×

The method of lower and upper solutions and the stability of periodic oscillations. (English) Zbl 1043.34044

This paper is devoted to the method of lower and upper solutions to solve the nonlinear periodic problem (1) \(\ddot x+g(t,x)=0\), \(x(0)= x(T)\), \(\dot x(0)= \dot x(T)\), where \(g\in \mathbb{C}^{0,4} (\mathbb{R}/T\mathbb{Z}\times \mathbb{R})\). A function \(\alpha\in \mathbb{C}^2(\mathbb{R}/T\mathbb{Z})\) is said to be a lower solution of (1), if \(\ddot\alpha(t)+ g(t,\alpha(t))\geq 0\) \(\forall\,t\in \mathbb{R}\). An upper solution \(\beta\) is defined in a similar way by reversing the above inequality.
C. de Coster and P. Habets showed that if \(\alpha(t)\leq \beta(t)\) \(\forall\,t\in\mathbb{R}\), then (1) has a solution laying between \(\alpha\) and \(\beta\). E. N. Dancer and R. Ortega [J. Dyn. Differ. Equ. 6, 631–637 (1994; Zbl 0811.34018)] have proved that, when \(\alpha\) and \(\beta\) are strict lower and upper solutions and the region \(\alpha\leq \beta\) contains a unique solution of (1), then this solution is unstable.
The main result of this paper gives sufficient conditions for the existence of a unique solution \(\varphi\in [\beta,\alpha]\) that is stable. Significant examples are considered.

MSC:

34C25 Periodic solutions to ordinary differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion

Citations:

Zbl 0811.34018
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Alonso, J.; Ortega, R., Boundedness and global asymptotic stability of a forced oscillator, Nonlinear anal. theory methods appl., 25, 297-309, (1995) · Zbl 0826.34043
[2] Arnold, V.I.; Avez, A., Ergodic problems of classical mechanics, (1968), Benjamin New York · Zbl 0167.22901
[3] H.W. Broer, G. Vegter, Bifurcational aspects of parametric resonance, in: Dynamical Reported, New Series, Vol. 1, Springer, Berlin, 1992, pp. 1-51. · Zbl 0893.58043
[4] Dancer, E.N.; Ortega, R., The index of Lyapunov stable fixed points in two dimensions, J. dyn. differential equations, 6, 631-637, (1994) · Zbl 0811.34018
[5] De Coster, C.; Habets, P., Upper and lower solutions in the theory of ODE boundary value problems: classical and recent results, () · Zbl 0889.34018
[6] Herman, M., Sur LES courbes invariantes par LES difféomorphismes de l’anneu I, Asterisque, 103-104, (1983)
[7] Liu, B., The stability of the equilibrium of a conservative system, J. math. anal. appl., 202, 133-149, (1996) · Zbl 0873.34042
[8] Magnus, W.; Winkler, S., Hill’s equation, (1979), Dover New York · Zbl 0158.09604
[9] Mather, J.N., Existence of quasi-periodic orbits for twist homeomorphisms of the annulus, Topology, 21, 457-467, (1982) · Zbl 0506.58032
[10] Moser, J., On invariant curves of area preserving mappings of an annulus, Nachr. akad. wiss. Göttingen math. phys. kl., II, 1-20, (1962) · Zbl 0107.29301
[11] Moser, J., An example of a schroedinger operator with almost periodic potential and nowhere dense spectrum, Comm. math. helv., 56, 198-224, (1986)
[12] Núñez, D.; Ortega, R., Parabolics fixed points and stability criteria for nonlinear Hill’s equation, Z. angew. math. phys., 51, 890-911, (2000) · Zbl 0973.34046
[13] Ortega, R., Topological degree and stability of periodic solutions for certain differential equations, J. London math. soc., 42, 505-516, (1990) · Zbl 0677.34042
[14] Ortega, R., The twist coefficient of periodic solutions of a time-dependent Newton’s equations, J. dyn. differential equations, 4, 651-665, (1992) · Zbl 0761.34036
[15] Ortega, R., The stability of the equilibrium of a non linear Hill’s equation, SIAM J. math. anal., 25, 1393-1401, (1994) · Zbl 0807.34065
[16] Ortega, R., Periodic solutions of a Newtonian equation: stability by the third approximation, J. differential equations, 128, 491-518, (1996) · Zbl 0855.34058
[17] Pliss, V., Nonlocal problems of the theory of oscillations, (1966), Academic Press New York · Zbl 0151.12104
[18] Siegel, C.L.; Moser, J.K., Lectures on celestial mechanics, (1971), Springer-Verlag New York, Berlin · Zbl 0312.70017
[19] Starzinskii, V.M., Surveys of works on conditions of stability of the trivial solution of a system of linear differential equations with periodic coefficients, Amer. math. soc. transl. ser. 2, Vol. 1, (1955), Dordrecht Providence, RI
[20] P. Torres, Existence and Uniqueness of Elliptic Periodic Solutions of The Brillouin Electron Beam Focusing System, Math. Met. Appl. Sci. 23 (2000)1139-1143, accepted for publication. · Zbl 0966.34038
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.