Adaptive reconstruction of chaotic attractors out of their single trajectories

Signal Processing

Volumn 80, Issue 6, 2000, Pages 1099-1113

  Zóltowski, Mariusz ^a  

^a GDANSK UNIVERSITY OF TECHNOLOGY   (Poland)

Author keywords

[No Author keywords available]

Indexed keywords

CHAOS THEORY; ITERATIVE METHODS; LEAST SQUARES APPROXIMATIONS; LYAPUNOV METHODS; MATRIX ALGEBRA; SECURITY OF DATA;

CHAOTIC ATTRACTORS;

ADAPTIVE FILTERING;

EID: 0342472228     PISSN: 01651684     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0165-1684(00)00022-0     Document Type: Article

References (53)

1. Abarbanel H.D.I., Kennel M.B. Local false nearest neighbours and dynamical dimensions from observed chaotic data. Phys. Rev. E. 47:1993;3057-3068.
2. Akaike H. Maximum likelihood identification of Gaussian autoregressive moving average models. Biometrika. 60:1973;255-265.
3. Akaike H. A new look at the statistical model identification. IEEE Trans. Automat. Control. AC-19:1974;716-723.
4. M. Bellanger, Adaptive Digital Filtering, ECCTD, Paris, 1987.
5. S. Boccaletti et al., Adaptive strategies for recognition, control and synchronization of chaos, Chaos Solitons Fractals 8 (9) (1997) 1431-1448.
6. Boccaletti S.et al. Adaptive synchronization of chaos for secure communication. Phys. Rev. E. 55(5):1997;4979-4981.
7. Brown P.et al. Computing the Lyapunov spectrum of a dynamical system from an observed time series. Phys. Rev. A. 42:1991;2787-2806.
8. P. Celka, Synchronization of chaotic systems through parameter adaptation, ISCAS 95, Seatle, 1995, pp. 962-695.
9. Corron N.J. A new approach to communications using chaotic signals. IEEE Trans. Circuits Systems-I. 44(5):May 1997;373-382.
10. J.Z. Cypkin, Podstawy teorii ukladów ucza̧cych siȩ, WNT, Warszawa, 1973.
11. Eckmann S.O.et al. Liapunov exponents from time series. Phys. Rev. A. 34:1986;4971-4979.
12. P.E. Eykhoff, Identyfikacja w ukladach dynamicznych, PWN, Warszawa, 1980.
13. Fraser A.M. Information and entropy in strange attractors. IEEE Trans. Inform. Theory. IT-35:1989;245-262.
14. S.W. Fomin, I.P. Kornfeld, J.G. Sinaj, Ergodic Theory, PWN, Warszawa, 1978.
15. Grassberger P., Procaccia I. Characterization of strange attractors. Phys. Rev. Lett. 50(5):January 1983;346-349.
16. Guckenheimer J., Holmes P. Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields. 1983;Springer, Berlin.
17. I. Gumovski, C. Mira, Dynamique Chaotique, Cepadues Editions, Toulouse, 1980.
18. Hammel S.M. A noise reduction method for chaotic systems. Phys. Lett. A. 148(8,9):September 1990;421-428.
19. Haykin S. Adaptive Filter Theory. 3rd Edition:1996;Prentice-Hall, Englewood Cliffs.
20. Jaźwiński A.H. Stochastic Processes and Filtering Theory. 1970;Academic Press, New York.
21. Judd K., Mees A. On selecting models for nonlinear time series. Physica D. 82:1995;426-444.
22. Kay S.M. Modern Spectral Estimation: Theory and Applications. 1988;Prentice-Hall, Englewood Cliffs, NJ.
23. Leung H., Lam J. Design of demodulator of the chaotic modulation communication system. IEEE Trans. Circuit Systems-I. 44(3):March 1977;262-267.
24. A.J. Lichtenberg, M.A. Lieberman, Regular and Stochastic Motion, Springer, Berlin, 1991, pp. 597-613.
25. E.N. Lorentz, Deterministic nonperiodic flow, J. Atmos. Sci. 20 (1993) 130.
