ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

For citation:

Kuznetsov A. P., Savin A. V., Sataev I. R. On the critical behavior of non-identical asymmetrically coupled Chua’s circuits. Izvestiya VUZ. Applied Nonlinear Dynamics, 2007, vol. 15, iss. 2, pp. 3-13. DOI: 10.18500/0869-6632-2007-15-2-3-13

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text PDF(Ru):
(downloads: 115)
Article type: 

On the critical behavior of non-identical asymmetrically coupled Chua’s circuits

Kuznetsov Aleksandr Petrovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Savin Aleksej Vladimirovich, Saratov State University
Sataev Igor Rustamovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences

The complex dynamics and the peculiarities of the transition to chaos in two coupled flow systems – Chua’s circuits are investigated. It is shown that this system demonstrates more complicated behavior at the onset of chaos than the discrete maps. In particular, the codimension of the critical behavior increases in such system.  

Key words: 
  1. Feigenbaum MJ. Quantitative universality for a class of nonlinear transformations. J. of Stat. Phys. 1978;19(1):25–52. DOI: 10.1007/BF01020332.
  2. Feigenbaum MJ. The universal metric properties of nonlinear transformations. J. of Stat. Phys. 1979;26(6):669–706. DOI: 10.1007/BF01107909.
  3. Chang SJ, Wortis M, Wright JA. Iterative properties of a one-dimensional quartic map: Critical lines and tricritical behavior. Phys. Rev. A. 1981;24:2669–2684.
  4. MacKey RS, Tresser C. Some flesh on the skeleton: The bifurcation structure of bimodal maps. Physica D. 1987;27(3):412–422.
  5. MacKey RS, Tresser C. Boundary of topological chaos for bimodal maps of the interval. J. London Math. Soc. 1988;37(1):164–181. DOI: 10.1112/jlms/s2-37.121.164.
  6. Schell M, Fraser S, Kapral R. Subharmonic bifurcations in the sine map: An infinite of bifurcations. Phys. Rev. A. 1983;28(1):373–378. DOI: 10.1103/PhysRevA.28.373.
  7. Mackey RS, van Zeijts JBJ. Period doubling for bimodal maps: A horseshoe for a renormalization operator. Nonlinearity. 1988;1:253–277. DOI:10.1088/0951-7715/1/1/011.
  8. Kuznetsov AP, Kuznetsov SP, Sataev IR. Critical dynamics for one-dimensional maps. Part II. Two-parametre transition to chaos. Izvestiya VUZ. Applied Nonlinear Dynamics, 1993;1(3):17–35.
  9. Kuznetsov AP, Kuznetsov SP, Sataev IR. Codimension and typicity in a context of description of transition to chaos via period-doubling in dissipative dynamical systems. Regul. Chaotic Dyn. 1997;2(3-4):90–105.
  10. Feigenbaum MJ, Kadanoff LP, Shenker SJ. Quasiperiodicity in dissipative systems: A renormalization group analysis. Physica D. 1982;5(2-3):370–386. DOI: 10.1016/0167-2789(82)90030-6.
  11. Shenker SJ. Scaling behavior in a map of a circle onto itself: Empirical results. Physica D. 1982;5(2-3):405–411. DOI: 10.1016/0167-2789(82)90033-1.
  12. Rand D, Ostlund S, Sethna J, Siggia ED. Universal transition from quasiperiodicity to chaos in dissipative systems. Phys. Rev. Lett. 1982;49(2):132–135. DOI: 10.1103/PhysRevLett.49.132.
  13. Ostlund S, Rand D, Sethna J, Siggia ED. Universal properties of the transition from quasiperiodicity to chaos in dissipative systems. Physica D. 1983;8(3):303–342. DOI: 10.1016/0167-2789(83)90229-4.
  14. Rand D. Existence, non-existence and universal breakdown of dissipative goldeninvariant tori. I. Golden critical circle maps. Nonlinearity. 1992;5:639–662. DOI: 10.1088/0951-7715/5/3/002.
  15. Kuznetsov SP. A variety of critical phenomena associated with the golden mean quasiperiodicity. Izvestiya VUZ. Applied Nonlinear Dynamics. 2002;10(3):22.
  16. Kuznetsov SP, Sataev IR. Period-doubling for two-dimensional non-invertible maps: Renormalization group analysis and quantitative universality. Physica D. 1997;101(3-4):249–269. DOI: 10.1016/S0167-2789(96)00237-0.
