Adaptive set-oriented computation of topological horseshoe factors in area-and volume preserving maps

SIAM Journal on Applied Dynamical Systems

Volumn 9, Issue 4, 2010, Pages 1164-1200

  James, J D Mireles ^a  

^a RUTGERS UNIVERSITY   (United States)

Author keywords

Automatic chaos verification; Connecting orbits; Conservative dynamics; Set oriented numerics

Indexed keywords

AUTOMATIC CHAOS VERIFICATION; CONNECTING ORBITS; CONSERVATIVE DYNAMICS; DIFFEOMORPHISMS; EXISTING METHOD; GRAPH-THEORETIC; ITERATIVE ALGORITHM; PERIODIC POINTS; SET-ORIENTED NUMERICS; TOPOLOGICAL HORSESHOES;

ALGORITHMS; GRAPH THEORY; NUMERICAL METHODS;

CHAOS THEORY;

EID: 79251554983     PISSN: None     EISSN: 15360040     Source Type: Journal    
DOI: 10.1137/090776330     Document Type: Article

References (44)

1. Z. Arai, W. Kalies, H. Kokubu, K. Mischaikow, H. Oka, and P. Pilarczyk, A database schema for the analysis of global dynamics of multiparameter systems, SIAM J. Appl. Dyn. Syst., 8(2009), pp. 757-789.
2. E. Bollt and E. Kostelich, Optimal targeting of chaos, Phys. Lett. A, 245(1998), pp. 399-406.
3. E. Bollt and J. Meiss, Targeting chaotic orbits to the Moon through recurrence, Phys. Lett. A, 204(1995), pp. 373-378.
4. E. Bollt and J. Meiss, Controlling chaotic transport through recurrence, Phys. D, 81(1995), pp. 280-294.
5. K. Burns and H. Weiss, A geometric criterion for positive topological entropy, Comm. Math. Phys., 172(1995), pp. 95-118.
6. T. Cormen, C. Leiserson, R. Rivest, and C. Stein, Introduction to Algorithms, 2nd ed., MIT Press, Cambridge, MA, 2001.
7. S. Day, R. Frongillo, and R. Treviño, Algorithms for rigorous entropy bounds and symbolic dynamics, SIAM J. Appl. Dyn. Syst., 7(2008), pp. 1477-1506.
8. S. Day, O. Junge, and K. Mischaikow, A rigor ous numerical method for the global analysis of infinitedimensional discrete dynamical systems, SIAM J. Appl. Dyn. Syst., 3(2004), pp. 117-160.
9. S. Day, O. Junge, and K. Mischaikow, Towards automated chaos verification, in EQUADIFF 2003, World Scientific Publishing, Hackensack, NJ, 2005, pp. 157-162.
10. M. Dellnitz, GAIO-Global Analysis of Invariant Objects, software described at http://www2.math. uni-paderborn. de/ags/ag-dellnitz/software.html.
11. M. Dellnitz and A. Hohmann, The computation of unstable manifolds using subdivision and continuation, in Nonlinear Dynamical Systems and Chaos (Groningen, 1995), Progr. Nonlinear Differential Equations Appl. 19, Birkhäuser-Verlag, Basel, Switzerland, 1996, pp. 449-459.
12. M. Dellnitz and A. Hohmann, A subdivision algorithm for the computation of unstable manifolds and global attractors, Numer. Math., 75(1997), pp. 293-317. (Pubitemid 127372558)
13. M. Dellnitz, O. Junge, W. Koon, F. Lekien, M. Lo, J. E. Marsden, K. Padberg, R. Preis, S. D. Ross, and B. Thiere, Transport in d ynamical astronomy and multibody problems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15(2005), pp. 699-727.
14. M. Dellnitz, O. Junge, M. Post, and B. Thiere, On target for Venus-Set oriented computation of energy efficient low thrust trajectories, Celestial Mech. Dynam. Astronom., 95(2006), pp. 357-370.
15. M. Dellnitz, O. Junge, and B. Thiere, The numerical detection of connecting orbits, Discrete Contin. Dyn. Syst. Ser. B, 1(2001), pp. 125-135.
16. M. Feingold, L. Kadanoff, and O. Piro, Passive scalars, three-dimensional volume-preserving maps, and chaos, J. Statist. Phys., 50(1988), pp. 529-565.
17. M. Gidea and R. de la Llave, Arnold Diffusion with Optimal Time in the Large Gap Problem, preprint.
18. M. Gidea and R. de la Llave, Topological methods in the instability problem of Hamiltonian systems, Discrete Contin. Dynam. Systems, 14(2006), pp. 295-328.
19. M. Gidea and C. Robinson, Symbolic dynamics for transition tori. II, in New Advances in Celestial Mechanics and Hamiltonian Systems, Kluwer/Plenum, New York, 2004, pp. 95-108.