26. Milnor J. On the concept of attractor. Commun. Math. Phys. 99:1985;175-195.
27. Núňéz-Yépez H.N.et al. Nonlinear behaviour in a dripping faucet. Cordero P., Nachtergaele B. Nonlinear Phenomena in Fluids, Solids and other Complex Systems. 1991;Elsevier, Amsterdam.
28. Ott E. Recent developments in chaotic dynamics. IEEE Trans. Plasma Sci. 22(1):February 1994;43-46.
29. Ott E. A physical system with qualitatively uncertain dynamics. Nature. 365(9):1993;138-140.
30. Packard N.et al. Geometry from a time series. Phys. Rev. Lett. 45:1980;712-716.
31. Parker B. Einstein's Dream, The Search for a Unified Theory of the Universe. 1987;Plenum Press, New York.
32. Parker T.S., Chua L.O. Practical Numerical Algorithms for Chaotic Systems. 1989;Springer, New York.
33. F.J. Pineda, J.C. Sommerer, A fast algorithm for estimating the generalised dimension and choosing time delays, in: Time Series Prediction, A.S. Weigend, N.A. Gershenfeld (Eds.), Addison-Wesley, Reading, MA, 1994, pp. 367-385.
34. Priestley M.B. Spectral Analysis and Time Series, Vols. 1-2. 1981;Academic Press, New York.
35. Rissanen J. Modeling by shortest data description. Automatica. 14:1978;465-471.
36. Rissanen J. Universal Coding, Information, Prediction, and Estimation. IEEE Trans. Inform. Theory. IT-30(4):July 1984;629-636.
37. Rissanen J. Stochastic complexity and modeling. Ann. Stat. 14:1986;1080-1100.
38. Rössler O. Haken H. Synergetics: a Workshop. 1977;Springer, Berlin.
39. Sauer T., Yorke J.A., Casdagli M. Embodology. J. Stat. Phys. 65:1991;579-616.
40. H.G. Schuster, Chaos deterministyczny, Wprowadzenie, wyd.II, PWN, Warszawa, 1995.
41. Schwartz G. Estimating the dimension of a model. Ann. Stat. 6:1978;461-464.
42. F.C. Schweppe, Uklady dynamiczne w warunkach losowych, WNT, Warszawa, 1978.
43. Shaw R. Strange attractors, chaotic behavior, and information flow. Z. Naturforsch. 36a:1981;80-112.
44. Scott I., Mulgrew B. Nonlinear system identification and prediction using orthonormal functions. IEEE Trans. Signal Process. 45:1997;1842-1853.
45. Stark J. Recursive prediction of chaotic time series. J. Nonlinear Sci. 3:1993;197-223.
46. W. Szlenk, Wstep do teorii gladkich ukladów dynamicznych, PWN, Warszawa, 1982.
47. Takens F. On the numerical determination of the dimension of attractor. Rand D., Young L.S. Dyn. Syst. and Turbul., Warwick 1980, Lecture Notes in Mathematics, Vol. 898. 1981;366-381 Springer, Berlin.
48. Wax M., Kaillath T. Detection of signals by information theoretic criteria. IEEE Trans. Acoust., Speech, Signal Process. ASSP-33(2):April 1985;387-392.
49. Whitney H. Differential manifolds. Ann. Math. 37:1936;645-680.
50. Yang T.et al. Cryptography based on chaotic systems. IEEE Trans. Circuits Systems - I. 44(5):May 1997;469-477.
51. Young L-S. Entropy, Liapunov exponents, and Hausdorff dimension in differentiable dynamical systems. IEEE Trans. Circuits Systems. CAS-30:1983;599-606.
52. Zhou H., Ling X-T. Problems with the chaotic inverse system encryption approach. IEEE Trans. Circuits Systems-I. 44(3):March 1997;268-271.
53. M. Żól towski, Why do optimal forgetting RLS-s exhibit long term divergence and how can this be avoided, Signal Processing 68 1998 195-218.