Advertisement

Journal of Engineering Mathematics

, Volume 53, Issue 3–4, pp 301–336 | Cite as

Working with multiscale asymptotics

Solving Weakly nonlinear oscillator equations on long-time intervals
  • Blessing Mudavanhu
  • Robert E. O’MalleyJr
  • David B. Williams
Article

Abstract

This paper surveys, compares and updates techniques to obtain the asymptotic solution of the weakly nonlinear oscillator equation ÿ + y +ε f(y, \(\dot{y}\)) =0 as ε → 0 and for corresponding first-order vector systems. The solutions found by the regular perturbation method generally feature resonance and so break down as t → ∞. The classical methods of averaging and multiple scales eliminate such secular behavior and provide asymptotic solutions valid for time intervals of length t=O1). The renormalization group method proposed by Chen et al. [Phys. Rev. E 54 (1996) 376–394] gives equivalent results. Several well-known examples are solved with these methods to demonstrate the respective techniques and the equivalency of the approximations produced. Finally, an amplitude-equation method is derived that incorporates the best features of all these techniques. This method is both straightforward to automate with a computer-algebra system and flexible enough to allow the forcing f to depend on the small parameter

Keywords

amplitude equation averaging multiple scales oscillations renormalization singular perturbations 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    N. Minorsky, Introduction to Non-Linear Mechanics: Topological Methods, Analytical Methods, Non-Linear Resonance, Relaxation Oscillations. Ann Arbor: J. W. Edwards (1947) xiv + 464 pp.Google Scholar
  2. 2.
    Hoppensteadt F.C. (2000). Analysis and Simulation of Chaotic Systems. Volume 94 of Applied Mathematical Sciences 2nd edition. Springer-Verlag, New York xx + 315 pp.Google Scholar
  3. 3.
    Smith D.R. (1985). Singular-Perturbation Theory: An Introduction with Applications. Cambridge University Press, Cambridge, xvi + 500 pp.MATHGoogle Scholar
  4. 4.
    Murdock J. (1991). Perturbations: Theory and Methods. John Wiley & Sons Inc, New York, xviii + 509 ppMATHGoogle Scholar
  5. 5.
    Sanders J.A., Verhulst F. (1985). Averaging Methods in Nonlinear Dynamical Systems Volume 59 of Applied Mathematical Sciences. Springer-Verlag, New York, x + 247 pp.Google Scholar
  6. 6.
    O’Malley R.E. Jr. (1991). Singular Perturbation Methods for Ordinary Differential Equations Volume 89 of Applied Mathematical Sciences. Springer-Verlag, New York, viii + 225 pp.Google Scholar
  7. 7.
    Wasow W. (1965). Asymptotic Expansions for Ordinary Differential Equations. Wiley Interscience, New York, ix + 362 pp.MATHGoogle Scholar
  8. 8.
    Hardy G.H. (1949). Divergent Series. Clarendon Press, Oxford, xvi + 396pp.MATHGoogle Scholar
  9. 9.
    Copson E.T. (1965). Asymptotic Expansions Volume 55 of Cambridge Tracts in Mathematics and Mathematical Physics. Cambridge University Press, New York, vii + 120 ppGoogle Scholar
  10. 10.
    Ramis J.-P. (1993). Séries Divergentes et Théories Asymptotiques Volume 121 of Panoramas et Synthèses: Suppl. au bulletin da la SMF. Société Mathématique de France, Paris, 74 pp.Google Scholar
  11. 11.
    Lagrange J.-L. Méchanique Analitique. Paris: Desaint (1788). English translation: Analytical Mechanics. Dordrecht (Neth.): Kluwer Academic Publishers (1997) xli + 592 ppGoogle Scholar
  12. 12.
    Poincaré H. (1993). New Methods of Celestial Mechanics, Vols I–III Volume 13 of History of Modern Physics and Astronomy. Am. Inst. Physics, New York, xxiv + 1078 pp.Google Scholar
  13. 13.
    Lindstedt A. (1882). Über die Integration einer für die Storungstheorie wichigen Differentialgleichungen. Astron. Nachr. 103:211–220CrossRefADSGoogle Scholar
  14. 14.
