Skip to main content
SpringerLink
Log in

Near-Identity Averaging Transformations: Transient and Sustained Resonance

  • Chapter
Multiple Scale and Singular Perturbation Methods

Part of the book series: Applied Mathematical Sciences ((AMS,volume 114))

  • 1938 Accesses

Abstract

In this chapter, we study another approach for calculating asymptotic solutions for systems in the standard form (4.5.1)

$$ \frac{{d{p_m}}}{{dt}} = \in {F_m}\left( {{p_i},{q_i},\tilde{t}; \in } \right),m = 1,2,...,M, $$
(5.1.1a)
$$ \frac{{d{q_n}}}{{dt}} = {\omega_n}\left( {{p_i},\tilde{t}} \right) + \in {G_n}\left( {{p_i},{q_i},\tilde{t}; \in } \right),n = 1,2,...,N. $$
(5.1.1b)

. Recall that 0<∈≪1,t̃= ≪t, and the subscript i indicates that all M of the p i or all N of the q i are present in the argument. We assume that for each m = 1,..., M and each n = 1,..., N the F m and G n are O(1) as ≪→ 0, and that the F m and G m are 2π-periodic functions of each of the q i .

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. D.L. Bosley, “An improved matching procedure for transient resonance layers in weakly nonlinear oscillatory systems,” SIAM J. Appl. Math., 56, 1996.

    Google Scholar 

  2. D.L. Bosley and J. Kevorkian, “On the asymptotic solution of non-Hamiltonian systems exhibiting sustained resonance,” Stud. Appl. Math., 94, 1995, pp. 83–130.

    MathSciNet  MATH  Google Scholar 

  3. D.L. Bosley and J. Kevorkian, “Adiabatic invariance and transient resonance in very slowly varying Hamiltonian systems,” SIAM J. Appl. Math., 52, 1992, pp. 494–527.

    Article  MathSciNet  MATH  Google Scholar 

  4. D.L. Bosley and J. Kevorkian, “Free-electron lasers with very slow wiggler taper,” IEEE J. Quantum Electron., 27, 1991, pp. 1078–1089.

    Article  Google Scholar 

  5. D.L. Bosley and J. Kevorkian, “Sustained resonance in very slowly varying oscillatory Hamiltonian systems,” SIAM J. Appl Math., 51, 1991, pp. 439–471.

    Article  MathSciNet  MATH  Google Scholar 

  6. F.J. Bourland and R. Haberman, “Separatrix crossing: time-invariant potentials with dissipation,” SIAM J. Appl. Math., 50, 1990, pp. 1716–1744.

    Article  MathSciNet  MATH  Google Scholar 

  7. B.V. Chirikov, “The passage of a nonlinear oscillating system through resonance,” Sov. Phys. Dokl, 4, 1959, pp. 390–394.

    MathSciNet  MATH  Google Scholar 

  8. H. Goldstein, Classical Mechanics, 2nd ed., Addison-Wesley, Reading, MA, 1980.

    MATH  Google Scholar 

  9. R. Haberman, “Energy bounds for the slow capture by a center in sustained resonance” SIAM J. Appl. Math., 43, 1983, pp. 244–256.

    Article  MathSciNet  MATH  Google Scholar 

  10. W.L. Kath, “Conditions for sustained resonance II,” SIAM J. Appl. Math., 43, 1983, pp. 579–583.

    Article  MathSciNet  MATH  Google Scholar 

  11. W.L. Kath, “Necessary conditions for sustained resonance,” SIAM J. Appl. Math., 43, 1983, pp. 314–324.

    Article  MathSciNet  MATH  Google Scholar 

  12. J. Kevorkian, Partial Differential Equations: Analytical Solution Techniques, Chapman and Hall, New York, London, 1990, 1993.

    MATH  Google Scholar 

  13. J. Kevorkian, “Perturbation techniques for oscillatory systems with slowly varying coefficients,” SIAM Rev., 29, 1987, pp. 391–461.

