Eigenvalue bifurcation in multiparameter families of non-self-adjoint operator matrices

  • O. N. KirillovEmail author


We consider two-point non-self-adjoint boundary eigenvalue problems for linear matrix differential operators. The coefficient matrices in the differential expressions and the matrix boundary conditions are assumed to depend analytically on the complex spectral parameter λ and on the vector of real physical parameters p. We study perturbations of semi-simple multiple eigenvalues as well as perturbations of non-derogatory eigenvalues under small variations of p. Explicit formulae describing the bifurcation of the eigenvalues are derived. Application to the problem of excitation of unstable modes in rotating continua such as spherically symmetric MHD α 2-dynamo and circular string demonstrates the efficiency and applicability of the approach.

Mathematics Subject Classification (2000)

Primary 34B08 Secondary 34D10 


Operator matrix Non-self-adjoint boundary eigenvalue problem Keldysh chain Multiple eigenvalue Diabolical point Exceptional point Perturbation Bifurcation Stability Veering Spectral mesh Rotating continua 


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  1. 1.
    Pedersen P.: Influence of boundary conditions on the stability of a column under non-conservative load. Int. J. Solids Struct. 13, 445–455 (1977)zbMATHCrossRefGoogle Scholar
  2. 2.
    Bou-Rabee N.M., Romero L.A., Salinger A.G.: A multiparameter, numerical stability analysis of a standing cantilever conveying fluid. SIAM J. Appl. Dyn. Syst. 1(2), 190–214 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Stefani F., Gailitis A., Gerbeth G.: Magnetohydrodynamic experiments on cosmic magnetic fields. Z. Angew. Math. Mech. 88(12), 930–954 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    von Neumann J., Wigner E.P.: Über das Verhalten von Eigenwerten bei adiabatischen Prozessen. Z. Phys. 30, 467–470 (1929)Google Scholar
  5. 5.
    Arnold V.I.: Geometrical Methods in the Theory of Ordinary Differential Equations. Springer, Berlin (1983)zbMATHGoogle Scholar
  6. 6.
    Teytel M.: How rare are multiple eigenvalues?. Comm. Pure Appl. Math. 52, 917–934 (1999)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Günther U., Kirillov O.N.: A Krein space related perturbation theory for MHD α 2-dynamos and resonant unfolding of diabolical points. J. Phys. A 39, 10057–10076 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Leissa A.W.: On a curve veering aberration. Z. Angew. Math. Phys. 25, 99–111 (1974)zbMATHCrossRefGoogle Scholar
  9. 9.
    Yang L., Hutton S.G.: Interactions between an idealized rotating string and stationary constraints. J. Sound Vibr. 185(1), 139–154 (1995)zbMATHCrossRefGoogle Scholar
  10. 10.
    Vidoli S., Vestroni F.: Veering phenomena in systems with gyroscopic coupling. Trans. ASME J. Appl. Mech. 72, 641–647 (2005)zbMATHGoogle Scholar
  11. 11.
    Rellich F.: Störungstheorie der Spektralzerlegung. Math. Ann. 113, 600–619 (1937)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Holopainen E.O.: On the effect of friction in baroclinic waves. Tellus 13(3), 363–367 (1961)CrossRefGoogle Scholar
  13. 13.
    Hoveijn I., Ruijgrok M.: The stability of parametrically forced coupled oscillators in sum resonance. Z. Angew. Math. Phys. 46, 384–392 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Crandall S.H.: The effect of damping on the stability of gyroscopic pendulums. Z. Angew. Math. Phys. 46, S761–S780 (1995)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Kirillov O.N.: A theory of the destabilization paradox in non-conservative systems. Acta Mech. 174(3–4), 145–166 (2005)zbMATHCrossRefGoogle Scholar
  16. 16.
    Kirillov O.N., Seyranian A.P.: Instability of distributed non-conservative systems caused by weak dissipation. Dokl. Math. 71(3), 470–475 (2005)MathSciNetGoogle Scholar
  17. 17.
    Kirillov O.N., Seyranian A.O.: The effect of small internal and external damping on the stability of distributed non-conservative systems. J. Appl. Math. Mech. 69(4), 529–552 (2005)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Kirillov O.N.: Gyroscopic stabilization of non-conservative systems. Phys. Lett. A 359(3), 204–210 (2006)zbMATHCrossRefGoogle Scholar
  19. 19.
    Krechetnikov R., Marsden J.E.: Dissipation-induced instabilities in finite dimensions. Rev. Mod. Phys. 79, 519–553 (2007)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Mailybaev A.A., Kirillov O.N., Seyranian A.P.: Geometric phase around exceptional points. Phys. Rev. A 72, 014104 (2005)CrossRefGoogle Scholar
  21. 21.
    Stefani F., Gerbeth G., Günther U., Xu M.: Why dynamos are prone to reversals. Earth Planet. Sci. Lett. 243, 828–840 (2006)CrossRefGoogle Scholar
  22. 22.
    Spelsberg-Korspeter G., Kirillov O.N., Hagedorn P.: Modeling and stability analysis of an axially moving beam with frictional contact. Trans. ASME J. Appl. Mech. 75(3), 031001 (2008)Google Scholar
