Journal of Statistical Physics

, Volume 121, Issue 5–6, pp 697–748 | Cite as

Multiparameter Critical Situations, Universality and Scaling in Two-Dimensional Period-Doubling Maps

  • S. P. KuznetsovEmail author
  • A. P. Kuznetsov
  • I. R. Sataev


We review critical situations, linked with period-doubling transition to chaos, which require using at least two-dimensional maps as models representing the universality classes. Each of them corresponds to a saddle solution of the two-dimensional generalization of Feigenbaum-Cvitanović equation and is characterized by a set of distinct universal constants analogous to Feigenbaum’s α and δ. One type of criticality designated H was discovered by several authors in 80-th in the context of period doubling in conservative dynamics, but occurs as well in dissipative dynamics, as a phenomenon of codimension 2. Second is bicritical behavior, which takes place in systems allowing decomposition onto two dissipative period-doubling subsystems, each of which is brought by parameter tuning onto a threshold of chaos. Types of criticality designated as FQ and C occur in non-invertible two-dimensional maps. We present and discuss a number of realistic systems manifesting those types of critical behavior and point out some relevant conditions of their potential observation in physical systems. In particular, we indicate a possibility for realization of the H type criticality without vanishing dissipation, but with its compensation in a self-oscillatory system. Next, we present a number of examples (coupled Hénon-like maps, coupled driven oscillators, coupled chaotic self-oscillators), which manifest bicritical behavior. For FQ-type we indicate possibility to arrange it in non-symmetric systems of coupled period-doubling subsystems, e.g. in Hénon-like maps and in Chua’s circuits. For C-type we present examples of its appearance in a driven Rössler oscillator at the period-doubling accumulation on the edge of syncronization tongue and in a model map with the Neimark–Sacker bifurcation


Period-doubling onset of chaos renormalization group universality scaling coupled maps coupled oscillators multi-parameter analysis 


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  1. 1.
    Feigenbaum, M.J. 1978Quantitative universality for a class of nonlinear transformationsJ. Stat. Phys.192552CrossRefzbMATHMathSciNetADSGoogle Scholar
  2. 2.
    Feigenbaum, M.J. 1979The universal metric properties of nonlinear transformationsJ. Stat. Phys.21669706CrossRefzbMATHMathSciNetADSGoogle Scholar
  3. 3.
    Cvitanović, P. eds. 1989Universality in Chaos2Adam BilgerBostonzbMATHGoogle Scholar
  4. 4.
    Collet, P., Eckmann, J.-P., Koch, H. 1981Period Doubling Bifurcations for Families of Maps on R n J. Stat. Phys.25114CrossRefMathSciNetzbMATHADSGoogle Scholar
  5. 5.
    Vul, E.B., Sinai, Ya.G., Khanin, K.M. 1984Feigenbaum universality and thermodynamic formalismRuss. Math. Surv.39140CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Crutchfield, J.P., Huberman, B.A. 1980Fluctuations and the onset of chaosPhys. Lett. A77407410CrossRefADSMathSciNetGoogle Scholar
  7. 7.
    Lichtenberg A.J., Lieberman M.A. Regular and Chaotic Dynamics 2nd Edn. (Springer-Verlag, 1992).Google Scholar
  8. 8.
    Gallas, J.A.C. 1993Structure of the parameter space of the Hénon mapPhys. Rev. Lett.7027142717CrossRefADSGoogle Scholar
  9. 9.
    Carmichael, H.J., Snapp, R.R., Schieve, W.C. 1982Oscillatory instabilities leading to “optical turbulence” in a bistable ring cavityPhys. Rev. A2634083422CrossRefADSGoogle Scholar
  10. 10.
    Glass, L., Perez, R. 1982Fine Structure of Phase LockingPhys. Rev. Lett.4817721775CrossRefADSMathSciNetGoogle Scholar
  11. 11.
    Novak, S.R.G. 1982Frehlich Transition to chaos in the Duffing oscillatorPhys. Rev. A2636603663CrossRefADSMathSciNetGoogle Scholar
  12. 12.
