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Foundations of Computational Mathematics

, Volume 11, Issue 2, pp 211–239 | Cite as

Finite Resolution Dynamics

  • Stefano Luzzatto
  • Paweł Pilarczyk
Article

Abstract

We develop a new mathematical model for describing a dynamical system at limited resolution (or finite scale), and we give precise meaning to the notion of a dynamical system having some property at all resolutions coarser than a given number. Open covers are used to approximate the topology of the phase space in a finite way, and the dynamical system is represented by means of a combinatorial multivalued map. We formulate notions of transitivity and mixing in the finite resolution setting in a computable and consistent way. Moreover, we formulate equivalent conditions for these properties in terms of graphs, and provide effective algorithms for their verification. As an application we show that the Hénon attractor is mixing at all resolutions coarser than 10−5.

Keywords

Dynamical system Finite resolution Open cover Combinatorial dynamics Rigorous numerics Directed graph Transitivity Mixing Algorithm 

Mathematics Subject Classification (2000)

37M99 65P20 65G20 

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References

  1. 1.
    Z. Arai, On hyperbolic plateaus of the Hénon map, Exp. Math. 16, 181–188 (2007). MathSciNetzbMATHGoogle Scholar
  2. 2.
    Z. Arai, K. Mischaikow, Rigorous computations of homoclinic tangencies, SIAM J. Appl. Dyn. Syst. 5, 280–292 (2006). MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Z. Arai, W. Kalies, H. Kokubu, K. Mischaikow, H. Oka, P. Pilarczyk, A database schema for the analysis of global dynamics of multiparameter systems, SIAM J. Appl. Dyn. Syst. 8, 757–789 (2009). MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    A. Arbieto, C. Matheus, Decidability of chaos for some families of dynamical systems, Found. Comput. Math., 269–275 (2004). Google Scholar
  5. 5.
    M. Benedicks, L. Carleson, The dynamics of the Hénon map, Ann. Math. 133, 73–169 (1991). MathSciNetCrossRefGoogle Scholar
  6. 6.
    M. Benedicks, L.-S. Young, Sinai–Bowen–Ruelle measures for certain Hénon maps, Invent. Math. 112, 541–576 (1993). MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    R. Bowen, Markov partitions for Axiom A diffeomorphisms, Am. J. Math. 92, 725–747 (1970). MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, vol. 470 (Springer, Berlin, 1975). zbMATHGoogle Scholar
  9. 9.
    R. Bowen, D. Ruelle, The ergodic theory of Axiom A flows, Invent. Math. 29, 181–202 (1975). MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    E. Colli, Infinitely many coexisting strange attractors, Ann. Inst. H. Poincaré Anal. Non Linéaire 15(5), 539–579 (1998). MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    T.H. Cormen, C.E. Leiserson, R.L. Rivest, C. Stein, Introduction to Algorithms, 2nd edn. (MIT Press, Cambridge, 2001). zbMATHGoogle Scholar
  12. 12.
    J.-P. Eckmann, H. Koch, P. Wittwer, A Computer-Assisted Proof of Universality for Area-Preserving Maps, Mem. Amer. Math. Soc., vol. 47(289) (Springer, Berlin, 1984), vi+122. Google Scholar
  13. 13.
    Z. Galias, Rigorous investigation of the Ikeda map by means of interval arithmetic, Nonlinearity 15, 1759–1779 (2002). MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    S.V. Gonchenko, D.V. Turaev, L.P. Shilprimenikov, On the dynamic properties of diffeomorphisms with homoclinic tangencies, Sovrem. Mat. Prilozh. 7, 91–117 (2003). Google Scholar
  15. 15.
    M. Hénon, A two-dimensional mapping with a strange attractor, Commun. Math. Phys. 50, 69–77 (1976). zbMATHCrossRefGoogle Scholar
  16. 16.
    S. Lynch Hruska, A numerical method for constructing the hyperbolic structure of complex Hénon mappings, Found. Comput. Math. 6, 427–455 (2006). MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    D. Jabłoński, M. Kulczycki, Topological transitivity, mixing and nonwandering set of subshifts of finite type—a numerical approach, Int. J. Comput. Math. 80, 671–677 (2003). MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    M.V. Jakobson, Absolutely continuous invariant measures for one-parameter families of one-dimensional maps, Commun. Math. Phys. 81(1), 39–88 (1981). MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    W.D. Kalies, K. Mischaikow, R.C.A.M. VanderVorst, An algorithmic approach to chain recurrence, Found. Comput. Math. 5, 409–449 (2005). MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    T. Kapela, C. Simó, Computer assisted proofs for nonsymmetric planar choreographies and for stability of the eight, Nonlinearity 20, 1241–1255 (2007). MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    T. Kapela, P. Zgliczyński, The existence of simple choreographies for the N-body problem—a computer-assisted proof, Nonlinearity 16, 1899–1918 (2003). MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    O. Lanford, A computer-assisted proof of the Feigenbaum conjectures, Bull. Am. Math. Soc. 6, 427–434 (1982). MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    S. Luzzatto, H. Takahasi, Computable conditions for the occurrence of non-uniform hyperbolicity in families of one-dimensional maps, Nonlinearity 19, 1657–1695 (2006). MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    S. Luzzatto, W. Tucker, Non-uniformly expanding dynamics in maps with singularities and criticalities, Inst. Ht. Etudes Sci. Publ. Math. 89, 179–226 (1999). MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    S. Luzzatto, M. Viana, Positive Lyapunov exponents for Lorenz-like families with criticalities, Astérisque 261(xiii), 201–237 (2000). MathSciNetGoogle Scholar
