Differential Equations

, Volume 53, Issue 13, pp 1703–1714 | Cite as

Existence and Dimension Properties of a Global B-Pullback Attractor for a Cocycle Generated by a Discrete Control System

  • A. A. Maltseva
  • V. Reitmann
Control Theory


We consider cocycles on finite-dimensional manifolds generated by discrete-time control systems. Frequency conditions for the existence of a global B-pullback attractor for such cocycles considered over a general base system on a metric space are given. Upper bounds for the Hausdorff dimension of the global B-pullback attractor of a discrete cocycle are obtained using the transfer function of the linear part of the cocycle and the discrete Kalman–Yakubovich–Popov frequency theorem.


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  1. 1.
    Dmitriev, Yu.A., Frequency conditions for the dissipativity and existence of periodic solutions of sampled-data automatic control systems with one nonlinear element, Dokl. Akad. Nauk SSSR, 1965, vol. 164, no. 1, pp. 28–31.MathSciNetGoogle Scholar
  2. 2.
    Kruk, A.V., Malykh, A.E., and Reitmann, V., Upper bounds for the Hausdorff dimension and the stratification of an invariant set of an evolution system on a Hilbert manifold, Differ. Equations, 2017, vol. 53, no. 13, pp. 1715–1733.CrossRefGoogle Scholar
  3. 3.
    Leonov, G.A., Reitmann, V., and Slepukhin, A.S., Upper estimates for the Hausdorff dimension of negatively invariant sets of local cocycles, Dokl. Math., 2011, vol. 84, no. 1, pp. 551–554.MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Maricheva, A.V., Estimate for the Hausdorff dimension of cocycles on a finite-dimensional Riemannian manifold, Diploma Thesis, St. Petersburg: St. Petersburg State Univ., 2015.Google Scholar
  5. 5.
    Reitmann, V., Dinamicheskie sistemy, attraktory i otsenki ikh razmernosti (Dynamical Systems, Attractors, and Estimates for Their Dimension), St. Petersburg: S.-Peterb. Gos. Univ., 2013.Google Scholar
  6. 6.
    Yakubovich, V.A., The matrix-inequality method in the theory of the stability of nonlinear control systems. I: The absolute stability of forced vibrations, Autom. Remote Control, 1965, vol. 25 (1964), pp. 905–917.MATHGoogle Scholar
  7. 7.
    Yakubovich, V.A., A frequency theorem in control theory, Sib. Math. J., 1973, vol. 14, no. 2, pp. 265–289.MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Boichenko, V.A., Leonov, G.A., and Reitmann, V., Dimension Theory for Ordinary Differential Equations, Stuttgart: Teubner, 2005.CrossRefMATHGoogle Scholar
  9. 9.
    Douady, A. and Oesterlé, J., Dimension de Hausdorff des attracteurs, C. R. Seances Acad. Sci. Ser. A, 1980, vol. 290, no. 24, pp. 1135–1138.MathSciNetMATHGoogle Scholar
  10. 10.
    Kloeden, P.E. and Schmalfuss, B., Nonautonomous systems, cocycle attractors, and variable time-step discretization, Numer. Algorithms, 1997, vol. 14, no. 1–3, pp. 141–152.MATHGoogle Scholar
  11. 11.
    Kuznetsov, N.V., The Lyapunov dimension and its estimation via the Leonov method, Phys. Lett. A, 2016, vol. 380, no. 25–26, pp. 2142–2149.MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Kuznetsov, N.V., Alexeeva, T.A., and Leonov, G.A., Invariance of Lyapunov exponents and Lyapunov dimension for regular and irregular linearizations, Nonlinear Dynam., 2016, vol. 85, no. 1, pp. 195–201.MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Leonov, G.A., Alexeeva, T.A., and Kuznetsov, N.V., Analytic exact upper bound for the Lyapunov dimension of the Shimizu–Morioka system, Entropy, 2015, vol. 17, no. 7, pp. 5101–5116.CrossRefGoogle Scholar
  14. 14.
    Leonov, G.A., Kuznetsov, N.V., Korzhemanova, N.A., and Kusakin, D.V., Lyapunov dimension formula for the global attractor of the Lorenz system, Commun. Nonlinear Sci. Numer. Simul., 2016, vol. 41, pp. 84–103.MathSciNetCrossRefGoogle Scholar
  15. 15.
    Leonov, G.A., On estimations of Hausdorff dimension of attractors, Vestn. Leningr. Univ. Ser. 1: Mat. Mekh. Astron., 1991, no. 3, pp. 41–44.MATHGoogle Scholar
  16. 16.
    Maltseva, A. and Reitmann, V., Bifurcations of invariant measures in discrete-time parameter dependent cocycles, Math. Bohem., 2015, vol. 140, no. 2, pp. 205–213.MathSciNetMATHGoogle Scholar
  17. 17.
    Maltseva, A. and Reitmann, V., Global B-pullback attractors for cocycles generated by discrete-time cardiac conduction models, Proc. 11th AIMS Conf. on Dynam. Syst., Differ. Equations and Appl., Orlando, 2016.Google Scholar
  18. 18.
    Maltseva, A. and Reitmann, V., Global stability and bifurcations of invariant measures for the discrete cocycles of the cardiac conduction system’s equations, Differ. Equations, 2014, vol. 50, no. 13, pp. 1718–1732.MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Noack, A., Hausdorff dimension estimates for time-discrete feedback control systems, Z. Angew. Math. Mech., 1997, vol. 77, no. 12, pp. 891–899.MathSciNetCrossRefMATHGoogle Scholar

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© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Faculty of Mathematics and MechanicsSaint Petersburg State UniversityPeterhofRussia

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