  17. Kuznetsov AP, Kuznetsov SP, Sataev IR. A variety of period-doubling universality classes in multiparameter analysis of transition to chaos. Physica D. 1997;109:91–112. DOI:10.1016/S0167-2789(97)00162-0.
  18. Kuznetsov SP, Sataev IR. Universality and scaling in non-invertible two-dimensional maps. Physica Scripta. 1996;67:184–187. DOI: 10.1088/0031-8949/1996/T67/035.
  19. Kuznetsov SP. Tricriticality in two-dimensional maps. Phys. Lett. A. 1992;169:438–444.
  20. Kuznetsov AP, Kuznetsov SP, Mosekilde E, Turukina LV. Two-parameter analysis of the scaling behavior at the onset of chaos: Tricritical and pseudo-tricritical points. Physica A. 2001;300:367–385. DOI: 10.1016/S0378-4371(01)00368-5.
  21. Kuznetsov AP, Savin AV. On some type of transition from order to chaos in the system of coupled maps with period doubling. Izvestiya VUZ. Applied Nonlinear Dynamics. 2003;11(6):16–31.
  22. Kuznetsov SP. Critical behavior of one-dimensional chains. Tech. Phys. Lett. 1983;9(2):94–98.
  23. Isaeva OB. On possibility of realization of the phenomena of complex analytical dynamics for the physical systems, built up of coupled elements, which demonstrate period-doublings. Izvestiya VUZ. Applied Nonlinear Dynamics. 2001;9(6):129–146.
  24. Matsumoto T, Chua LO, Komuro M. The double scroll. IEEE Transactions on Circuits and Systems. 1985;32(8):797–818. DOI: 10.1109/TCS.1985.1085791.
  25. Chua LO, Komuro M, Matsumoto T. The double scroll family. IEEE Transactions on Circuits and Systems. 1986;33(11):1073–1118. DOI: 10.1109/TCS.1986.1085869.
  26. Matsumoto T, Chua LO, Ayaki K. Reality of chaos in the double scroll circuit: A computer-assisted proof. IEEE Transactions on Circuits and Systems. 1988;35(7):909. DOI: 10.1109/31.1836.
  27. Komuro M, Tokunaga R, Matsumoto T, Hotta A. Global bifurcation analysis of the double scroll circuit. Int. J. of Bif. and Chaos. 1991;1(1):139–182. DOI: 10.1142/S0218127491000105.
  28. Kahlert C. Heteroclinic orbits and scaled similar structures in the parameter space of the Chua oscillator. In Chaotic Hierarchy. Singapure, World Scientific. 1991:209–234. DOI: 10.1142/9789814503372_0011.
  29. Lozi R, Ushiki S. Confinors and bounded-time patterns in Chua’s circuit and the double scroll family. Int. J. of Bif. and Chaos. 1991;1(1):119–138. DOI: 10.1142/S0218127491000099.
  30. Genot M. Application of 1D Chua’s map from Chua’s circuit: A pictorial guide. J. of Circuits, Systems and Computers. 1993;3(2):375–409. DOI: 10.1142/S0218126693000241.
  31. Kuznetsov AP, Kuznetsov SP, Sataev IR, Chua LO. Self-similarity and universality in Chua’s circuit via the approximate Chua’s 1D map. Journal on Circuits, Systems and Computers. 1993;3(2):431–440. DOI: 10.1142/S0218126693000265.
  32. Kuznetsov AP, Kuznetsov SP, Sataev IR, Chua LO. Two-parameter study of transition to chaos in Chua’s circuit: Renormalization group, universality and scaling. Int. J. of Bif. and Chaos. 1993;3(4):943–962. DOI: 10.1142/S0218127493000799.
  33. Kennedy MP. Robust OP Amp realization of Chua’s circuit. Frequenz. 1992;46(3-4):66–80. DOI: 10.1515/FREQ.1992.46.3-4.66.
  34. Zhong GQ. Implementation of Chua’s circuit with a cubic nonlinearity. IEEE Transactions on Circuits and Systems – I: Fundamental Theory and Applications. 1994;41(12):934–941. DOI: 10.1109/81.340866.
  35. Zhong GQ, Ayrom F. Experimental confirmation of chaos from Chua’s circuit. Int. J. Circuit Theory Appl. 1985;13(11):93-98.
  36. Chua LO, Itoh M, Kocarev L, Eckert K. Chaos synchronization in Chua’s circuit. Elecrtonics research laboratory, University of California, Berkeley, Memorandum UCB/ERL M92/111; 1 May 1992.
Short text (in English):
(downloads: 92)