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

Abstract | References | Similar Articles | Additional Information

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.

**[1]**Jose Miguel Alonso and Rafael Ortega,*Boundedness and global asymptotic stability of a forced oscillator*, Nonlinear Anal.**25**(1995), no. 3, 297–309. MR**1336527**, 10.1016/0362-546X(94)00140-D**[2]**R. W. Brockett,*Variational methods for stability of periodic equations*, Differential Equations and Dynamical Systems (Proc. Internat. Sympos., Mayaguez, P.R., 1965) Academic Press, New York, 1967, pp. 299–308. MR**0218709****[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), no. 2, 721–739. MR**979963**, 10.1090/S0002-9947-1989-0979963-1**[4]**A. C. Lazer and P. J. McKenna,*On the existence of stable periodic solutions of differential equations of Duffing type*, Proc. Amer. Math. Soc.**110**(1990), no. 1, 125–133. MR**1013974**, 10.1090/S0002-9939-1990-1013974-9**[5]**E. B. Lee and L. Markus,*Foundations of optimal control theory*, John Wiley & Sons, Inc., New York-London-Sydney, 1967. MR**0220537****[6]**Wilhelm Magnus and Stanley Winkler,*Hill’s equation*, Dover Publications, Inc., New York, 1979. Corrected reprint of the 1966 edition. MR**559928****[7]**Rafael Ortega,*The first interval of stability of a periodic equation of Duffing type*, Proc. Amer. Math. Soc.**115**(1992), no. 4, 1061–1067. MR**1092925**, 10.1090/S0002-9939-1992-1092925-7**[8]**T. Yoshizawa,*Stability theory and the existence of periodic solutions and almost periodic solutions*, Springer-Verlag, New York-Heidelberg, 1975. Applied Mathematical Sciences, Vol. 14. MR**0466797**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
34C25,
34D05,
70K20,
70K40

Retrieve articles in all journals with MSC: 34C25, 34D05, 70K20, 70K40

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1995-1301005-7

Article copyright:
© Copyright 1995
American Mathematical Society