Abstract
This paper is a self-contained exposition of the theory of averaging for periodic and quasiperiodic systems, with the emphasis being on the author’s research (part of it joint work with Clark Robinson) on qualitative aspects of nonlinear resonance. Many topics in averaging theory are not covered, among them: averaging for systems more general than quasiperiodic; relations between averaging and multiple time-scale methods; Eckhaus’s approach to averaging; combinations of averaging with matching of asymptotic expansions. The principal question which is addressed is: when does averaging (to first or higher order) lead to an accurate qualitative description of the solutions of the original (unaveraged) equation? By qualitative description we mean both locally (existence and stability of certain invariant sets such as periodic orbits, or almost invariant ‘lingering’ and globally (connecting orbits between invariant sets, or claims that in certain large regions all orbits drift in a certain direction).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
E. Akin, untitled manuscript in progress, private communication.
V. I. Arnol’d, Mathematical Methods of Classical Mechanics. Springer-Verlag, NY, 1978.
N. N. Bogoliubov and Y. A. Mitropolskii, Asymptotic Methods in the Theory of Nonlinear Oscillations. Gordon and Breach, NY, 1961.
H. Bohr, Almost Periodic Functions. Chelsea, NY, 1947.
T. J. Burns, On the rotation of Mercury. Celestial Mechanics, 19 (1979), 297–313.
T. J. Burns, On a dissipative model of the spin-orbit resonance of Mercury, unpublished manuscript.
C. Conley, Isolated Invariant Sets and the Morse Index. A.M.S., Providence, 1978.
C. C. Counselman and I. I. Shapiro, Spin-orbit resonance of Mercury. Symposia Mathematica, 3 (1970), 121–69.
A. M. Fink, Almost Periodic Differential Equations. Springer-Verlag, Berlin, 1974.
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer-Verlag, NY, 1983.
J. K. Hale, Ordinary Differential Equations. Wiley, NY, 1969.
J. K. Hale and L. C. Pavlu, Dynamic behavior from asymptotic expansions. Quart. Appl. Math., 41 (1983–84), 161–8.
M. Hall, Jr., The Theory of Groups. Macmillan, NY, 1959.
S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol. I. Wiley, NY, 1963.
N. Krylov and N.N. Bogoliubov, Introduction to Nonlinear Mechanics. Annals of Mathematics Studies, No. 11. Princeton Univ. Press, Princeton, NJ, 1947.
J. T. Montogomery, Existence and stability of periodic motion under higher order averaging. J. Diff. Eq., 64 (1986), 67–78.
J. Moser, Stable and Random Motions in Dynamical Systems. Annals of Mathematics Studies, No. 77. Princeton Univ. Press, Princeton, NJ, 1973.
J. Murdock, Nearly Hamiltonian systems in nonlinear mechanics: averaging and energy methods. Indiana U. Math. J., 25 (1976), 499–523.
J. Murdock, Nested attractors near nonlinear centers. J. Diff. Eq., 25 (1977), 115–29.
J. Murdock, Resonance capture in certain nearly Hamiltonian systems. J. Diff Eq., 17 (1975), 361–74.
J. Murdock, Some asymptotic estimates for higher order averaging and a comparison with iterated averaging. SIAM J. Math. Anal., 14 (1983), 421–4.
J. Murdock, Some mathematical aspects of spin-orbit resonance. Celestial Mechanics, 18 (1978), 237–53.
J. Murdock and C. Robinson, A note on the asymptotic expansion of eigen-values. SIAM J. Math. Anal., 11 (1980), 458–9.
J. Murdock and C. Robinson, Qualitative dynamics from asymptotic expansions: local theory. J. Diff. Eq., 36 (1980), 425–41.
A. Nayfeh, Perturbation Methods. Wiley, NY, 1973.
Z. Nitecki, Differentiable Dynamics. MIT Press, Mass., 1971.
J. Palis Jr. and W. deMelo, Geometric Theory of Dynamical Systems: An Introduction. Springer-Verlag, NY, 1982.
L. M. Perko, Higher order averaging and related methods for perturbed periodic and quasiperiodic systems. SIAM J. Appl. Math., 17 (1968), 698–724.
C. Robinson, Stability of periodic solutions from asymptotic expansions, in Classical Mechanics and Dynamical Systems (Medford, Mass., 1079) Lecture Notes in Pure and Appl. Math., 70, Dekker, NY, 1981, 173–85.
C. Robinson, Structural stability on manifolds with boundary. J. Diff. Eq., 37 (1980), 1–11.
C. Robinson and J. Murdock, Some mathematical aspects of spin–orbit resonance. II. Celestial Mechanics, 24 (1981), 83–107.
J. A. Sanders and F. Verhulst, Averaging Methods in Nonlinear Dynamical Systems. Springer-Verlag, NY, 1985.
S. Smale, On gradient dynamical systems. Ann. Math., 74 (1961), 199–206.
S. Sternberg, Celestial Mechanics, Part I. Benjamin, NY, 1969.
J. J. Stoker, Nonlinear Vibrations in Mechanical and Electrical Systems. Wiley, NY, 1950.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1988 John Wiley & Sons and B. G. Teubner
About this chapter
Cite this chapter
Murdock, J. (1988). Qualitative Theory of Nonlinear Resonance by Averaging and Dynamical Systems Methods. In: Kirchgraber, U., Walther, HO. (eds) Dynamics Reported. Dynamics Reported, vol 1. Vieweg+Teubner Verlag, Wiesbaden. https://doi.org/10.1007/978-3-322-96656-8_3
Download citation
DOI: https://doi.org/10.1007/978-3-322-96656-8_3
Publisher Name: Vieweg+Teubner Verlag, Wiesbaden
Print ISBN: 978-3-519-02150-6
Online ISBN: 978-3-322-96656-8
eBook Packages: Springer Book Archive