20. M. Gidea and C. Robinson, Obstruction argument for transition chains of tori interspersed with gaps, Discrete Contin. Dynam. Systems, 2(2009), pp. 393-416.
21. L. Grüne and O. Junge, A set oriented appro ach to optimal feedback stabilization, Systems Control Lett., 54(2005), pp. 169-180.
22. M.-R. Herman, R. McGehee, and E. Zehnder, eds., Charles Conley Memorial Volume, Special Issue of Ergodic Theory and Dynamical Systems, Vol. 8, Cambridge University Press, Cambridge, UK, 1988.
23. O. Junge, An adaptive subdivision technique for the approximation of attractors and invariant measures: Proof of convergence, Dyn. Syst., 16(2001), pp. 213-222.
24. T. Kaczynski, K. Mischaikow, and M. Mrozek, Computational Homology, Appl. Math. Sci. 157, Springer-Verlag, New York, 2004.
25. W. Kalies and H. Ban, A computational approach to Conley's decomposition theorem, J. Comput. Nonlinear Dyn., 1(2006), pp. 312-319.
26. W. D. Kalies, K. Mischaikow, and R. C. A. M. Vander Vorst, An algorithmic approach to chain recurrence, Found. Comput. Math., 5(2005), pp. 409-449.
27. A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia Math. Appl. 54, Cambridge University Press, Cambridge, UK, 1995.
28. M. Lee, F. Musser, P. Plauger, and A. Stepanov, The C++ Standard Template Library, User Guide and Reference Manual, Prentice-Hall, Englewood Cliffs, NJ, 2000.
29. A. Lumsdaine, J. Siek, and L. Lee, The Boost Graph Library, User Guide and Reference Manual, Addison-Wesley Professional, Reading, MA, 2001.
30. J. D. Mireles James, ACVLCM++: Automated chaos verification library for conservative maps (dynamical systems software in C++), online at www.math.utexas.edu/users/jjames/codes/ACVLCM++.
31. K. Mischaikow, The Conley index theory: A brie f introduction, in Conley Index Theory (Warsaw, 1997), Banach Center Publ. 47, Polish Academy of S cience, Warsaw, 1999, pp. 9-19.
32. K. Mischaikow and M. Mrozek, Chaos in the L orenz equations: A computer assisted proof. II. Details, Math. Comp., 67(1998), pp. 1023-1046.
33. K. Mischaikow, M. Mrozek, and P. Pilarczyk, Graph approach to the computation of the homology of continuous maps, Found. Comput. Math., 5(2005), pp. 199-229.
34. G. Osipenko, Dynamical Systems, Graphs, and Algorithms, Lecture Notes in Math. 1889, Springer-Verlag, Berlin, 2007 (Appendix A by N. B. Ampilova and Appendix B by D. Fundinger).
35. J. Palis, Jr., and W. de Melo, Geometric Theory of Dynamical Systems, An Introduction (translated from the Portuguese by A. K. Manning), Springer-Verlag, New York, 1982.
36. P. Pilarczyk, The CHomP Software, available at the website of the Computational Homology Project, http://chomp.rutgers.edu/, 1997.
37. P. Pilarczyk and K. Stolot, Excision-preserving cubical approach to the algorithmic computation of the discrete Conley index, Topology Appl., 155(2008), pp. 1149-1162.
38. C. Robinson, Symbolic dynamics for transition tori, in Celestial Mechanics (Evanston, IL, 1999), Contemp. Math. 292, Amer. Math. Soc., Providence, RI, 2002, pp. 199-208.
39. S. Smale, Diffeomorphisms with many periodic points, in Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton University Press, Princeton, NJ, 1965, pp. 63-80.
40. R. Srzednicki, A generalization of the Lefschetz fixed point theorem and detection of chaos, Proc. Amer. Math. Soc., 128(2000), pp. 1231-1239.
41. Y. B. Suris, A discrete-time Garnier system, Phys. Lett. A, 189(1994), pp. 281-289.
42. A. Szymczak, The Conley index for discrete semidynamical systems, Topology Appl., 66(1995), pp. 215-240.
43. A. Szymczak, A combinatorial procedure for findi ng isolating neighbourhoods and index pairs, Proc. Roy. Soc. Edinburgh Sect. A, 127(1997), pp. 1075-1088.
44. D. Yang, C++ and Object-oriented Numeric Computing for Scientists and Engineers, Springer, New York, 2000.