Mathematics of Complexity and Dynamical Systems

2011 Edition
| Editors: Robert A. Meyers (Editor-in-Chief)

Perturbation Analysis of Parametric Resonance

  • Ferdinand Verhulst
Reference work entry
DOI: https://doi.org/10.1007/978-1-4614-1806-1_76

Article Outline

Glossary

Definition of the Subject

Introduction

Perturbation Techniques

Parametric Excitation of Linear Systems

Nonlinear Parametric Excitation

Applications

Future Directions

Acknowledgment

Bibliography

Keywords

Manifold Alan 
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Notes

Acknowledgment

A number of improvements and clarifications were suggested by the editor, Giuseppe Gaeta. Additional references were obtained from Henk Broer and Fadi Dohnal.

Bibliography

Primary Literature

  1. 1.
    Arnold VI (1983) Geometrical methods in the theory of ordinary differential equations. Springer, New YorkMATHCrossRefGoogle Scholar
  2. 2.
    Banichuk NV, Bratus AS, Myshkis AD (1989) Stabilizing and destabilizing effects in nonconservative systems. PMM USSR 53(2):158–164MathSciNetMATHGoogle Scholar
  3. 3.
    Bogoliubov NN, Mitropolskii Yu A (1961) Asymptotic methods in the theory of nonlinear oscillations. Gordon and Breach, New YorkGoogle Scholar
  4. 4.
    Bolotin VV (1963) Non‐conservative problems of the theory of elastic stability. Pergamon Press, OxfordGoogle Scholar
  5. 5.
    Broer HW, Vegter G (1992) Bifurcational aspects of parametric resonance, vol 1. In: Jones CKRT, Kirchgraber U, Walther HO (eds) Expositions in dynamical systems. Springer, Berlin, pp 1–51Google Scholar
  6. 6.
    Broer HW, Levi M (1995) Geometrical aspects of stability theory for Hill's equation. Arch Rat Mech Anal 131:225–240MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Broer HW, Simó C (1998) Hill's equation with quasi‐periodic forcing: resonance tongues, instability pockets and global phenomena. Bol Soc Brasil Mat 29:253–293Google Scholar
  8. 8.
    Broer HW, Hoveijn I, Van Noort M (1998) A reversible bifurcation analysis of the inverted pendulum. Physica D 112:50–63MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Broer HW, Hoveijn I, Van Noort M, Vegter G (1999) The inverted pendulum: a singularity theory approach. J Diff Eqs 157:120–149MATHCrossRefGoogle Scholar
  10. 10.
    Broer HW, Simó C (2000) Resonance tongues in Hill's equations: a geometric approach. J Differ Equ 166:290–327Google Scholar
  11. 11.
    Broer HW, Puig J, Simó C (2003) Resonance tongues and instability pockets in the quasi‐periodic Hill‐Schrödinger equation. Commun Math Phys 241:467–503Google Scholar
  12. 12.
    Broer HW, Hoveijn I, Van Noort M, Simó C, Vegter G (2005) The parametrically forced pendulum: a case study in \({1 \frac{1}{2}}\) degree of freedom. J Dyn Diff Equ 16:897–947Google Scholar
  13. 13.
    Cicogna G, Gaeta G (1999) Symmetry and perturbation theory in nonlinear dynamics. Lecture Notes Physics, vol 57. Springer, BerlinGoogle Scholar
  14. 14.
    Fatimah S, Ruijgrok M (2002) Bifurcation in an autoparametric system in 1:1 internal resonance with parametric excitation. Int J Non‐Linear Mech 37:297–308MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Golubitsky M, Schaeffer D (1985) Singularities and groups in bifurcation theory. Springer, New YorkMATHGoogle Scholar
  16. 16.
    Hale J (1963) Oscillation in nonlinear systems. McGraw‐Hill, New York, 1963; Dover, New York, 1992Google Scholar
  17. 17.
    Hoveijn I, Ruijgrok M (1995) The stability of parametrically forced coupled oscillators in sum resonance. ZAMP 46:383–392MathSciNetCrossRefGoogle Scholar
  18. 18.
    Iooss G, Adelmeyer M (1992) Topics in bifurcation theory. World Scientific, SingaporeMATHGoogle Scholar
  19. 19.
    Kuznetsov Yu A (2004) Elements of applied bifurcation theory, 3rd edn. Springer, New YorkMATHCrossRefGoogle Scholar
  20. 20.
    Krupa M (1997) Robust heteroclinic cycles. J Nonlinear Sci 7:129–176MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Len JL, Rand RH (1988) Lie transforms applied to a non‐linear parametric excitation problem. Int J Non‐linear Mech 23:297–313MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Levy DM, Keller JB (1963) Instability intervals of Hill's equation. Comm Pure Appl Math 16:469–476MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Magnus W, Winkler S (1966) Hill's equation. Interscience‐John Wiley, New YorkMATHGoogle Scholar