    Schmidt H. (1937). Beiträge zu einer Theorie der allgemeinen asymptotischen Darstellungen. Math. Ann. 113:629–656CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    Olver F.W.J. (1974). Asymptotics and Special Functions. Academic Press, New York, xvi + 572 pp.Google Scholar
  16. 16.
    Chen L.-Y., Goldenfeld N., Oono Y. (1996). Renormalization group and singular perturbations: Multiple scales, boundary layers, and reductive perturbation theory. Phys. Rev. E 54:376–394CrossRefADSGoogle Scholar
  17. 17.
    O’Malley R.E. Jr., Williams D.B. Deriving amplitude equations for weakly-nonlinear oscillators and their generalizations. J. Comput. Appl. Math. (2005) In PressGoogle Scholar
  18. 18.
    Kevorkian J., Cole J.D. (1996). Multiple Scale and Singular Perturbation Methods Volume 114 of Applied Mathematical Sciences. Springer-Verlag, New York, viii + 632 pp.Google Scholar
  19. 19.
    Bluman G., Cook L.P., Flaherty J., Kevorkian J., Malmuth N., O’Malley R.E. Jr., Schwendeman D.W., Tulin M., Julian D. (2000). Cole (1925–1999). Notices Amer. Math. Soc. 47:466–473MathSciNetMATHGoogle Scholar
  20. 20.
    Kuzmak G.E. (1959). Asymptotic solutions of nonlinear second order differential equations with variable coefficients. J. Appl. Math. Mech. 23:730–744CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Morrison J.A. (1966). Comparison of the modified method of averaging and the two variable expansion procedure. SIAM Rev. 8 :66–85CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Mudavanhu B. A New Renormalization Method for the Asymptotic Solution of Multiple-Scale Singular Perturbation Problems. PhD thesis, University of Washington (2002) v + 114 ppGoogle Scholar
  23. 23.
    Murdock J., Wang L.-C. (1996). Validity of the multiple scale method for very long intervals. Z. Angew. Math. Phys. 47: 760–789CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    van der Pol B. (1926). On relaxation oscillations. Phil. Mag. 2:978–992Google Scholar
  25. 25.
    Cesari L. (1963). Asymptotic Behavior and Stability Problems in Ordinary Differential Equations. Academic Press, New York, viii + 271 pp.MATHGoogle Scholar
  26. 26.
    Grimshaw R. (1990). Nonlinear Ordinary Differential Equations. Volume 2 of Applied Mathematics and Engineering Science Texts. Blackwell Scientific Publications Ltd, OxfordGoogle Scholar
  27. 27.
    Woodruff S.L. (1993). The use of an invariance condition in the solution of multiple-scale singular perturbation problems: Ordinary differential equations. Stud. Appl. Math. 90:225–248MATHMathSciNetGoogle Scholar
  28. 28.
    Woodruff S.L. (1995). A uniformly valid asymptotic solution to a matrix system of ordinary differential equations and a proof of its validity. Stud. Appl. Math. 94:393–413MATHMathSciNetGoogle Scholar
  29. 29.
    Moise I., Ziane M. (2001). Renormalization group method. Applications to partial differential equations. J. Dynam. Diff. Equations 13:275–321CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Arnold V.I., Kozlov V.V., Neishtadt A.I. (1997). Mathematical Aspects of Classical and Celestial Mechanics. Springer-Verlag, Berlin, xiv + 291 pp. Translated from the 1985 Russian original by A. IacobMATHGoogle Scholar
  31. 31.
    Grebenikov E.A., Mitropolsky Y.A., Ryabov Y.A. (2004). Asymptotic Methods in Resonance Analytical Dynamics Volume 21 of Stability and Control: Theory, Methods and Applications. Chapman & Hall/CRC, Boca Raton, xx + 255 pp.Google Scholar
  32. 32.
    Reid W.T. (1971). Ordinary Differential Equations. Wiley-Interscience, New York, xv + 553 pp.MATHGoogle Scholar
  33. 33.
    Bogoliubov N.N., Mitropolsky Y.A. (1961). Asymptotic Methods in the Theory of Non-Linear Oscillations. International Monographs on Advanced Mathematics and Physics. Delhi, Gordon and Breach Science Publishers, New York, x + 537 pp.Google Scholar
  34. 34.
    Verhulst F. (2005). Methods and Applications of Singular Perturbations. Volume 50 of Texts in Applied Mathematics. Springer, New York, xv+ 324 pp.Google Scholar
