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

Computational hyperbolicity

Abstract Related Papers Cited by
  • Using semihyperbolicity as a basic tool, we provide a general computer assisted method for verifying hyperbolicity of a given set. As a consequence we obtain that the Hénon attractor is hyperbolic for some parameter values.
    Mathematics Subject Classification: Primary: 37D05, 37D20, Secondary: 37M99.

    Citation:

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

    A. Al-Nayef, P. Diamond, P. Kloeden, V. Kozyakin and A. Pokrovskii, Bi-shadowing and delay equations, Dynam. Stability Systems, 10 (1996), 121-134.

    [2]

    A. A. Al-Nayef, P. E. Kloeden and A. V. Pokrovskii, Semi-hyperbolic mappings, condensing operators, and neutral delay equations, J. Differential Equations, 137 (1997), 320-339.doi: 10.1006/jdeq.1997.3262.

    [3]

    Z. Arai, On Hyperbolic Plateaus of the Hénon Maps, Experiment. Math., 16 (2007), 181-188.

    [4]
    [5]

    R. Sacker and G. Sell, Existence of dichotomies and invariant splittings for linear differential systems. I, J. Differential Equations, 15 (1974), 429-458.doi: 10.1016/0022-0396(74)90067-9.

    [6]

    M. Davis, R. MacKay and A. Sannami, Markov shifts in the Hénon family, Phys. D, 52 (1991), 171-178.doi: 10.1016/0167-2789(91)90119-T.

    [7]

    S. Day, H. Kokubu, S. Luzzatto, K. Mischaikow, H. Oka and P. Pilarczyk, Quantitative hyperbolicity estimates in one-dimensional dynamics, Nonlinearity, 21 (2008), 1967-1987.doi: 10.1088/0951-7715/21/9/002.

    [8]

    R. Devaney and Z. Nitecki, Shift automorphisms in the Hénon mapping, Comm. Math. Phys., 67 (1979), 137-146.doi: 10.1007/BF01221362.

    [9]

    P. Diamond, P. Kloeden, V. Kozyakin and A. Pokrovskii, Semihyperbolic mappings, J. Nonlinear Sci., 5 (1995), 419-431.doi: 10.1007/BF01212908.

    [10]

    P. Diamond, P. E. Kloeden, V. S. Kozyakin and A. V. Pokrovskii, "Semi-Hyperbolicity and Bi-Shadowing,'' Manuscript.

    [11]

    Z. Galias and P. Zgliczyński, Infinite-dimensional Krawczyk operator for finding periodic orbits of discrete dynamical systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 4261-4272.doi: 10.1142/S0218127407019937.

    [12]

    S. Hruska, A numerical method for constructing the hyperbolic structure of complex Hénon mappings, Found. Comput. Math., 6 (2006), 427-455.doi: 10.1007/s10208-006-0141-2.

    [13]

    A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,'' Cambridge University Press, Cambridge, 1995.

    [14]

    M. Mazur, On some useful conditions on hyperbolicity, Trends in Math., 10 (2008), 57-64.

    [15]

    M. Mazur, J. Tabor and P. Kościelniak, Semi-hyperbolicity and hyperbolicity, Discrete Contin. Dynam. Syst., 20 (2008), 1029-1038.doi: 10.3934/dcds.2008.20.1029.

    [16]

    M. Mazur, J. Tabor and K. Stolot, Semi-hyperbolicity implies hyperbolicity in the linear case, Proceedings of the Conference "Topological Methods in Differential Equations and Dynamical Systems'' (Krakow-Przegorzaly, 1996), Univ. Iagell. Acta Math., 36 (1998), 121-126.

    [17]

    M. Mazur, J. Tabor, T. Kułaga and P. KościelniakComputational hyperbolicity group, http://www.im.uj.edu.pl/MarcinMazur/comphyp.

    [18]

    K. Mischaikow and M. Mrozek, Chaos in the Lorenz equations: A computer-assisted proof, Bull. Amer. Math. Soc. (N.S.), 32 (1995), 66-72.

    [19]

    S. Newhouse, Cone-fields, domination, and hyperbolicity, in "Modern Dynamical Systems and Applications,'' Cambridge Univ. Press, Cambridge, (2004), 419-432.

    [20]

    K. Palmer, "Shadowing in Dynamical Systems. Theory and Applications,'' Kluwer Academic Publishers, Dordrecht, 2000.

    [21]

    F. Riesz and B. Sz.-Nagy, "Functional Analysis,'' Frederick Ungar Publishing Co., New York, 1955.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(70) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return