  24. 24.
    McLaughlin JB (1981) Period‐doubling bifurcations and chaotic motion for a parametrically forced pendulum. J Stat Phys 24:375–388MathSciNetCrossRefGoogle Scholar
  25. 25.
    Ng L, Rand RH (2002) Bifurcations in a Mathieu equation with cubic nonlinearities. Chaos Solitons Fractals 14:173–181MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Ng L, Rand RH (2003) Nonlinear effects on coexistence phenomenon in parametric excitation. Nonlinear Dyn 31:73–89MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Pikovsky AS, Feudel U (1995) Characterizing strange nonchaotic attractors. Chaos 5:253–260MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Ramani DV, Keith WL, Rand RH (2004) Perturbation solution for secondary bifurcation in the quadratically‐damped Mathieu equation. Int J Non‐linear Mech 39:491–502MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Recktenwald G, Rand RH (2005) Coexistence phenomenon in autoparametric excitation of two degree of freedom systems. Int J Non‐linear Mech 40:1160–1170MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Roseau M (1966) Vibrations nonlinéaires et théorie de la stabilité. Springer, BerlinGoogle Scholar
  31. 31.
    Ruijgrok M (1995) Studies in parametric and autoparametric resonance. Thesis, Utrecht University, UtrechtGoogle Scholar
  32. 32.
    Ruijgrok M, Verhulst F (1996) Parametric and autoparametric resonance. Prog Nonlinear Differ Equ Their Appl 19:279–298MathSciNetGoogle Scholar
  33. 33.
    Ruijgrok M, Tondl A, Verhulst F (1993) Resonance in a rigid rotor with elastic support. ZAMM 73:255–263MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Sanders JA, Verhulst F, Murdock J (2007) Averaging methods in nonlinear dynamical systems, rev edn. Appl Math Sci, vol 59. Springer, New YorkGoogle Scholar
  35. 35.
    Seyranian AP (2001) Resonance domains for the Hill equation with allowance for damping. Phys Dokl 46:41–44MathSciNetCrossRefGoogle Scholar
  36. 36.
    Seyranian AP, Mailybaev AA (2003) Multiparameter stability theory with mechanical applications. Series A, vol 13. World Scientific, SingaporeGoogle Scholar
  37. 37.
    Seyranian AA, Seyranian AP (2006) The stability of an inverted pendulum with a vibrating suspension point. J Appl Math Mech 70:754–761MathSciNetCrossRefGoogle Scholar
  38. 38.
    Stoker JJ (1950) Nonlinear vibrations in mechanical and electrical systems. Interscience, New York, 1950; Wiley, New York, 1992Google Scholar
  39. 39.
    Strutt MJO (1932) Lamé-sche, Mathieu‐sche und verwandte Funktionen. Springer, BerlinCrossRefGoogle Scholar
  40. 40.
    Szemplinska‐Stupnicka W (1990) The behaviour of nonlinear vibrating systems, vol 2. Kluwer, DordrechtGoogle Scholar
  41. 41.
    Tondl A (1991) Quenching of self‐excited vibrations. Elsevier, AmsterdamGoogle Scholar
  42. 42.
    Tondl A, Ruijgrok M, Verhulst F, Nabergoj R (2000) Autoparametric resonance in mechanical systems. Cambridge University Press, New YorkMATHGoogle Scholar
  43. 43.
    Van der Pol B, Strutt MJO (1928) On the stability of the solutions of Mathieu's equation. Phil Mag Lond Edinb Dublin 7(5):18–38Google Scholar
  44. 44.
    Verhulst F (1996) Nonlinear differential equations and dynamical systems. Springer, New YorkMATHCrossRefGoogle Scholar
  45. 45.
    Verhulst F (2005) Invariant manifolds in dissipative dynamical systems. Acta Appl Math 87:229–244MathSciNetMATHCrossRefGoogle Scholar
  46. 46.
    Verhulst F (2005) Methods and applications of singular perturbations. Springer, New YorkMATHCrossRefGoogle Scholar
  47. 47.
    Wiggins S (1988) Global Bifurcation and Chaos. Appl Math Sci, vol 73. Springer, New YorkCrossRefGoogle Scholar
  48. 48.
    Yakubovich VA, Starzhinskii VM (1975) Linear differential equations with periodic coefficients, vols 1 and 2. Wiley, New YorkGoogle Scholar
  49. 49.
    Zounes RS, Rand RH (1998) Transition curves for the quasi‐periodic Mathieu equation. SIAM J Appl Math 58:1094–1115MathSciNetMATHCrossRefGoogle Scholar
  50. 50.
    Zounes RS, Rand RH (2002) Global behavior of a nonlinear quasi‐periodic Mathieu equation. Nonlinear Dyn 27:87–105MathSciNetMATHCrossRefGoogle Scholar

