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Identification and reconstruction of chaotic systems using multiresolution wavelet decompositions. (English) Zbl 1085.93022
Summary: A new modelling framework for identifying and reconstructing chaotic systems is developed based on multiresolution wavelet decompositions. Qualitative model validation is used to compare the multiresolution wavelet models and it is shown that the dynamical features of chaotic systems can be captured by the identified models provided the wavelet basis functions are properly selected. Two basis selection algorithms, orthogonal least squares and a new matching pursuit orthogonal least squares, are considered and compared. Several examples are included to illustrate the results.

MSC:
93D10 Popov-type stability of feedback systems
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
65T60 Numerical methods for wavelets
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References:
[1] Abarbanel HDI, Physics Letters 138 pp 401– (1989)
[2] Abarbanel HDI, Physical Review 41 pp 1782– (1990)
[3] Adamatzky A, Identification of Cellular Automata (1994)
[4] DOI: 10.1142/S0218127495000363 · Zbl 0886.58100
[5] DOI: 10.1142/S0218127494000095 · Zbl 0876.58028
[6] Aguirre LA, Physica 80 pp 26– (1995)
[7] DOI: 10.1142/S0218127496000059 · Zbl 0870.58091
[8] Albano AM, Physica 58 pp 1– (1992)
[9] DOI: 10.1142/S0218127498001789 · Zbl 0984.37102
[10] Adamatzky A, Identification of Cellular Automata (1994)
[11] Bagarinao E, Physical Review 60 pp 1073– (1999)
[12] Billings SA, International Journal of Control 49 pp 2157– (1989)
[13] DOI: 10.1142/S0218127499000894 · Zbl 0955.93502
[14] DOI: 10.1080/00207728808964057 · Zbl 0669.93015
[15] Billings SA, 6th IFAC Symposium on Identification and Systems Parameter Estimation pp pp. 427–432– (1982)
[16] DOI: 10.1080/00207178608933633 · Zbl 0597.93058
[17] Billings SA, International Journal of Systems Science (2003)
[18] Billings SA, IEEE Transactions on Systems, Man and Cybernetics 33 pp 332– (2003)
[19] DOI: 10.1006/mssp.1998.0189
[20] Brown R, Physical Review 43 pp 2787– (1991)
[21] Brown M, UKACC International Conference on Control ’98 pp pp. 709–714– (1998)
[22] Cao LY, Physica 85 pp 225– (1995)
[23] DOI: 10.1142/S0218127497001394 · Zbl 0899.62120
[24] Casdagli M, Physica 35 pp 335– (1989)
[25] DOI: 10.1080/00207178908953472 · Zbl 0686.93093
[26] DOI: 10.1109/72.80341
[27] DOI: 10.1142/S0218127493001112 · Zbl 0886.58076
[28] Chen ZH, Journal of the Royal Statistical Society, Series B (Methodological) 55 pp 473– (1993)
[29] Chui CK, An Introduction to Wavelets (1992)
[30] DOI: 10.2307/2153941 · Zbl 0759.41008
[31] DOI: 10.1142/S0218127497000066 · Zbl 0925.93153
[32] Correa MV, International Journal of Bifurcation and Chaos 10 pp 1019– (2000)
[33] Crutchfield JP, Complex Systems 1 pp 417– (1987)
[34] DOI: 10.1103/RevModPhys.57.617 · Zbl 0989.37516
[35] Eckmann J-P, Physical Review 34 pp 4971– (1986)
[36] DOI: 10.1103/PhysRevLett.59.845
[37] DOI: 10.1214/aos/1176347963 · Zbl 0765.62064
[38] DOI: 10.1109/3477.775275
[39] Grassberger P, Physica 9 pp 189– (1983)
[40] Guckenheimer J, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (1983) · Zbl 0515.34001
[41] DOI: 10.1023/A:1013903804720 · Zbl 0998.68119
[42] Gutowitz HA, Physica 45 pp vii– (1990)
[43] Gouesbet G, Physical Review 44 pp 6264– (1991)
[44] Gouesbet G, Physical Review 46 pp 1784– (1992)
[45] Hayners BR, Nonlinear Dynamics 5 pp 93– (1994)
[46] DOI: 10.1007/BF01608556 · Zbl 0576.58018
[47] DOI: 10.1109/72.914539
[48] Ilachinski A, Cellular Automata: A Discrete Universe (2001)
[49] DOI: 10.1080/00207178808906169 · Zbl 0647.93062
[50] DOI: 10.1080/0020718508961129 · Zbl 0569.93011
[51] Linsay PS, Physics Letters 153 pp 353– (1991)
[52] DOI: 10.1021/jp010450t
[53] DOI: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2 · Zbl 1417.37129
[54] DOI: 10.1109/34.192463 · Zbl 0709.94650
[55] DOI: 10.1109/78.258082 · Zbl 0842.94004
[56] DOI: 10.1142/S021812749300057X · Zbl 0875.62426
[57] Menard O, Physical Review 62 pp 6325– (2000)
[58] DOI: 10.1142/S0218127497001758 · Zbl 0976.93501
[59] DOI: 10.1142/S0218127498000346 · Zbl 0933.37055
[60] DOI: 10.1007/BF01212280 · Zbl 0595.58028
[61] Moon FC, Chaotic Vibrations: An Introduction for Applied Scientists and Engineers (1987)
[62] DOI: 10.1103/PhysRevLett.45.712
[63] Parker TS, Practical Numerical Algorithms for Chaotic Systems (1989)
[64] DOI: 10.1016/0959-1524(95)00014-H
[65] Pearson RK, Discrete-time Dynamic Models (1999)
[66] DOI: 10.1142/S0218127492000598 · Zbl 0900.62497
[67] Smith LA, Physica 58 pp 50– (1992)
[68] Takens F, Lecture Notes in Mathematics 898 (1981)
[69] DOI: 10.1109/TCS.1981.1084975
[70] DOI: 10.1109/72.159070
[71] DOI: 10.1080/0020717042000197622 · Zbl 1143.93312
[72] Wolf A, Physics 16 pp 285– (1985)
[73] DOI: 10.1103/RevModPhys.55.601 · Zbl 1174.82319
[74] DOI: 10.1016/S0925-2312(01)00638-5 · Zbl 1006.68742
[75] DOI: 10.1109/72.557660
[76] DOI: 10.1080/002071799220065 · Zbl 0941.93511
[77] DOI: 10.1080/00207179608921662 · Zbl 0858.93069
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