Abstract
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.
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References
E.A. Coddington, N. Levinson, The Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955.
J.K. Hale, Ordinary Differential Equations, Wiley-Interscience, New York, 1971.
M. Hirsch, S. Smale, Differential Equations, Dynamical Systems and Linear Algebra, Academic Press, New York, 1974.
A. Denjoy, Sur les courbes definies par les equations differentielles a la surface du tor, J. Math. Pures Appl. 9 (1932): 333–375.
F.C. Hoppensteadt, Introduction to the Mathematics of Neurons, Cambridge University Press, New York, 1986.
P. Horowitz, W. Hill, The Art of Electronics, 2nd ed., Cambridge University Press, New York, 1989.
W. Chester, The forced oscillations of a simple pendulum, J. Inst. Maths. Appl., 15 (1975): 298–306.
M. Levi, F.C. Hoppensteadt, W.L. Miranker, Dynamics of the Josephson junction, Quart. Appl. Math. (July 1978): 167–198.
R. Courant, D. Hilbert, Methods of Mathematical Physics, Vol I, WileyInterscience, New York, 1968.
J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1983.
F.C. Hoppensteadt, Mathematical Methods of Population Biology, Cambridge University Press, New York, 1982.
A.T. Fomenko, Integrable systems on Lie algebras and symmetric spaces, Gordon and Breach, New York, 1988.
M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1972.
A.I. Khinchin, An Introduction to Information Theory, Dover, New York, 1965.
C.L. Siegel, J. Moser, Lectures in Celestial Mechanics, Springer-Verlag, New York, 1971.
P.R. Garabedian, Partial Differential Equations, Wiley, New York, 1964.
Preconditioned Conjugate Gradient Methods,Springer-Verlag, New York, 1990.
R. Thom, Structural Stability and Morphogenesis: An Outline of a General Theory of Models, W.A. Benjamin, Reading, Mass., 1975.
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.
T.Y. Li, J.A. Yorke, Period three implies chaos, Amer. Math. Montly, 82 (1975): 985–992.
S. Ulam, A Collection of Mathematical Problems, Wiley-Interscience, New York, 1960.
M.E. Munroe, Introduction to Measure and Integration. Addison-Wesley, Cambridge, Mass., 1953.
F.C. Hoppensteadt, J.M. Hyman, Periodic solutions of a logistic difference equation, SIAM J. Appl. Math. 58 (1977): 73–81.
K. Krohn, J.L. Rhodes, Algebraic Theory of Machines (M.A. Arbib, ed.), Academic Press, New York, 1968.
N.N. Minorsky, Nonlinear Oscillations, Van Nostrand, Princeton, 1962.
J. Moser, On the theory of quasi-periodic motions, SIAM Rev. 8(1966): 145171.
G.D. Birkhoff, Dynamical Systems, Vol. IX., American Mathematics Society, Providence, RI, 1966.
J. Hadamard, Sur l’iteration et les solutions asymptotiques des equations differentielles, Bull. Soc. Math. France 29 (1901): 224–228.
V.I. Arnol’d, A. Avez, Ergodic problems in classical mechanics, W.A. Benjamin, New York, 1968.
R. Bellman, K. Cooke, Differential-Difference Equations, Academic Press, New York, 1963.
P.E. Sobolevski, Equations of parabolic type in Banach space, AMS Translation, 49 (1966): 1–62.
H.T. Banks, F. Kappel, Spline approximations for functional differential equations, J. Differential Eqns., 34 (1979): 496–522.
R.D. Nussbaum, H.O. Peitgen, Special and spurious solutions of dx/dt —aF(x(t — 1)), preprint.
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Hoppensteadt, F.C. (1993). Free Oscillations. In: Analysis and Simulation of Chaotic Systems. Applied Mathematical Sciences, vol 94. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-2275-8_2
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DOI: https://doi.org/10.1007/978-1-4757-2275-8_2
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