Books and Reviews

  1. 51.
    Arnold VI (1977) Loss of stability of self‐oscillation close to resonance and versal deformation of equivariant vector fields. Funct Anal Appl 11:85–92CrossRefGoogle Scholar
  2. 52.
    Arscott FM (1964) Periodic differential equations. MacMillan, New YorkMATHGoogle Scholar
  3. 53.
    Cartmell M (1990) Introduction to linear, parametric and nonlinear vibrations. Chapman and Hall, LondonMATHGoogle Scholar
  4. 54.
    Dohnal F (2005) Damping of mechanical vibrations by parametric excitation. Ph D thesis, Vienna University of TechnologyGoogle Scholar
  5. 55.
    Dohnal F, Verhulst F (2008) Averaging in vibration suppression by parametric stiffness excitation. Nonlinear Dyn (accepted for publication)Google Scholar
  6. 56.
    Ecker H (2005) Suppression of self‐excited vibrations in mechanical systems by parametric stiffness excitation. Fortschrittsberichte Simulation Bd 11. Argesim/Asim Verlag, ViennaGoogle Scholar
  7. 57.
    Fatimah S (2002) Bifurcations in dynamical systems with parametric excitation. Thesis, University of UtrechtGoogle Scholar
  8. 58.
    Hale J (1969) Ordinary differential equations. Wiley, New YorkMATHGoogle Scholar
  9. 59.
    Kirillov ON (2007) Gyroscopic stabilization in the presence of nonconservative forces. Dokl Math 76:780–785; Orig Russian: (2007) Dokl Ak Nauk 416:451–456Google Scholar
  10. 60.
    Meixner J, Schäfke FW (1954) Mathieusche Funktionen und Sphäroidfunktionen. Springer, BerlinGoogle Scholar
  11. 61.
    Moon FC (1987) Chaotic vibrations: an introduction for applied scientists and engineers. Wiley, New YorkMATHGoogle Scholar
  12. 62.
    Nayfeh AH, Mook DT (1979) Nonlinear Oscillations. Wiley Interscience, New YorkMATHGoogle Scholar
  13. 63.
    Schmidt G (1975) Parametererregte Schwingungen. VEB Deutscher Verlag der Wissenschaften, BerlinMATHGoogle Scholar
  14. 64.
    Schmidt G, Tondl A (1986) Non‐linear vibrations. Akademie‐Verlag, BerlinCrossRefGoogle Scholar
  15. 65.
    Tondl A (1978) On the interaction between self‐excited and parametric vibrations. In: Monographs and Memoranda, vol 25. National Res Inst Běchovice, PragueGoogle Scholar
  16. 66.
    Tondl A (1991) On the stability of a rotor system. Acta Technica CSAV 36:331–338MATHGoogle Scholar
  17. 67.
    Tondl A (2003) Combination resonances and anti‐resonances in systems parametrically excited by harmonic variation of linear damping coefficients. Acta Technica CSAV 48:239–248Google Scholar
  18. 68.
    Van der Burgh AHP, Hartono (2004) Rain-wind induced vibrations of a simple oscillator. Int J Non‐Linear Mech 39:93–100MATHCrossRefGoogle Scholar
  19. 69.
    Van der Burgh AHP, Hartono, Abramian AK (2006) A new model for the study of rain-wind‐induced vibrations of a simple oscillator. Int J Non‐Linear Mech 41:345–358MATHCrossRefGoogle Scholar
  20. 70.
    Weinstein A, Keller JB (1985) Hill's equation with a large potential. SIAM J Appl Math 45:200–214MathSciNetMATHCrossRefGoogle Scholar
  21. 71.
    Weinstein A, Keller JB (1987) Asymptotic behaviour of stability regions for Hill's equation. SIAM J Appl Math 47:941–958MathSciNetMATHCrossRefGoogle Scholar
  22. 72.
    Wiggins S (1990) Introduction to applied nonlinear dynamical systems and chaos. Springer, New YorkMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Ferdinand Verhulst
    • 1
  1. 1.Mathematisch InstituutUniversity of UtrechtUtrechtThe Netherlands