    Moon, W., Holmes, P.J. 1979A magnetoelastic strange attractorJ. Sound Vib.65285296CrossRefADSGoogle Scholar
  13. 13.
    Testa, J., Perez, J., Jeffries, C. 1982Evidence for universal chaotic behavior of a driven nonlinear oscillatorPhys. Rev. Lett.48714717CrossRefADSMathSciNetGoogle Scholar
  14. 14.
    Beiersdorfer, P., Wersinger, J.M. 1983Topology of the Invariant Manifolds of a Period-doubling Attractors for Some Forced Nonlinear OscillatorsPhys. Lett. A96269272CrossRefADSMathSciNetGoogle Scholar
  15. 15.
    M. Schanz and A. Pelster, On the Period-Doubling Scenario in Dynamical Systems with Time Delay, Proceedings of the 15th IMACS World Congress 1997 I:215–220 (Wissenschaft and Technik, 1997).Google Scholar
  16. 16.
    Anishchenko, V.S. 1989Dynamical Chaos in Physical Systems: Experimental Investigation of Self-Oscillating CircuitsWorld Sci. Publ.SingaporeGoogle Scholar
  17. 17.
    A. S. Dmitriev and V. Ya. Kislov, Stochastic oscillations in radiophysics and electronics (Moscow, Nauka, 1989). (In Russian.)Google Scholar
  18. 18.
    Komura, M., Tokunaga, R., Matsumoto, T., Chua, L.O. 1991Global bifurcation analysis of the double scroll circuitInt. J. Bifurcation Chaos1139182Google Scholar
  19. 19.
    Ryskin, N.M., Titov, V.N., Trubetskov, D.I. 1998Transition to the Chaotic Regime in a System Composed of an Electron Beam and an Inverse Electromagnetic WaveDoklady Physics439093ADSGoogle Scholar
  20. 20.
    Maurer, J., Libchaber, A. 1979Rayleigh-Bénard experiment in liquid helium: frequency locking and the onset of turbulenceJ. de Physique Lettres40419423Google Scholar
  21. 21.
    Libchaber, A., Laroche, C., Fauve, S. 1982Period doubling in mercury, a quantative measurementJ. de Phys. Lett.43211216Google Scholar
  22. 22.
    Lauterborn, W., Schmitz, E., Judt, A. 1993Experimental approach to a complex acoustic systemInt. J. Bifurcation Chaos3635642zbMATHGoogle Scholar
  23. 23.
    Arecci, F.T., Meucci, R., Puccioni, G., Tredicce, J. 1982Experimental Evidence of Subharmonic Bifurcations, Multistability, and Turbulence in a Q-Switched Gas LaserPhys. Rev. Lett.4912171220ADSGoogle Scholar
  24. 24.
    Vallée, R., Delisle, C. 1985Route to chaos in an acousto-optic bistable devicePhys. Rev. A3123902396CrossRefADSGoogle Scholar
  25. 25.
    T. Poston and I. Stewart, Catastrophe Theory and its Application (Dover Publications, 1997).Google Scholar
  26. 26.
    V. I. Arnol’d and G. S. Wassermann, Catastrophe Theory (Springer-Verlag, 1992).Google Scholar
  27. 27.
    Guckenheimer, J., Holmes, P. 1997Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector FieldsSpringer-VerlagNew YorkGoogle Scholar
  28. 28.
    Kuznetsov, Y.A. 1995Elements of Applied Bifurcation TheorySpringer-VerlagNew YorkzbMATHGoogle Scholar
  29. 29.
    Eckmann, J.-P., Koch, H., Wittwer, P. 1982Existence of a fixed point of the doubling transformation for area-preserving maps of the planePhys. Rev. A26720722CrossRefADSMathSciNetGoogle Scholar
  30. 30.
    Quispel, G.R.W. 1986Universal functional equation for period doubling in constant-Jacobian mapsPhys. Lett. A118457462CrossRefADSMathSciNetGoogle Scholar
  31. 31.
