A small twist theorem and boundedness of solutions for semilinear Duffing equations at resonance

Nonlinear Analysis, Theory, Methods and Applications

Volumn 67, Issue 1, 2007, Pages 200-237

  Ma, Shiwang ^a   Wu, Jianhong ^b  

^a NANKAI UNIVERSITY   (China)

^b YORK UNIVERSITY   (Canada)

Author keywords

Boundedness; Moser's small twist theorem; Resonance; Semilinear Duffing equation

Indexed keywords

ASYMPTOTIC STABILITY; FUNCTIONS; LINEAR EQUATIONS; RESONANCE;

BOUNDED FUNCTION; MOSER'S SMALL TWIST THEOREM; SEMILINEAR DUFFING EQUATION;

THEOREM PROVING;

EID: 34047116444     PISSN: 0362546X     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.na.2006.04.023     Document Type: Article

References (19)

1. Arnold V.I. Mathematical Methods of Classical Mechanics (1989), Springer-Verlag, New York
2. Arnold V.I., Kozlov V.V., and Neishtadt A.V. Mathematical Aspects of Classical and Celestial Mechanics (1997), Springer-Verlag, Berlin
3. Dieckerhoff R., and Zehnder E. Boundedness of solutions via the twist theorem. Ann. Scula. Norm. Sup. Pisa. Cl. Sci. 14 1 (1987) 79-95
4. Herman M. Sur les courbes invariantes per les difféomorphismes de l'anneau I. Astérisque (1983) 103-104
5. Hao D.Y., and Ma S.W. Semilinear Duffing equations crossing resonance points. J. Differential Equations 133 (1997) 98-116
6. Lazer A.C., and Leach D.E. Bounded perturbations of forced harmonic oscillators at resonance. Ann. Mat. Pura. Appl. 82 (1969) 49-68
7. Levi M. Quasiperiodic motions in superquadratic time-periodic potentials. Commun. Math. Phys. 143 (1991) 43-83
8. Littlewood J.E. Unbounded solutions of an equation over(y, ̈) + g (y) = p (t), with p (t) periodic and bounded, and g (y) / y → ∞ as y → ± ∞. J. London Math. Soc. 41 (1966) 497-507
9. Liu B. Boundedness of solutions for semilinear Duffing equations. J. Differential Equations 145 (1998) 119-144
10. Liu B. Boundedness in nonlinear oscillations at resonance. J. Differential Equations 153 (1999) 142-174
11. S.W. Ma, Existence and multiplicity of periodic solutions for semilinear Duffing equation at resonance (in press)
12. Markus I. Lectures in Differentiable Dynamics (1969), American Mathematical Society, Providence, RI
13. Massera J.L. The existence of periodic solutions of differential equations. Duke Math. J. 17 (1950) 457-475
14. Morris G.R. A case of boundedness of Littlewood's problem on oscillatory differential equations. Bull. Austral. Math. Soc. 14 (1976) 71-93
15. Moser J. On invariant curves of area-preserving mappings of annul us. Nachr. Akad. Wiss. Gottingen Math. Phys. KI. II (1962) 1-20
16. Moser J. Stable and Random Motion in Dynamical Systems (1973), Princeton University Press, Princeton, NJ
17. Ortega R. Asymmetric oscillators and twist mappings. J. London Math. Soc. 53 2 (1996) 325-342
18. Ortega R. Boundedness in a piecewise linear oscillator and a variant of the small twist theorem. Proc. London Math. Soc. 79 3 (1999) 381-431
19. Wang Z.H. Multiplicity of periodic solutions of semilinear Duffing's equation at resonance. J. Math. Anal. Appl. 237 (1999) 166-187