  23. 23.
    Kirillov O.N.: Subcritical flutter in the acoustics of friction. Proc. R. Soc. A 464(2097), 2321–2339 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  24. 24.
    Kirillov O.N.: Campbell diagrams of weakly anisotropic flexible rotors. Proc. R. Soc. A 465(2109), 2703–2723 (2009)CrossRefMathSciNetzbMATHGoogle Scholar
  25. 25.
    Seyranian, A.P., Mailybaev, A.A.: Multiparameter Stability Theory with Mechanical Applications. World Scientific, Singapore (2003)Google Scholar
  26. 26.
    MacKay R.S.: Stability of equilibria of Hamiltonian systems. In: Sarkar, S. (eds) Nonlinear Phenomena and Chaos., pp. 254–270. Adam Hilger, Bristol (1986)Google Scholar
  27. 27.
    Krein M.G.: A generalization of some investigations of linear differential equations with periodic coefficients. Dokl. Akad. Nauk SSSR N.S. 73, 445–448 (1950)MathSciNetGoogle Scholar
  28. 28.
    Vishik M.I., Lyusternik L.A.: Solution of some perturbation problems in the case of matrices and selfadjoint or non-selfadjoint equations. Russ. Math. Surv. 15, 1–73 (1960)zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Lidskii V.B.: Perturbation theory of non-conjugate operators. U.S.S.R. Comput. Math. Math. Phys. 1, 73–85 (1965)MathSciNetGoogle Scholar
  30. 30.
    Rellich F.: Perturbation Theory of Eigenvalue Problems. Gordon and Breach, New York (1968)Google Scholar
  31. 31.
    Kato T.: Perturbation Theory for Linear Operators. Springer, Berlin (1966)zbMATHGoogle Scholar
  32. 32.
    Baumgärtel H.: Analytic Perturbation Theory for Matrices and Operators. Akademie-Verlag, Berlin (1984)Google Scholar
  33. 33.
    Sun J.G.: Eigenvalues and eigenvectors of a matrix dependent on several parameters. J. Comput. Math. 3(4), 351–364 (1985)zbMATHMathSciNetGoogle Scholar
  34. 34.
    Sun J.G.: Multiple eigenvalue sensitivity analysis. Linear Algebra Appl. 137/138, 183–211 (1990)CrossRefGoogle Scholar
  35. 35.
    Seyranian A.P., Kirillov O.N., Mailybaev A.A.: Coupling of eigenvalues of complex matrices at diabolic and exceptional points. J. Phys. A 38(8), 1723–1740 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Kirillov O.N., Mailybaev A.A., Seyranian A.P.: Unfolding of eigenvalue surfaces near a diabolic point due to a complex perturbation. J. Phys. A: Math. Gen. 38(24), 5531–5546 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Gohberg I., Lancaster P., Rodman L.: Perturbation of analytic Hermitian matrix functions. Appl. Anal. 20, 23–48 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Naiman B.: Remarks on the perturbation of analytic matrix functions. Int. Equ. Oper. Theor. 9, 593–599 (1986)Google Scholar
  39. 39.
    Langer H., Naiman B.: Remarks on the perturbation of analytic matrix functions II. Int. Equ. Oper. Theor. 12, 392–407 (1989)zbMATHCrossRefGoogle Scholar
  40. 40.
    Hryniv R., Lancaster P.: On the perturbation of analytic matrix functions. Int. Equ. Oper. Theor. 34(3), 325–338 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Lancaster P., Markus A.S., Zhou F.: Perturbation theory for analytic matrix functions: the semisimple case. SIAM J. Matrix Anal. Appl. 25(3), 606–626 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    Kirillov O.N., Seyranian A.P.: Collapse of Keldysh chains and the stability of non-conservative systems. Dokl. Math. 66(1), 127–131 (2002)Google Scholar
  43. 43.
    Kirillov O.N., Seyranian A.P.: Collapse of the Keldysh chains and stability of continuous nonconservative systems. SIAM J. Appl. Math. 64(4), 1383–1407 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    Goodman J., Ji H.: Magnetorotational instability of dissipative Couette flow. J. Fluid Mech. 462, 365–382 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    Naimark M.A.: Linear Differential Operators. Frederick Ungar Publishing, New York (1967)zbMATHGoogle Scholar
  46. 46.
    Mennicken R., Möller M.: Non-Self-Adjoint Boundary Eigenvalue Problems. Elsevier, Amsterdam (2003)Google Scholar
  47. 47.
    Stefani F., Gerbeth G.: Oscillatory mean-field dynamos with a spherically symmetric, isotropic helical turbulence parameter α. Phys. Rev. E 67, 027302 (2003)CrossRefGoogle Scholar
  48. 48.
    Kirillov O.N., Günther U., Stefani F.: Determining role of Krein signature for three dimensional Arnold tongues of oscillatory dynamos. Phys. Rev. E 79(1), 016205 (2009)CrossRefMathSciNetGoogle Scholar
  49. 49.
    Günther U., Kirillov O.N.: Asymptotic methods for spherically symmetric MHD α 2-dynamos. Proc. Appl. Math. Mech. 7(1), 4140023–4140024 (2007)CrossRefGoogle Scholar
  50. 50.
    Chen J.-S., Bogy D.B.: Mathematical structure of modal interactions in a spinning disk-stationary load system. Trans. ASME J. Appl. Mech. 59, 390–397 (1992)MathSciNetCrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  1. 1.Dynamics and Vibrations Group, Department of Mechanical EngineeringTechnische Universität DarmstadtDarmstadtGermany

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