\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

On the control of stability of periodic orbits of completely integrable systems

Abstract Related Papers Cited by
  • We provide a constructive method designed in order to control the stability of a given periodic orbit of a general completely integrable system. The method consists of a specific type of perturbation, such that the resulting perturbed system becomes a codimension-one dissipative dynamical system which also admits that orbit as a periodic orbit, but whose stability can be a-priori prescribed. The main results are illustrated in the case of a three dimensional dissipative perturbation of the harmonic oscillator, and respectively Euler's equations form the free rigid body dynamics.
    Mathematics Subject Classification: Primary: 37C27, 37C75; Secondary: 34C25.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    P. Birtea, M. Boleanţu, M. Puta and R. M. Tudoran, Asymptotic stability for a class of metriplectic systems, J. Math. Phys., 48 (2007), 082703, 7pp.doi: 10.1063/1.2771420.

    [2]

    P. Birtea and D. Comănescu, Asymptotic stability of dissipated Hamilton-Poisson systems, SIAM J. Appl. Dyn. Syst., 8 (2009), 967-976.doi: 10.1137/080735217.

    [3]

    C. Dăniasă, A. Gîrban and R. M. Tudoran, New aspects on the geometry and dynamics of quadratic Hamiltonian systems on $(\mathfrak{so}(3))^{*}$, Int. J. Geom. Methods Mod. Phys., 8 (2011), 1695-1721.doi: 10.1142/S0219887811005889.

    [4]

    A. Gasull, H. Giacomini and M. Grau, On the stability of periodic orbits for differential systems on $\mathbbR^n$, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 495-509.doi: 10.3934/dcdsb.2008.10.495.

    [5]

    P. Hartman, Ordinary Differential Equations, Classics in Applied Mathematics, 38, SIAM, Philadelphia, 2002.doi: 10.1137/1.9780898719222.

    [6]

    J. Moser and E. Zehnder, Notes on Dynamical Systems, Courant Lecture Notes in Mathematics, 12, American Mathematical Society, Providence, 2005.

    [7]

    T. S. Ratiu, R. M. Tudoran, L. Sbano, E. Sousa Dias and G. Terra, II: A crash course in geometric mechanics, in Geometric Mechanics and Symmetry (eds. J. Montaldi and T. S. Ratiu), London Mathematical Society Lecture Notes Series, 306, Cambridge University Press, Cambridge, 2005, 23-156.doi: 10.1017/CBO9780511526367.003.

    [8]

    R. M. Tudoran, Affine Distributions on Riemannian Manifolds with Applications to Dissipative Dynamics, J. Geom. Phys., (2015).doi: 10.1016/j.geomphys.2015.01.017.

    [9]

    F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, $2^{nd}$ edition, Springer-Verlag, Berlin, 1996.doi: 10.1007/978-3-642-61453-8.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(95) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return