the higher derivatives. Akademiia Nauk SSSR, Doklady, 209:576–579, 1973.Google Scholar
  69. [SM96]
    P.E. Strizhak and M. Menzinger. Slow passage through a supercritical Hopf bifurcation: time-delayed response in the Belousov–Zhabotinsky reaction in a batch reactor. J. Chem. Phys., 105:10905–10910, 1996.CrossRefGoogle Scholar
  70. [SSB+87]
    W. Scharpf, M. Squicciarini, D. Bromley, C. Green, J.R. Tredicce, and L.M. Narducci. Experimental observation of a delayed bifurcation at the threshold of an argon laser. Optics Commun., 63(5):344–348, 1987.CrossRefGoogle Scholar
  71. [Sti98b]
    M. Stiefenhofer. Singular perturbation with Hopf points in the fast dynamics. Z. Angew. Math. Phys., 49(4):602–629, 1998.CrossRefzbMATHMathSciNetGoogle Scholar
  72. [Su96a]
    J. Su. Delayed bifurcation properties in the FitzHugh–Nagumo equation with periodic forcing. Differen. Integral Equat., 9(3):527–539, 1996.zbMATHGoogle Scholar
  73. [Su96b]
    J. Su. Persistent unstable periodic motions, I. J. Math. Anal. Appl., 198(3):796–825, 1996.CrossRefzbMATHMathSciNetGoogle Scholar
  74. [Su97]
    J. Su. Effects of periodic forcing on delayed bifurcations. J. Dyn. Differen. Equat., 9(4):561–625, 1997.CrossRefzbMATHGoogle Scholar
  75. [Su01]
    J. Su. The phenomenon of delayed bifurcation and its analysis. In C.K.R.T. Jones, editor, Multiple Time Scale Dynamical Systems, volume 122 of The IMA Volumes in Mathematics and its Applications, pages 203–214. Springer, 2001.Google Scholar
  76. [Var76]
    A.N. Varchenko. Newton polyhedra and estimation of oscillating integrals. Functional Anal., 10(3): 175–196, 1976.CrossRefMathSciNetGoogle Scholar
  77. [ZKSW11]
    W. Zhang, V. Kirk, J. Sneyd, and M. Wechselberger. Changes in the criticality of Hopf bifurcations due to certain model reduction techniques in systems with multiple timescales. J. Math. Neurosci., 1:9, 2011.CrossRefMathSciNetGoogle Scholar
  78. [ZW10a]
    Y.G. Zheng and Z.H. Wang. Delayed Hopf bifurcation in time-delayed slow–fast systems. Science China - Technological Sciences, 53(3):656–663, 2010.CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Christian Kuehn
    • 1
  1. 1.Institute for Analysis and Scientific ComputingVienna University of TechnologyViennaAustria

Personalised recommendations