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Communications in Mathematical Physics

, Volume 48, Issue 1, pp 53–79 | Cite as

On resonant non linearly coupled oscillators with two equal frequencies

  • Martin Kummer
Article

Abstract

This paper contains a detailed study of the flow that the classical Hamiltonian
$$H = \tfrac{1}{2}(x\begin{array}{*{20}c} 2 \\ 1 \\ \end{array} + y\begin{array}{*{20}c} 2 \\ 1 \\ \end{array} ) + \tfrac{1}{2}(x\begin{array}{*{20}c} 2 \\ 2 \\ \end{array} + y\begin{array}{*{20}c} 2 \\ 2 \\ \end{array} ) + \mathcal{O}_3 $$
induces inR4,\(\mathcal{O}_3 \) representing a convergent power series that begins with a third order term.

In particular the existence and stability of periodic orbits is investigated.

Keywords

Neural Network Statistical Physic Complex System Periodic Orbit Nonlinear Dynamics 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Braun, M.: On the applicability of the third integral of motion. J. Diff. Eq.13, 300–318 (1973)Google Scholar
  2. 2.
    Gustavson, F.: On constructing formal integrals of a Hamiltonian system near an equilibrium point. Astron. J.71, 670–686 (1966)Google Scholar
  3. 3.
    Moser, J. K.: Lectures on Hamiltonian systems. Mem. Amer. Math. Soc.81, 000–000 (1968)Google Scholar
  4. 4.
    Siegel, C. L., Moser, J. K.: Lectures on celestial mechanics. Berlin-Heidelberg-New York: Springer 1971Google Scholar
  5. 5.
    Moser, J. K.: On invariant curves of area preserving mappings of the annulus. Nachr. Akad. Wiss.1, 1–20 (1962)Google Scholar
  6. 6.
    Hénon, M., Heiles, C.: The applicability of the third integral of motion; some numerical experiments. Astron. J.69, 73–79 (1964)Google Scholar
  7. 7.
    Arnold, V. I.: Small denominators and problems of stability of motion in classical and celestial mechanics. Usp. Math. Nauk.18, 91–196 (1963); Russian Math. Surv.18, 85–193 (1963)Google Scholar
  8. 8.
    Arnold, V. I.: The stability of the equilibrium position of a Hamiltonian system of ordinary differential equations in the general elliptic case. Dokl. Akad. Nauk. USSR137, 255–257 (1961); Soviet Math.2, 247 (1961)Google Scholar
  9. 9.
    Moser, J.: Regularization of Kepler's problem and the averaging method on a manifold. Commun. Pure Appl. Math.23, 609–636 (1970)Google Scholar
  10. 10.
    Schmidt, D., Sweet, D.: A unifying theory in determining periodic families for Hamiltonian systems at resonance. J. Diff. Eq.14, 597–609 (1973)Google Scholar
  11. 11.
    Roels, J.: An extension to resonant case of Liapunov's theorem concerning periodic solutions near a Hamiltonian equilibrium. J. Diff. Eq.9, 300–324 (1971)Google Scholar
  12. 12.
    Weinstein, A.: Normal modes for nonlinear Hamiltonian systems. Inventiones math.20, 47–57 (1973)Google Scholar
  13. 13.
    Weinstein, A.: Lagrangian submanifolds and Hamiltonian systems. Ann. Math.98, 377–410 (1973)Google Scholar
  14. 14.
    Kummer, M.: An interaction of three resonant modes in a nonlinear lattice. J. Math. Anal. Appl.52, 64–104 (1975)Google Scholar

Copyright information

© Springer-Verlag 1976

Authors and Affiliations

  • Martin Kummer
    • 1
  1. 1.Department of MathematicsThe University of ToledoToledoUSA

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