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Optimal intervals of stability of a forced oscillator
Author:
Jose Miguel Alonso
Journal:
Proc. Amer. Math. Soc. 123 (1995), 2031-2040
MSC:
Primary 34C25; Secondary 34D05, 70K20, 70K40
MathSciNet review:
1301005
Full-text PDF Free Access
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Abstract: Consider the differential equation of a nonlinear oscillator with linear friction and a T-periodic external force. We find optimal bounds on the derivative of the restoring force and on the period T in order to obtain a unique T-periodic solution that is asymptotically stable.
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Jose
Miguel Alonso and Rafael
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979963 (90a:34011), http://dx.doi.org/10.1090/S0002-9947-1989-0979963-1
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C. Lazer and P.
J. McKenna, On the existence of stable periodic
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Applied Mathematical Sciences, Vol. 14. MR 0466797
(57 #6673)
- [1]
- J. M. Alonso and R. Ortega, Boundedness and global asymptotic stability of a forced oscillator, Nonlinear Anal. TMA (to appear). MR 1336527 (96e:34076)
- [2]
- R. W. Brockett, Variational methods for stability of periodic equations, Differential Equations and Dynamical Systems (J. Hale and J. Lasalle, eds.), Academic Press, New York, 1967, pp. 299-308. MR 0218709 (36:1793)
- [3]
- A. C. Lazer and P. J. McKenna, Existence, uniqueness, and stability of oscillations in differential equations with asymmetric nonlinearities, Trans. Amer. Math. Soc. 315 (1989), 721-739. MR 979963 (90a:34011)
- [4]
- -, On the existence of stable periodic solutions of differential equations of Duffing type, Proc. Amer. Math. Soc. 110 (1990), 125-133. MR 1013974 (90m:34093)
- [5]
- E. Lee and L. Markus, Foundations of optimal control theory, Wiley, New York, 1967. MR 0220537 (36:3596)
- [6]
- W. Magnus and S. Winkler, Hill's equation, Dover, New York, 1979. MR 559928 (80k:34001)
- [7]
- R. Ortega, The first interval of stability of a periodic equation of Duffing type, Proc. Amer. Math. Soc. 115 (1992), 1061-1067. MR 1092925 (92j:34090)
- [8]
- T. Yoshizawa, Stability theory and the existence of periodic solutions and almost periodic solutions, Springer-Verlag, Berlin, 1975. MR 0466797 (57:6673)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002-9939-1995-1301005-7
PII:
S 0002-9939(1995)1301005-7
Article copyright:
© Copyright 1995 American Mathematical Society
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