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Fundamentals of synchronization in chaotic systems, concepts, and applications. (English) Zbl 0933.37030

Summary: The field of chaotic synchronization has grown considerably since its advent in 1990. Several subdisciplines and “cottage industries” have emerged that have taken on bona fide lives of their own. Our purpose in this paper is to collect results from these various areas in a review article format with a tutorial emphasis. Fundamentals of chaotic synchronization are reviewed first with emphases on the geometry of synchronization and stability criteria. Several widely used coupling configurations are examined and, when available, experimental demonstrations of their success (generally with chaotic circuit systems) are described. Particular focus is given to the recent notion of synchronous substitution – a method to synchronize chaotic systems using a larger class of scalar chaotic coupling signals than previously thought possible. Connections between this technique and well-known control theory results are also outlined. Extensions of the technique are presented that allow so-called hyperchaotic systems (systems with more than one positive Lyapunov exponent) to be synchronized. Several proposals for “secure” communication schemes have been advanced; major ones are reviewed and their strengths and weaknesses are touched upon. Arrays of coupled chaotic systems have received a great deal of attention lately and have spawned a host of interesting and, in some cases, counterintuitive phenomena including bursting above synchronization thresholds, destabilizing transitions as coupling increases (short-wave length bifurcations), and riddled basins. In addition, a general mathematical framework for analyzing the stability of arrays with arbitrary coupling configurations is outlined. Finally, the topic of generalized synchronization is discussed, along with data analysis techniques that can be used to decide whether two systems satisfy the mathematical requirements of generalized synchronization.

MSC:

37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37N99 Applications of dynamical systems
34C60 Qualitative investigation and simulation of ordinary differential equation models

References:

