×

The twist coefficient of periodic solutions of a time-dependent Newton’s equation. (English) Zbl 0761.34036

A \(T\)-periodic solution \(\varphi\) of a \(T\)-periodic equation \(x''+f(t,x)=0\) is considered. It is supposed that \(\varphi\) is generic non zero twist type up to the 4th order terms. If \(f(t,x)=a(t)x+b(t)x^ 2+c(t)x^ 3+\dots\) and \(\varphi(t)\equiv 0\) then conditions for the coefficients (boundedness and sign type) are formulated, which guarantee that the trivial solution is of twist type and as a consequence Lyapunov stable. The results are illustrated by examples including the pendulum of variable length.

MSC:

34C25 Periodic solutions to ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Ambrosetti, A., and Prodi, G. (1972). On the inversion of some differentiable mappings with singularities between Banach spaces.Ann. Mat. Pura Appl. 93, 231-246. · Zbl 0288.35020
[2] Arnold, V. I., and Avez, A. (1968).Ergodic Problems of Classical Mechanics, Benjamin, New York. · Zbl 0167.22901
[3] Arrowsmith, D. K., and Place, C. M. (1990).An Introduction to Dynamical Systems, Cambridge University Press, Cambridge. · Zbl 0702.58002
[4] Cesari, L. (1971).Asymptotic Behavior and Stability Problems in Ordinary Differential Equations, Springer Verlag, Berlin. · Zbl 0215.13802
[5] Ekeland, I. (1990).Convexity Methods in Hamiltonian Mechanics, Springer Verlag, Berlin. · Zbl 0707.70003
[6] Fabry, C., Mawhin, J., and Nkashama, N. (1986). A multiplicity result for periodic solutions of forced nonlinear second order ordinary differential equations.Bull. London Math. Soc. 18, 173-180. · Zbl 0586.34038
[7] Fonda, A., Ramos, M., and Willem, M. (1989). Subharmonic solutions for second order differential equations. Preprint. · Zbl 0803.34029
[8] Ioos, G. (1979).Bifurcation of Maps and Applications, North-Holland, Amsterdam.
[9] Krasnoselskii, M. (1968).Translations Along Trajectories of Differential Equations, American Mathematical Society, Providence, RI.
[10] Magnus, W., and Winkler, S. (1979).Hill’s Equation, Dover, New York. · Zbl 0158.09604
[11] Markus, L., and Meyer, K. R. (1980). Periodic orbits and solenoids in generic hamiltonian dynamical systems.Am. J. Math. 102, 25-92. · Zbl 0438.58013
[12] Moeckel, R. (1990). Generic bifurcations of the twist coefficient.Ergod. Theor. Dynam. Syst. 10, 185-195. · Zbl 0734.58021
[13] Ortega, R. (1989). Stability and index of periodic solutions of an equation of Duffing type.Boll. Un. Mat. Ital. 3-B, 533-546. · Zbl 0686.34052
[14] Ortega, R. (1990). Stability of a periodic problem of Ambrosetti-Prodi type.Diff. Integral Eqs. 3, 275-284. · Zbl 0724.34059
[15] Siegel, C. L., and Moser, J. (1971).Lectures on Celestial Mechanics, Springer Verlag, Berlin. · Zbl 0312.70017
[16] Wan, Y. (1978). Computation of the stability condition for the Hopf bifurcation of diffeomorphisms on ?2.Siam J. Appl. Math. 34, 167-175. · Zbl 0389.58008
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.