On the approximation of complicated dynamical behavior

SIAM Journal on Numerical Analysis

Volumn 36, Issue 2, 1999, Pages 491-515

  Dellnitz, Michael ^a   Junge, Oliver ^a  

^a UNIVERSITY OF BAYREUTH   (Germany)

Author keywords

Almost invariant set; Approximation of the Frobenius Perron operator; Computation of invariant measures; Computation of SRB measures; Cyclic behavior

Indexed keywords

EID: 0000371241     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142996313002     Document Type: Article

References (28)

1. M. BENEDICKS AND L.-S. YOUNG, Sinai-Bowen-Ruelle measures for certain Hénon maps, Invent. Math., 112 (1993), pp. 541-576.
2. M. BLANK AND G. KELLER, Random perturbations of chaotic dynamical systems: Stability of the spectrum, Nonlinearity, 11 (1998), pp. 1351-1364.
3. R. BOWEN AND D. RUELLE, The ergodic theory of axiom a flows, Invent. Math., 29 (1975), pp. 181-202.
4. P. CHOSSAT AND M. GOLUBITSKY, Symmetry-increasing bifurcation of chaotic attractors, Phys. D, 32 (1988), pp. 423-436.
5. M. DELLNITZ AND A. HOHMANN, The computation of unstable manifolds using subdivision and continuation, in Nonlinear Dynamical Systems and Chaos, Progr. Nonlinear Differential Equations Appl. 19, H. W. Broer, S. A. van Gils, I. Hoveijn, and F. Takens, eds., Birkhäuser, Basel, 1996, pp. 449-459.
6. M. DELLNITZ AND A. HOHMANN, A subdivision algorithm for the computation of unstable manifolds and global attractors, Numer. Math., 75 (1997), pp. 293-317.
7. M. DELLNITZ, A. HOHMANN, O. JUNGE, AND M. RUMPF, Exploring invariant sets and invariant measures, Chaos, 7 (1997), p. 221.
8. M. DELLNITZ AND O. JUNGE, Almost invariant sets in Chua's circuit, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 7 (1997), pp. 2475-2485.
9. M. DELLNITZ AND O. JUNGE, An adaptive subdivision technique for the approximation of attractors and invariant measures, Comput. Visual. Sci., 1 (1998), pp. 63-68.
10. J. DING, Q. DU, AND T. Y. LI, High order approximation of the Frobenius-Perron operator, Appl. Math. Comput., 53 (1993), pp. 151-171.
11. J. L. DOOB, Stochastic Processes, John Wiley, New York, 1960.
12. M. EIDENSCHINK, Exploring Global Dynamics: A Numerical Algorithm Based on the Conley Index Theory, Ph.D. thesis, Georgia Institute of Technology, Atlanta, GA, 1995.
13. R. GUDER, M. DELLNITZ, AND E. KREUZER, An adaptive method for the approximation of the generalized cell mapping, Chaos Solitons Fractals, 8 (1997), pp. 525-534.
14. H. HSU, Global analysis by cell mapping, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2 (1992), pp. 727-771.
15. F. Y. HUNT, Approximating the invariant measures of randomly perturbed dissipative maps, J. Math. Anal. Appl., 198 (1996), pp. 534-551.
16. A. KHIBNIK, D. ROOSE, AND L. O. CHUA, On periodic orbits and homoclinic bifurcations in Chua's circuit with a smooth nonlinearity, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 3 (1993), pp. 363-384.
17. YU. KIFER, General random perturbations of hyperbolic and expanding transformations, J. Anal. Math., 47 (1986), pp. 111-150.
18. YU. KIFER, Random Perturbations of Dynamical Systems, Birkhäuser, Basel, 1988.
19. YU. KIFER, Computations in dynamical systems via random perturbations, Discrete Continuous Dynam. Systems, 3 (1997), pp. 457-476.
20. E. KREUZER, Numerische Untersuchung nichtlinearer dynamischer Systeme, Springer-Verlag, New York, 1987.
21. A. LASOTA AND M. C. MACKEY, Chaos, Fractals and Noise, Springer-Verlag, New York, 1994.
22. R. B. LEHOUCQ, Analysis and Implementation of an Implicitly Restarted Arnoldi Iteration, Rice University Technical Report TR95-13, Houston, TX, 1995.
23. T. MATSUMOTO, L. O. CHUA, AND M. KOMURO, The double scroll, IEEE Trans. Circuits Systems, 32 (1985), pp. 797-818.
24. S. P. MEYN AND R. L. TWEEDIE, Markov Chains and Stochastic Stability, Springer-Verlag, New York, 1993.
25. J. E. OSBORN, Spectral approximation for compact operators, Math. Comp., 29 (1975), pp. 712-725.
26. D. RUELLE, A measure associated with axiom a attractors, Amer. J. Math., 98 (1976), pp. 619-654.
27. Y. G. SINAI, Gibbs measures in ergodic theory, Russian Math. Surveys, 166 (1972), pp. 21-69.
28. K. YOSIDA, Functional Analysis, Springer-Verlag, New York, 1980.