  35. 35.
    Kolesov A.Y., Mishchenko E.F., Rozov N.K. (1999). Solution of singularly perturbed boundary value problems by the “duck hunting” [chasse aux canards] method. Proc. Steklov Inst. Math. 224:169–188MathSciNetGoogle Scholar
  36. 36.
    Perko L.M. (1969). Higher order averaging and related methods for perturbed periodic and quasi-periodic systems. SIAM J. Appl. Math. 17:698–724CrossRefMATHMathSciNetGoogle Scholar
  37. 37.
    Murdock J. (1988). Qualitative theory of nonlinear resonance by averaging and dynamical systems methods. In: Kirch-graber U., Walther W.O (eds). Dynamics Reported, Volume 1. John Wiley & Sons, New York, pp. 91–172Google Scholar
  38. 38.
    Bakhvalov N.S., Panasenko G.P., Shtaras A.L. (1999). The averaging method for partial differential equations (homogenization) and its applications. In: Egorov Y.V., Shubin M.A (eds). Partial Differential Equations V Volume 34 of Encyclopaedia Math. Sci. Springer-Verlag, Berlin, pp. 211–247Google Scholar
  39. 39.
    Ramis J.-P., Schäfke R. (1996). Gevrey separation of fast and slow variables. Nonlinearity 9:353–384CrossRefMATHADSMathSciNetGoogle Scholar
  40. 40.
    R.E.L. DeVille, A. Harkin, K. Josic and T.J. Kaper, Applications of asymptotic normal form theory and its connections with the renormalization group method. Preprint (2003)Google Scholar
  41. 41.
    Ei S.-I., Fujii K., Kunihiro T. (2000). Renormalization-group method for reduction of evolution equations; invariant manifolds and envelopes. Ann. Physics 280:236–298CrossRefMATHADSMathSciNetGoogle Scholar
  42. 42.
    Oono Y. (2000). Renormalization and asymptotics. Int. J. Modern Physics B 14:1327–1361ADSMathSciNetGoogle Scholar
  43. 43.
    Nozaki K., Oono Y. (2001). Renormalization-group theoretical reduction. Phys. Rev. E 63:046101CrossRefADSGoogle Scholar
  44. 44.
    Promislow K. (2002). A renormalization method for modulational stability of quasi-steady patterns in dispersive systems. SIAM J. Math. Anal. 33:1455–1482CrossRefMATHMathSciNetGoogle Scholar
  45. 45.
    Wirosoetisno D., Shepherd T.G., Temam R.M. (2002). Free gravity waves and balanced dynamics. J. Atmos. Sci. 59: 3382–3398CrossRefADSMathSciNetGoogle Scholar
  46. 46.
    van der Corput J.G. (1959/1960). Introduction to the neutrix calculus. J. Analyse Math. 7:281–399MathSciNetGoogle Scholar
  47. 47.
    Peskin M.E., Schroeder D.V. (1995). Introduction to Quantum Field Theory. Addison-Wesley, Reading, xxii + 842 pp.Google Scholar
  48. 48.
    Whitham G.B. (1974). Linear and Nonlinear Waves. John Wiley & Sons, New York, xvi + 636 ppMATHGoogle Scholar
  49. 49.
    Mudavanhu B., O’Malley R.E. Jr. (2001). A renormalization group method for nonlinear oscillators. Stud. Appl. Math. 107:63–79CrossRefMATHMathSciNetGoogle Scholar
  50. 50.
    Tsien H.S. (1956). The Poincaré-Lighthill-Kuo method. In: Advances in Applied Mechanics Vol IV. Academic Press, New York, pp. 281–349Google Scholar
  51. 51.
    Coullet P.H., Spiegel E.A. (1983). Amplitude equations for systems with competing instabilities. SIAM J. Appl. Math. 43:776–821CrossRefMATHMathSciNetGoogle Scholar
  52. 52.
    Eckhaus W. (1992). On modulation equations of the Ginzburg-Landau type. In: O’Malley R.E. Jr (eds). ICIAM 91 (Washington, DC, 1991). SIAM, Philadelphia, pp. 83–98Google Scholar
  53. 53.
    Budd C.J., Hunt G.W., Kuske R. (2001). Asymptotics of cellular buckling close to the Maxwell load. R. Soc. London Proc. Ser. A Math. Phys. Eng. Sci. 457:2935–2964CrossRefMATHADSMathSciNetGoogle Scholar
  54. 54.
    Pedlosky J. (1987). Geophysical Fluid Dynamics. Springer-Verlag, New York, xiv + 710 pp.MATHGoogle Scholar
  55. 55.
    Fujimura K. (1989). The equivalence between two perturbation methods in weakly nonlinear stability theory for parallel shear flows. Proc. R. Soc. London A 424:373–392MATHADSMathSciNetCrossRefGoogle Scholar