[1] Roux J.-C., Physica D 8 pp 257– (1983) · Zbl 0538.58024 · doi:10.1016/0167-2789(83)90323-8
[2] Yamada T., Prog. Theor. Phys. 70 pp 1240– (1983) · Zbl 1171.70307 · doi:10.1143/PTP.70.1240
[3] Yamada T., Prog. Theor. Phys. 72 pp 885– (1984) · doi:10.1143/PTP.72.885
[4] Carroll T. L., Physica D 67 pp 126– (1993) · Zbl 0800.94100 · doi:10.1016/0167-2789(93)90201-B
[5] DOI: 10.1103/PhysRevLett.64.821 · Zbl 0938.37019 · doi:10.1103/PhysRevLett.64.821
[6] DOI: 10.1103/PhysRevA.44.2374 · doi:10.1103/PhysRevA.44.2374
[7] DOI: 10.1103/PhysRevE.50.1874 · doi:10.1103/PhysRevE.50.1874
[8] Tresser C., Chaos 5 pp 693– (1995) · Zbl 1055.34506 · doi:10.1063/1.166101
[9] Pecora L. M., Phys. Rev. Lett. 67 pp 945– (1991) · Zbl 0990.37504 · doi:10.1103/PhysRevLett.67.945
[10] Lorenz E. N., Physica D 13 pp 90– (1984) · Zbl 0588.58037 · doi:10.1016/0167-2789(84)90272-0
[11] DOI: 10.1142/S0218127492000781 · Zbl 0875.94137 · doi:10.1142/S0218127492000781
[12] Anishchenko V. S., Sov. J. Commun. Technol. Electron. 36 pp 23– (1991)
[13] Ding M., Phys. Rev. E 49 pp R945– (1994) · doi:10.1103/PhysRevE.49.R945
[14] Pyragas K., Phys. Lett. A 181 pp 203– (1993) · doi:10.1016/0375-9601(93)90640-L
[15] Wu C. W., Int. J. Bifurcation Chaos 4 pp 979– (1994) · Zbl 0875.93445 · doi:10.1142/S0218127494000691
[16] Heagy J. F., Phys. Rev. Lett. 74 pp 4185– (1994) · doi:10.1103/PhysRevLett.74.4185
[17] Heagy J. F., Phys. Rev. Lett. 73 pp 3528– (1995) · doi:10.1103/PhysRevLett.73.3528
[18] Newcomb R. W., Circuits Syst. Signal Process. 5 pp 321– (1986) · Zbl 0608.58033 · doi:10.1007/BF01600066
[19] Tamasevicius A., Electron. Lett. 32 pp 1536– (1996) · doi:10.1049/el:19961066
[20] Carroll T., Am. J. Phys. 63 pp 377– (1995) · doi:10.1119/1.17923
[21] DOI: 10.1103/PhysRevLett.71.65 · doi:10.1103/PhysRevLett.71.65
[22] DOI: 10.1109/82.246163 · doi:10.1109/82.246163
[23] Kocarev Lj., Int. J. Bifurcation Chaos 2 pp 709– (1992) · Zbl 0875.94134 · doi:10.1142/S0218127492000823
[24] Murali K., Phys. Rev. E 48 pp R1624– (1993) · doi:10.1103/PhysRevE.48.R1624
[25] Parlitz U., Int. J. Bifurcation Chaos 2 pp 973– (1992) · Zbl 0870.94011 · doi:10.1142/S0218127492000562
[26] DOI: 10.1103/PhysRevLett.74.1970 · doi:10.1103/PhysRevLett.74.1970
[27] Short K. M., Int. J. Bifurcation Chaos 4 pp 959– (1994) · Zbl 0875.94002 · doi:10.1142/S021812749400068X
[28] MacDonals A. H., Phys. Rev. B 27 pp 201– (1983) · doi:10.1103/PhysRevB.27.201
[29] Brauer E., Int. J. Bifurcation Chaos 4 pp 1031– (1993) · Zbl 0875.94130 · doi:10.1142/S0218127494000745
[30] Tankara S., Physica D 28 pp 317– (1987) · doi:10.1016/0167-2789(87)90022-4
[31] Grebogi C., Physica D 15 pp 354– (1985) · Zbl 0577.58023 · doi:10.1016/S0167-2789(85)80004-X
[32] D’Humieres D., Phys. Rev. A 26 pp 3483– (1982) · doi:10.1103/PhysRevA.26.3483
[33] Guemez J., Phys. Rev. E 52 pp 2145– (1995) · doi:10.1103/PhysRevE.52.R2145
[34] DOI: 10.1103/PhysRevE.47.3889 · doi:10.1103/PhysRevE.47.3889
[35] Stojanovski T., Phys. Rev. E 54 pp 2128– (1996) · doi:10.1103/PhysRevE.54.2128
[36] DOI: 10.1103/PhysRevE.55.4035 · doi:10.1103/PhysRevE.55.4035
[37] Carroll T. L., Phys. Rev. E 50 pp 2580– (1994) · doi:10.1103/PhysRevE.50.2580
[38] Carroll T. L., IEEE Trans. Circuits Syst. 42 pp 105– (1995) · doi:10.1109/81.372850
[39] DOI: 10.1103/PhysRevLett.74.5028 · doi:10.1103/PhysRevLett.74.5028
[40] Carroll T. L., Phys. Rev. E 54 (5) pp 4676– (1996) · doi:10.1103/PhysRevE.54.4676
[41] di Bernardo M., Int. J. Bifurcation Chaos 6 pp 557– (1996) · Zbl 0900.70413 · doi:10.1142/S0218127496000254
[42] di Bernardo M., Phys. Lett. A 214 pp 139– (1996) · Zbl 0972.93509 · doi:10.1016/0375-9601(96)00182-X
[43] Chen C.-C., Phys. Lett. A 213 pp 148– (1996) · Zbl 0972.93506 · doi:10.1016/0375-9601(96)00101-6
[44] Chen G., Int. J. Bifurcation Chaos 6 pp 1341– (1996) · Zbl 0875.93157 · doi:10.1142/S021812749600076X