    Article  MathSciNet  MATH  Google Scholar 

  14. J. Kevorkian, “Adiabatic invariance and passage through resonance for nearly periodic Hamiltonian systems,” Stud. Appl. Math., 66, 1982, pp. 95–119.

    MathSciNet  MATH  Google Scholar 

  15. J. Kevorkian, “On a model for re-entry roll resonance,” SIAM J. Appl. Math., 26, 1974, pp. 638–669.

    Article  MATH  Google Scholar 

  16. J. Kevorkian, “Passage through resonance for a one-dimensional oscillator with slowly varying frequency,” SIAM J. Appl. Math., 20, 1971, pp. 364–373. See also Errata in 26, 1974, p. 686.

    Article  MATH  Google Scholar 

  17. J. Kevorkian and H.K. Li, “Resonant modal interactions and adiabatic invariance for a nonlinear wave equation in a variable domain,” Stud. Appl. Math., 71, 1984, pp. 1–64.

    MathSciNet  MATH  Google Scholar 

  18. N.M. Krylov and N.N. Bogoliubov, Introduction to Nonlinear Mechanics, Acad. Sci., Ukrain, S.S.R., 1937. Translated by S. Lefschetz, Princeton University Press, Princeton, NJ, 1947.

    Google Scholar 

  19. L. Lewin and J. Kevorkian, “On the problem of sustained resonance,” SIAM J. Appl. Math., 35, 1978, pp. 738–754.

    Article  MATH  Google Scholar 

  20. Y.P. Li and J. Kevorkian, “The effects of wiggler taper rate and signal field gain rate in free-electron lasers,” IEEE J. Quantum Electron., 24, 1988, pp. 598–608.

    Article  Google Scholar 

  21. A.J. Lichtenberg and M.A. Lieberman, Regular and Chaotic Dynamics, Springer-Verlag, New York, 1992.

    MATH  Google Scholar 

  22. Y.A. Mitropolski, Problèmes de la Théorie Asymptotique des Oscillations Non Stationnaires, Gauthier-Villars, Paris, 1966. Translated from the Russian.

    Google Scholar 

  23. J.A. Morrison, “Generalized method of averaging and the von Zeipel method,” Progress in Astronautics and Aeronautics 17, Methods in Astrodynamics and Celestial Mechanics, R.L. Duncombe and V.G. Szebehely, Eds., Academic Press, New York, 1966, pp. 117–138.

    Google Scholar 

  24. J.C. Neu, “The method of near-identity transformations and its applications,” SIAM J. Appl. Math., 38, 1980, pp. 189–208.

    Article  MathSciNet  MATH  Google Scholar 

  25. V.M. Volosov, “Averaging in systems of ordinary differential equations,” Russ. Math. Surveys, 17, 1963, pp.1–126.

    Article  Google Scholar 

  26. H. von Zeipel, “Recherche sur le mouvement des petites planètes,” Ark. Astron. Mat. Fys., 11–13, 1916.

    Google Scholar 

  27. L. Wang, D.L. Bosley, and J. Kevorkian, “Asymptotic analysis of a class of three-degree-of-freedom Hamiltonian systems near stable equilibria,” Physica D, 88, 1995, pp. 87–115.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, University of Washington, Seattle, WA, 98195, USA

    J. Kevorkian

  2. Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY, 12181, USA

    J. D. Cole

Authors
  1. J. Kevorkian
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. D. Cole
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 1996 Springer-Verlag New York, Inc.

About this chapter

Cite this chapter

Kevorkian, J., Cole, J.D. (1996). Near-Identity Averaging Transformations: Transient and Sustained Resonance. In: Multiple Scale and Singular Perturbation Methods. Applied Mathematical Sciences, vol 114. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-3968-0_5

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-3968-0_5

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-8452-9

  • Online ISBN: 978-1-4612-3968-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation