Finite Resolution Dynamics

Foundations of Computational Mathematics

Volumn 11, Issue 2, 2011, Pages 211-239

  Luzzatto, Stefano ^a,b   Pilarczyk, Pawel ^c  

^a IMPERIAL COLLEGE LONDON   (United Kingdom)

^b ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS   (Italy)

^c UNIVERSITY OF MINHO   (Portugal)

Author keywords

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

Indexed keywords

EID: 79952485970     PISSN: 16153375     EISSN: 16153383     Source Type: Journal    
DOI: 10.1007/s10208-010-9083-z     Document Type: Article

References (49)

1. Z. Arai, On hyperbolic plateaus of the Hénon map, Exp. Math. 16, 181-188 (2007).
2. Z. Arai, K. Mischaikow, Rigorous computations of homoclinic tangencies, SIAM J. Appl. Dyn. Syst. 5, 280-292 (2006).
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).
4. A. Arbieto, C. Matheus, Decidability of chaos for some families of dynamical systems, Found. Comput. Math., 269-275 (2004).
5. M. Benedicks, L. Carleson, The dynamics of the Hénon map, Ann. Math. 133, 73-169 (1991).
6. M. Benedicks, L.-S. Young, Sinai-Bowen-Ruelle measures for certain Hénon maps, Invent. Math. 112, 541-576 (1993).
7. R. Bowen, Markov partitions for Axiom A diffeomorphisms, Am. J. Math. 92, 725-747 (1970).
8. R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, vol. 470 (Springer, Berlin, 1975).
9. R. Bowen, D. Ruelle, The ergodic theory of Axiom A flows, Invent. Math. 29, 181-202 (1975).
10. E. Colli, Infinitely many coexisting strange attractors, Ann. Inst. H. Poincaré Anal. Non Linéaire15(5), 539-579 (1998).
11. T. H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein, Introduction to Algorithms, 2nd edn. (MIT Press, Cambridge, 2001).
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.
13. Z. Galias, Rigorous investigation of the Ikeda map by means of interval arithmetic, Nonlinearity15, 1759-1779 (2002).
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).
15. M. Hénon, A two-dimensional mapping with a strange attractor, Commun. Math. Phys. 50, 69-77 (1976).
16. S. Lynch Hruska, A numerical method for constructing the hyperbolic structure of complex Hénon mappings, Found. Comput. Math. 6, 427-455 (2006).
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).
18. M. V. Jakobson, Absolutely continuous invariant measures for one-parameter families of one-dimensional maps, Commun. Math. Phys. 81(1), 39-88 (1981).
19. W. D. Kalies, K. Mischaikow, R. C. A. M. VanderVorst, An algorithmic approach to chain recurrence, Found. Comput. Math. 5, 409-449 (2005).
20. T. Kapela, C. Simó, Computer assisted proofs for nonsymmetric planar choreographies and for stability of the eight, Nonlinearity20, 1241-1255 (2007).
21. T. Kapela, P. Zgliczyński, The existence of simple choreographies for the N-body problem-a computer-assisted proof, Nonlinearity16, 1899-1918 (2003).
22. O. Lanford, A computer-assisted proof of the Feigenbaum conjectures, Bull. Am. Math. Soc. 6, 427-434 (1982).
23. S. Luzzatto, H. Takahasi, Computable conditions for the occurrence of non-uniform hyperbolicity in families of one-dimensional maps, Nonlinearity19, 1657-1695 (2006).
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).
25. S. Luzzatto, M. Viana, Positive Lyapunov exponents for Lorenz-like families with criticalities, Astérisque261(xiii), 201-237 (2000).
26. S. Luzzatto, I. Melbourne, F. Paccaut, The Lorenz attractor is mixing, Commun. Math. Phys. 260, 393-401 (2005).
27. K. Mischaikow, M. Mrozek, Chaos in Lorenz equations: A computer assisted proof, Bull. Am. Math. Soc. 33, 66-72 (1995).
28. R. E. Moore, Interval Analysis (Prentice-Hall, Inc., Englewood Cliffs, 1966).
29. M. Mrozek, Topological invariants, multivalued maps and computer assisted proofs in dynamics, Comput. Math. Appl. 32(4), 83-104 (1996).
30. M. Mrozek, An algorithm approach to the Conley index theory, J. Dyn. Differ. Equ. 11, 711-734 (1999).
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.
32. L. Mora, M. Viana, Abundance of strange attractors, Acta Math. 171, 1-71 (1993).
33. S. E. Newhouse, Diffeomorphisms with infinitely many sinks, Topology13, 9-18 (1974).
34. M. J. Pacifico, A. Rovella, M. Viana, Infinite-modal maps with global chaotic behavior, Ann. Math. 148, 441-484 (1998).
35. J. Palis, F. Takens, Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations (Cambridge University Press, Cambridge, 1993).
36. P. Pilarczyk, Computer assisted method for proving existence of periodic orbits, Topol. Methods Nonlinear Anal. 13, 365-377 (1999).
37. P. Pilarczyk, Finite resolution dynamics. Software and examples, http://www. pawelpilarczyk. com/finresdyn/.
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.
39. P. Pilarczyk, K. Stolot, Excision-preserving cubical approach to the algorithmic computation of the discrete Conley index, Topol. Appl. 155, 1149-1162 (2008).
40. J. G. Sinaǐ, Markov partitions and U-diffeomorphisms, Funkc. Anal. Prilozh. 2, 64-89 (1968).
41. A. Szymczak, A combinatorial procedure for finding isolating neighbourhoods and index pairs, Proc. R. Soc. Edinb. A127, 1075-1088 (1997).
42. R. Tarjan, Depth-first search and linear graph algorithms, SIAM J. Comput. 1, 146-160 (1972).
43. H. Thunberg, Positive exponent in families with flat critical point, Ergod. Theory Dyn. Syst. 19, 767-807 (1999).
44. M. Tsujii, Positive Lyapunov exponents in families of one-dimensional dynamical systems, Invent. Math. 111(1), 113-137 (1993).
45. W. Tucker, The Lorenz attractor exists, C. R. Acad. Sci. Paris, Sér. I328, 1197-1202 (1999).
46. Q. Wang, L.-S. Young, Strange attractors with one direction of instability, Commun. Math. Phys. 218, 1-97 (2001).
47. Wikipedia contributors, Aperiodic graph, Wikipedia, The Free Encyclopedia (2008), http://en. wikipedia. org/w/index. php?title=Aperiodic_graph& oldid=224030707.
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. 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).