  56. 56.
    Fujimura K. (1991). Methods of centre manifold and multiple scales in the theory of weakly nonlinear stability for fluid motions. Proc. R. Soc. London A 434:719–733MATHADSMathSciNetGoogle Scholar
  57. 57.
    Fujimura K. (1997). Centre manifold reduction and the Stuart-Landau equation for fluid motions. Proc. R. Soc. London A 453:181–203MATHADSMathSciNetGoogle Scholar
  58. 58.
    Nayfeh A.H. (1993). Method of Normal Forms. John Wiley & Sons Inc, New York, xii + 218 pp.Google Scholar
  59. 59.
    Cox S.M., Roberts A.J. (1995). Initial conditions for models of dynamical systems. Phys. D 85:126–141CrossRefMATHMathSciNetGoogle Scholar
  60. 60.
    Murdock J. Normal Forms and Unfoldings for Local Dynamical Systems. Springer Monographs in Mathematics, New York: Springer-Verlag (2003) xx + 494 pp.Google Scholar
  61. 61.
    Volosov V.M. (1962). Averaging in systems of ordinary differential equations. Russian Math. Surveys 17:3–126CrossRefMathSciNetGoogle Scholar
  62. 62.
    Sibuya Y. (2000). The Gevrey asymptotics in the case of singular perturbations. J. Diff. Equations 165:255–314CrossRefMATHMathSciNetGoogle Scholar
  63. 63.
    Temam R.M., Wirosoetisno D. (2003). On the solutions of the renormalized equations at all orders. Adv. Diff. Equations 8:1005–1024MATHMathSciNetGoogle Scholar
  64. 64.
    Andersen C.M., Geer J.F. (1982). Power series expansions for the frequency and period of the limit cycle of the van der Pol equation. SIAM J. Appl. Math. 42:678–693CrossRefMATHMathSciNetGoogle Scholar
  65. 65.
    Rubenfeld L.A. (1978). On a derivative-expansion technique and some comments on multiple scaling in the asymptotic approximation of solutions of certain differential equations. SIAM Rev. 20:79–105CrossRefMATHMathSciNetGoogle Scholar
  66. 66.
    Kuske R. (2003). Multi-scale analysis of noise-sensitivity near a bifurcation. In: Namachchivaya N.S., Lin Y.K (eds). IUTAM Symposium on Nonlinear Stochastic Dynamics Volume 110 Solid Mech. Appl., Kluwer Academic Publishers, Dordrecht, pp. 147–156Google Scholar
  67. 67.
    Nipp K. (1988). An algorithmic approach for solving singularly perturbed initial value problems. In: Kirchgraber U., Walther W.O (eds). Dynamics Reported, Volume 1. John Wiley & Sons, New York, pp. 173–263Google Scholar
  68. 68.
    Kreiss H.-O., Lorenz J. (1994). On the existence of slow manifolds for problems with different timescales. Philos. Trans. R. Soc. London A 346:159–171MATHCrossRefADSMathSciNetGoogle Scholar
  69. 69.
    Strygin V.V., Sobolev V.A. (1988). Separation of Motions by the Method of Integral Manifolds (in Russian). “Nauka”, Moscow 256 pp.Google Scholar
  70. 70.
    Gear C.W., Kaper T.J., Kevrekidis I.G., Zagaris A. Projecting to a slow manifold: Singularly perturbed systems and legacy codes. SIAM J. Appl. Dyn. Syst. (2005) in Press.Google Scholar
  71. 71.
    King A.C., Billingham J., Otto S.R. (2003). Differential Equations: Linear, Nonlinear, Ordinary, Partial. Cambridge University Press, Cambridge, xii + 541 pp.MATHGoogle Scholar
  72. 72.
    Haberman R. (2004). Applied Partial Differential Equations with Fourier Series and Boundary Value Problems, 4th edition. Pearson Prentice Hall, Upper Saddle River, xviii + 769 pp.Google Scholar
  73. 73.
    Hairer E., Lubich C., Wanner G. (2002). Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations Volume 31 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, xiv + 515 pp.Google Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  • Blessing Mudavanhu
    • 1
  • Robert E. O’MalleyJr
    • 2
  • David B. Williams
    • 3
  1. 1.Credit Risk AnalyticsMerrill Lynch & CoNew YorkUSA
  2. 2.Department of Applied MathematicsUniversity of WashingtonSeattleUSA
  3. 3.Department of MathematicsClayton State UniversityMorrowUSA

Personalised recommendations