Free Oscillations

  • Frank C. Hoppensteadt
Part of the Applied Mathematical Sciences book series (AMS, volume 94)


A distinction is usually made between systems that are isolated, known as free systems, and those that interact with the outside world, known as forced systems. Often we reduce forced systems to (apparently) free ones by looking at the system stroboscopically or by introducing extra variables to describe external influences. Often we reduce free problems to ones that appear to be forced. For example, systems in which energy is conserved can have dissipative components within them, which can be uncovered by finding a timelike variable among the variables of the system and using it to reduce the problem to a dissipative one of lower order.


Periodic Solution Hamiltonian System Phase Portrait Rotation Number Phase Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 2.1.
    E.A. Coddington, N. Levinson, The Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955.Google Scholar
  2. 2.2.
    J.K. Hale, Ordinary Differential Equations, Wiley-Interscience, New York, 1971.Google Scholar
  3. 2.3.
    M. Hirsch, S. Smale, Differential Equations, Dynamical Systems and Linear Algebra, Academic Press, New York, 1974.MATHGoogle Scholar
  4. 2.4.
    A. Denjoy, Sur les courbes definies par les equations differentielles a la surface du tor, J. Math. Pures Appl. 9 (1932): 333–375.Google Scholar
  5. 2.5.
    F.C. Hoppensteadt, Introduction to the Mathematics of Neurons, Cambridge University Press, New York, 1986.MATHGoogle Scholar
  6. 2.6.
    P. Horowitz, W. Hill, The Art of Electronics, 2nd ed., Cambridge University Press, New York, 1989.Google Scholar
  7. 2.7.
    W. Chester, The forced oscillations of a simple pendulum, J. Inst. Maths. Appl., 15 (1975): 298–306.MathSciNetCrossRefGoogle Scholar
  8. 2.8.
    M. Levi, F.C. Hoppensteadt, W.L. Miranker, Dynamics of the Josephson junction, Quart. Appl. Math. (July 1978): 167–198.Google Scholar
  9. 2.9.
    R. Courant, D. Hilbert, Methods of Mathematical Physics, Vol I, WileyInterscience, New York, 1968.Google Scholar
  10. 2.10.
    J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1983.MATHCrossRefGoogle Scholar
  11. 2.11.
    F.C. Hoppensteadt, Mathematical Methods of Population Biology, Cambridge University Press, New York, 1982.MATHCrossRefGoogle Scholar
  12. 2.12.
    A.T. Fomenko, Integrable systems on Lie algebras and symmetric spaces, Gordon and Breach, New York, 1988.Google Scholar
  13. 2.13.
    M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1972.MATHGoogle Scholar
  14. 2.14.
    A.I. Khinchin, An Introduction to Information Theory, Dover, New York, 1965.Google Scholar
  15. 2.15.
    C.L. Siegel, J. Moser, Lectures in Celestial Mechanics, Springer-Verlag, New York, 1971.CrossRefGoogle Scholar
  16. 2.16.
    P.R. Garabedian, Partial Differential Equations, Wiley, New York, 1964.MATHGoogle Scholar
  17. 2.17
    Preconditioned Conjugate Gradient Methods,Springer-Verlag, New York, 1990.Google Scholar
  18. 2.18.
    R. Thom, Structural Stability and Morphogenesis: An Outline of a General Theory of Models, W.A. Benjamin, Reading, Mass., 1975.Google Scholar
  19. 2.19.
    A.N. Sarkovski, Ukr. Math. Zh. 16(1964): 61–71. See also P. Stefan, A theorem of Sarkovskii on the existence of periodic orbits of continuous endomorphisms of the real line, Comm. Math. Phys. 54 (1977): 237–248.MathSciNetCrossRefGoogle Scholar
  20. 2.20.
    T.Y. Li, J.A. Yorke, Period three implies chaos, Amer. Math. Montly, 82 (1975): 985–992.MathSciNetMATHCrossRefGoogle Scholar
  21. 2.21.
    S. Ulam, A Collection of Mathematical Problems, Wiley-Interscience, New York, 1960.MATHGoogle Scholar
  22. 2.22.
    M.E. Munroe, Introduction to Measure and Integration. Addison-Wesley, Cambridge, Mass., 1953.MATHGoogle Scholar
  23. 2.23.
    F.C. Hoppensteadt, J.M. Hyman, Periodic solutions of a logistic difference equation, SIAM J. Appl. Math. 58 (1977): 73–81.MathSciNetCrossRefGoogle Scholar
  24. 2.24.
    K. Krohn, J.L. Rhodes, Algebraic Theory of Machines (M.A. Arbib, ed.), Academic Press, New York, 1968.Google Scholar
  25. 2.25.
    N.N. Minorsky, Nonlinear Oscillations, Van Nostrand, Princeton, 1962.MATHGoogle Scholar
  26. 2.26
    J. Moser, On the theory of quasi-periodic motions, SIAM Rev. 8(1966): 145171.Google Scholar
  27. 2.27.
    G.D. Birkhoff, Dynamical Systems, Vol. IX., American Mathematics Society, Providence, RI, 1966.MATHGoogle Scholar
  28. 2.28.
    J. Hadamard, Sur l’iteration et les solutions asymptotiques des equations differentielles, Bull. Soc. Math. France 29 (1901): 224–228.MATHGoogle Scholar
  29. 2.29.
    V.I. Arnol’d, A. Avez, Ergodic problems in classical mechanics, W.A. Benjamin, New York, 1968.Google Scholar
  30. 2.30.
    R. Bellman, K. Cooke, Differential-Difference Equations, Academic Press, New York, 1963.MATHGoogle Scholar
  31. 2.31.
    P.E. Sobolevski, Equations of parabolic type in Banach space, AMS Translation, 49 (1966): 1–62.Google Scholar
  32. 2.32.
    H.T. Banks, F. Kappel, Spline approximations for functional differential equations, J. Differential Eqns., 34 (1979): 496–522.MathSciNetMATHCrossRefGoogle Scholar
  33. 2.33
    R.D. Nussbaum, H.O. Peitgen, Special and spurious solutions of dx/dtaF(x(t — 1)), preprint.Google Scholar

Copyright information

© Springer Science+Business Media New York 1993

Authors and Affiliations

  • Frank C. Hoppensteadt
    • 1
  1. 1.College of Natural SciencesMichigan State UniversityEast LansingUSA

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