  26. 26.
    S. Luzzatto, I. Melbourne, F. Paccaut, The Lorenz attractor is mixing, Commun. Math. Phys. 260, 393–401 (2005). MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    K. Mischaikow, M. Mrozek, Chaos in Lorenz equations: A computer assisted proof, Bull. Am. Math. Soc. 33, 66–72 (1995). MathSciNetCrossRefGoogle Scholar
  28. 28.
    R.E. Moore, Interval Analysis (Prentice-Hall, Inc., Englewood Cliffs, 1966). zbMATHGoogle Scholar
  29. 29.
    M. Mrozek, Topological invariants, multivalued maps and computer assisted proofs in dynamics, Comput. Math. Appl. 32(4), 83–104 (1996). MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    M. Mrozek, An algorithm approach to the Conley index theory, J. Dyn. Differ. Equ. 11, 711–734 (1999). MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    M. Mrozek, P. Pilarczyk, The Conley index and rigorous numerics for attracting periodic orbits, in Proceedings of the Conference on Variational and Topological Methods in the Study of Nonlinear Phenomena, Pisa, 2000. Progr. Nonlinear Differential Equations Appl., vol. 49 (Birkhäuser, Boston, 2002), pp. 65–74. Google Scholar
  32. 32.
    L. Mora, M. Viana, Abundance of strange attractors, Acta Math. 171, 1–71 (1993). MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    S.E. Newhouse, Diffeomorphisms with infinitely many sinks, Topology 13, 9–18 (1974). MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    M.J. Pacifico, A. Rovella, M. Viana, Infinite-modal maps with global chaotic behavior, Ann. Math. 148, 441–484 (1998). MathSciNetCrossRefGoogle Scholar
  35. 35.
    J. Palis, F. Takens, Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations (Cambridge University Press, Cambridge, 1993). zbMATHGoogle Scholar
  36. 36.
    P. Pilarczyk, Computer assisted method for proving existence of periodic orbits, Topol. Methods Nonlinear Anal. 13, 365–377 (1999). MathSciNetzbMATHGoogle Scholar
  37. 37.
    P. Pilarczyk, Finite resolution dynamics. Software and examples, http://www.pawelpilarczyk.com/finresdyn/.
  38. 38.
    P. Pilarczyk, Topological-numerical approach to the existence of periodic trajectories in ODEs, Discrete and Continuous Dynamical Systems 2003, A Supplement Volume: Dynamical Systems and Differential Equations, pp. 701–708. Google Scholar
  39. 39.
    P. Pilarczyk, K. Stolot, Excision-preserving cubical approach to the algorithmic computation of the discrete Conley index, Topol. Appl. 155, 1149–1162 (2008). MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    J.G. Sinaĭ, Markov partitions and U-diffeomorphisms, Funkc. Anal. Prilozh. 2, 64–89 (1968). Google Scholar
  41. 41.
    A. Szymczak, A combinatorial procedure for finding isolating neighbourhoods and index pairs, Proc. R. Soc. Edinb. A 127, 1075–1088 (1997). MathSciNetzbMATHGoogle Scholar
  42. 42.
    R. Tarjan, Depth-first search and linear graph algorithms, SIAM J. Comput. 1, 146–160 (1972). MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    H. Thunberg, Positive exponent in families with flat critical point, Ergod. Theory Dyn. Syst. 19, 767–807 (1999). MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    M. Tsujii, Positive Lyapunov exponents in families of one-dimensional dynamical systems, Invent. Math. 111(1), 113–137 (1993). MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    W. Tucker, The Lorenz attractor exists, C.R. Acad. Sci. Paris, Sér. I 328, 1197–1202 (1999). zbMATHGoogle Scholar
  46. 46.
    Q. Wang, L.-S. Young, Strange attractors with one direction of instability, Commun. Math. Phys. 218, 1–97 (2001). MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Wikipedia contributors, Aperiodic graph, Wikipedia, The Free Encyclopedia (2008), http://en.wikipedia.org/w/index.php?title=Aperiodic_graph&oldid=224030707.
  48. 48.
    Wikipedia contributors, Tarjan’s strongly connected components algorithm, Wikipedia, The Free Encyclopedia (2009), http://en.wikipedia.org/w/index.php?title=Tarjan%27s_strongly_connected_components_algorithm&oldid=295022950.
  49. 49.
    P. Zgliczyński, Rigorous numerics for dissipative partial differential equations. II. Periodic orbit for the Kuramoto–Sivashinsky PDE—a computer-assisted proof, Found. Comput. Math. 4, 157–185 (2004). MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© SFoCM 2011

Authors and Affiliations

  1. 1.Mathematics DepartmentImperial CollegeLondonUK
  2. 2.Abdus Salam, International Centre for Theoretical PhysicsTriesteItaly
  3. 3.Centro de MatemáticaUniversidade do MinhoBragaPortugal

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