[45] DOI: 10.1103/PhysRevLett.76.904 · doi:10.1103/PhysRevLett.76.904
[46] Tsimring L. S., Phys. Lett. 213 pp 155– (1996) · doi:10.1016/0375-9601(96)00118-1
[47] Cenys A., Phys. Lett. A 213 pp 259– (1996) · doi:10.1016/0375-9601(96)00126-0
[48] DOI: 10.1016/0375-9601(94)90114-7 · doi:10.1016/0375-9601(94)90114-7
[49] DOI: 10.1103/PhysRevLett.71.4134 · Zbl 0972.37514 · doi:10.1103/PhysRevLett.71.4134
[50] So P., Phys. Lett. A 176 pp 421– (1993) · doi:10.1016/0375-9601(93)90472-C
[51] Ott E., Phys. Rev. Lett. 64 pp 1196– (1990) · Zbl 0964.37501 · doi:10.1103/PhysRevLett.64.1196
[52] DOI: 10.1103/PhysRevA.43.2787 · doi:10.1103/PhysRevA.43.2787
[53] DOI: 10.1103/PhysRevE.47.3962 · doi:10.1103/PhysRevE.47.3962
[54] Parlitz U., Phys. Rev. Lett. 76 pp 1232– (1996) · doi:10.1103/PhysRevLett.76.1232
[55] Heagy J. F., Chaos 4 pp 385– (1994) · doi:10.1063/1.166016
[56] Kocarev L., IEEE Trans. Circuits Syst. 42 pp 1009– (1995) · doi:10.1109/81.473587
[57] DOI: 10.1016/0375-9601(96)00326-X · doi:10.1016/0375-9601(96)00326-X
[58] DOI: 10.1103/PhysRevLett.72.2009 · doi:10.1103/PhysRevLett.72.2009
[59] Colet P., Opt. Lett. 19 pp 2056– (1994) · doi:10.1364/OL.19.002056
[60] Peterman D. W., Phys. Rev. Lett. 74 pp 1740– (1995) · doi:10.1103/PhysRevLett.74.1740
[61] Rul’kov N., Phys. Rev. E 51 pp 980– (1995) · doi:10.1103/PhysRevE.51.980
[62] Pecora L., Phys. Rev. E 52 pp 3420– (1995) · doi:10.1103/PhysRevE.52.3420
[63] DOI: 10.1103/PhysRevE.53.4528 · doi:10.1103/PhysRevE.53.4528
[64] Kocarev L., Phys. Rev. Lett. 76 pp 1816– (1996) · doi:10.1103/PhysRevLett.76.1816
[65] Badii R., Phys. Rev. Lett. 60 pp 979– (1988) · doi:10.1103/PhysRevLett.60.979
[66] Pecora L., Chaos 6 pp 432– (1996) · Zbl 1055.37544 · doi:10.1063/1.166186
[67] Campbell K. M., Nonlinearity 9 pp 801– (1996) · Zbl 0887.58028 · doi:10.1088/0951-7715/9/3/010
[68] Davies M. E., Nonlinearity 9 pp 487– (1996) · Zbl 0888.34043 · doi:10.1088/0951-7715/9/2/012
[69] Hunt B., Phys. Rev. E 55 pp 4029– (1997) · doi:10.1103/PhysRevE.55.4029
[70] Hunt B. R., Phys. Rev. E 54 (5) pp 4819– (1996) · doi:10.1103/PhysRevE.54.4819
[71] DOI: 10.1103/PhysRevE.49.1140 · doi:10.1103/PhysRevE.49.1140
[72] Platt N., Phys. Rev. Lett. 72 pp 3498– (1994) · doi:10.1103/PhysRevLett.72.3498
[73] DOI: 10.1103/PhysRevE.52.R1253 · doi:10.1103/PhysRevE.52.R1253
[74] Brown R., Chaos 7 pp 395– (1997) · Zbl 0933.37026 · doi:10.1063/1.166213
[75] DOI: 10.1103/PhysRevLett.77.1751 · doi:10.1103/PhysRevLett.77.1751
[76] Ashwin P., Nonlinearity 9 pp 703– (1994) · Zbl 0887.58034 · doi:10.1088/0951-7715/9/3/006
[77] Venkataramani S. C., Phys. Rev. E 54 pp 1346– (1996) · doi:10.1103/PhysRevE.54.1346
[78] Kapitaniak T., Int. J. Bifurcation Chaos 6 pp 211– (1996) · Zbl 0870.94003 · doi:10.1142/S021812749600196X
[79] Alexander J. C., Int. J. Bifurcation Chaos 2 pp 795– (1992) · Zbl 0870.58046 · doi:10.1142/S0218127492000446
[80] Kan I., Bull. Am. Math. Soc. 31 pp 68– (1994) · Zbl 0853.58077 · doi:10.1090/S0273-0979-1994-00507-5
[81] DOI: 10.1038/365138a0 · doi:10.1038/365138a0
[82] Lai Y-C., Phys. Rev. Lett. 72 pp 1640– (1994) · doi:10.1103/PhysRevLett.72.1640
[83] Parmenter R. H., Phys. Lett. A 189 pp 181– (1994) · doi:10.1016/0375-9601(94)90634-3
[84] DOI: 10.1016/0375-9601(94)90947-4 · Zbl 0959.37508 · doi:10.1016/0375-9601(94)90947-4
[85] DOI: 10.1016/0167-2789(83)90126-4 · Zbl 0561.58029 · doi:10.1016/0167-2789(83)90126-4
[86] Carroll T. L., Phys. Rev. Lett. 59 pp 2891– (1987) · doi:10.1103/PhysRevLett.59.2891
[87] McDonald S. W., Physica D 17 pp 125– (1985) · Zbl 0588.58033 · doi:10.1016/0167-2789(85)90001-6
[88] Park B.-S., Phys. Rev. A 40 pp 1576– (1989) · doi:10.1103/PhysRevA.40.1576
[89] Schiff S., Phys. Rev. E 54 pp 6708– (1996) · doi:10.1103/PhysRevE.54.6708
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