Hostname: page-component-76fb5796d-22dnz Total loading time: 0 Render date: 2024-04-26T05:45:25.093Z Has data issue: false hasContentIssue false
  • English
  • Français

Monotone twist maps and periodic solutions of systems of Duffing type

Published online by Cambridge University Press:  19 June 2014

ALBERTO BOSCAGGIN  and
RAFAEL ORTEGA
ALBERTO BOSCAGGIN
Affiliation:
Dipartimento di Matematica, Università di Torino, Via Carlo Alberto 10, I-10123 Torino, Italy. e-mail: alberto.boscaggin@unito.it
RAFAEL ORTEGA
Affiliation:
Departamento de Matemática Aplicada, Universidad de Granada, E-18071 Granada, Spain. e-mail: rortega@ugr.es
Get access Rights & Permissions [Opens in a new window]

Abstract

The theory of twist maps is applied to prove the existence of many harmonic and sub-harmonic solutions for certain Newtonian systems of differential equations. The method of proof leads to very precise information on the oscillatory properties of these solutions.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 157 , Issue 2 , September 2014 , pp. 279 - 296
Copyright
Copyright © Cambridge Philosophical Society 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Arnold, V. I.Mathematical Methods of Classical Mechanics (Springer-Verlag, New York-Heidelberg, 1978).Google Scholar
[2]Bahri, A. and Berestycki, A.Existence of forced oscillations for some nonlinear differential equations. Comm. Pure Appl. Math. 37 (1984), 403–442.Google Scholar
[3]Ding, T. and Zanolin, F.Periodic solutions of Duffing's equations with superquadratic potential. J. Differential Equations 97 (1992), 328–378.Google Scholar
[4]Jin, H., Liu, B. and Wang, Y.The existence of quasiperiodic solutions for coupled Duffing-type equations. J. Math. Anal. Appl. 374 (2011), 429–441.Google Scholar
[5]Kunze, M. and Ortega, R.Long-time stability estimates for the non-periodic Littlewood boundedness problem. Proc. London Math. Soc. 107 (2013), 39–75.Google Scholar
[6]Kunze, M. and Ortega, R.Twist mappings with non-periodic angles. Stability and bifurcation theory for non-autonomous differential equations. Lecture Notes in Math. 206 (Springer, 2013), 265–300.Google Scholar
[7]Lefschetz, S.Differential Equations: Geometric Theory (Dover Publications, Inc., New York, 1977).Google Scholar
[8]Lusternik, L. A. and Schnirelmann, L. G.Méthodes Topologiques dans les Problèmes Variationnels (Hermann, Paris, 1934).Google Scholar
[9]Morris, G. R.An infinite class of periodic solutions of x″ + 2x 3 = p(t). Proc. Camb. Phil. Soc. 61 (1965), 157–164.Google Scholar
[10]Moser, J. and Zehnder, E. J.Notes on Dynamical Systems. Courant Lecture Notes in Math. 12 (New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2005).Google Scholar
[11]Rouche, N. and Mawhin, J. Équations différentielles ordinaires, Tome II: Stabilité et solutions périodiques. Masson et Cie, Éditeurs, Paris, 1973.Google Scholar
[12]Terracini, S. and Verzini, G.Solutions of prescribed number of zeroes to a class of superlinear ODE's systems. NoDEA Nonlinear Differential Equations Appl. 8 (2001), 323–341.Google Scholar