    Kuznetsov, A.P., Kuznetsov, S.P., Sataev, I.R. 1997A variety of period-doubling universality classes in multi-parameter analysis of transition to chaosPhysica D10991112See also: Scholar
  32. 32.
    MacKay, R.S. 1983Period doubling as a universal route to stochasticityHorton, W.Reichl, L.E.Szebehely, V eds. book: Long time prediction in dynamics.J. Wiley & SonsNew York11Google Scholar
  33. 33.
    Chen, C., Gyorgyi, G., Schmidt, G. 1986Universal transition between Hamiltonian and dissipative chaosPhys. Rev. A3425682570ADSGoogle Scholar
  34. 34.
    Greene, J.M., MacKay, R.S., Vivaldi, F., Feigenbaum, M.J. 1981Universal behavior in families of area preserving mapsPhysica D3468486ADSMathSciNetGoogle Scholar
  35. 35.
    Widom, M., Kadanoff, L.P. 1982Renormalization group analysis of bifurcations in area-preserving mapsPhysica D5287292CrossRefADSMathSciNetGoogle Scholar
  36. 36.
    Heagy, J.F. 1992A physical interpretation of the Henon mapPhysica D57436446CrossRefADSzbMATHMathSciNetGoogle Scholar
  37. 37.
    Chen, C., Gyorgyi, G., Schmidt, G. 1987Universal scaling in dissipative systemsPhys. Rev. A3526602668ADSMathSciNetGoogle Scholar
  38. 38.
    Reick, C. 1992Universal corrections to parameter scaling in period-doubling systems: Multiple scaling and crossoverPhys. Rev. A45777792CrossRefADSGoogle Scholar
  39. 39.
    Bezruchko, B.P., Gulyaev, Yu.V., Kuznetsov, S.P., Seleznev, E.P. 1986New type of critical behavior of coupled systems at the transition to chaosSov. Phys. Dokl.31258260ADSGoogle Scholar
  40. 40.
    Kuznetsov, A.P., Kuznetsov, S.P., Sataev, I.R. 1991Bicritical dynamics of period-doubling systems with unidirectional couplingInt. J. Bifurcation Chaos1839848MathSciNetzbMATHGoogle Scholar
  41. 41.
    Kaneko, K. 1985Spatial period-doubling in open flowPhys. Lett. A111321325CrossRefADSMathSciNetGoogle Scholar
  42. 42.
    Aranson, I.S., Gaponov-Grekhov, A.V., Rabinovich, M.I. 1988The onset and spatial development of turbulence in flow systemsPhysica D33120CrossRefADSMathSciNetzbMATHGoogle Scholar
  43. 43.
    Philominathan, P., Neelamegam, P. 2001Characterization of chaotic attractors at bifurcations in Murali-Lakshmanan-Chua’s circuit and one-way coupled map lattice systemChaos, Soliton and Fractals1210051017zbMATHCrossRefGoogle Scholar
  44. 44.
    Heil, T., Mulet, J., Fischer, I., Mirasso, C.R., Peil, M., Colet, P., Elsäber, W. 2002ON/OFF Phase Shift Keying for Chaos-Encrypted Communication Using External-Cavity Semiconductor LasersIEEE J. Quant. Electronics3811621170ADSCrossRefGoogle Scholar
  45. 45.
    Wang, S., Kuang, J., Li, J., Luo, Y., Lu, H., Hu, G. 2002Chaos-based secure communications in a large communityPhys. Rev. E66065202ADSGoogle Scholar
  46. 46.
    Kuznetsov, A.P., Kuznetsov, S.P., Sataev, I.R. 1993Variety of types of critical behavior and multistability in period doubling systems with unidirectional coupling near the onset of chaosInt. J. Bifurcation Chaos3139152MathSciNetzbMATHGoogle Scholar
  47. 47.
    Kuznetsov, S.P., Sataev, I.R. 1997Period-doubling for two- dimensional non-invertible maps: Renormalization group analysis and quantitative universalityPhysica D101249269CrossRefADSMathSciNetzbMATHGoogle Scholar
  48. 48.
    Whitney, H. 1955On Singularities of Mappings of Euclidean Spaces I Mappings of the Plane into the PlaneAnn. Math.62374410zbMATHMathSciNetCrossRefGoogle Scholar
  49. 49.
    Flensberg, K., Svensmark, H. 1993Scaling relations for forced oscillators at the transition from a dissipative to a Hamiltonian systemPhys. Rev. E47289305CrossRefGoogle Scholar
  50. 50.
    Kuznetsov, S.P., Pikovsky, A.S. 1986Universality and scaling of period-doubling bifurcations in a dissipative distributed mediumPhysica D19384396CrossRefADSMathSciNetzbMATHGoogle Scholar
  51. 51.
    Kuznetsov, A.P., Kuznetsov, S.P., Sataev, I.R., Chua, L.O. 1996Multi-parameter criticality in Chua’s circuit at period-doubling transition to chaosInt. J. Bifurcation Chaos6119148MathSciNetzbMATHGoogle Scholar
  52. 52.
    Kim, S.-Y., Lim, W. 2001Bicritical scaling behavior in unidirectionally coupled oscillatorsPhys. Rev. E63036223ADSGoogle Scholar
  53. 53.
    A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization. A universal concept in nonlinear sciences (Cambridge Univ. Press, 2002).Google Scholar
  54. 54.
    Genot, M. 1993Application of 1D Chua’s map From Chua’s circuit: A pictorial guideJ. Circuits, Systems and Computers3431440MathSciNetGoogle Scholar
  55. 55.
    Kuznetsov, A.P., Kuznetsov, S.P., Sataev, I.R., Chua, L.O. 1993Two-parameter study of transition to chaos in Chua’s circuit: renormalization group, universality and scalingInt. J. Bifurcation Chaos3943962MathSciNetzbMATHGoogle Scholar
  56. 56.
    Kuznetsov, S.P., Sataev, I.R. 1992New types of critical dynamics for two-dimensional mapsPhys. Lett. A162236242ADSMathSciNetGoogle Scholar
  57. 57.
    Kuznetsov, A.P., Savin, A.V., Kim, S.-Y. 2004On the Criticality of the FQ-Type in the System of Coupled Maps with Period-DoublingNonlinear Phenomena in Complex Systems76977Google Scholar
  58. 58.
    Kuznetsov, S.P., Sataev, I.R. 2001Universality and scaling for the breakup of phase synchronization at the onset of chaos in a periodically driven Rössler oscillatorPhys. Rev. E64046214CrossRefADSGoogle Scholar
  59. 59.
    Landau, L.D. 1944On the problem of turbulenceDoklady Akademii Nauk SSSR44311(In Russian.)zbMATHGoogle Scholar
  60. 60.
    Hopf, E. 1948A mathematical example displaying features of turbulenceCommun. Appl. Math.1303322zbMATHMathSciNetGoogle Scholar
  61. 61.
    Ruelle, D., Takens, F. 1971On the nature of turbulenceCommun. Math. Phys.20167192CrossRefMathSciNetzbMATHADSGoogle Scholar
  62. 62.
    Thompson, J.M.T., Stewart, H.B. 1987Nonlinear Dynamics and Chaos. Geometrical Methods for Engineers and ScientistsJohn Wiley and SonsChichester – NY – Brisbane – Toronto –SingaporeGoogle Scholar
  63. 63.
    S. P. Kuznetsov, Dynamical Chaos (Moscow, Fizmatlit, 2001). (In Russian.)Google Scholar
  64. 64.
    Kuznetsov, A.P., Kuznetsova, A.Yu., Sataev, I.R. 2003On critical behavior of a map with Neimark–Sacker bifurcation at destruction of the phase syncronization at the limit point of the Feigenbaum cascadeIzvestija VUZov – Prikladnaja Nelineinaja Dinamika (Saratov)111218(In Russian.)zbMATHGoogle Scholar
  65. 65.
    Collet, P., Coullet, P., Tresser, C. 1985Scenarios under constraintJ. Physique Lett.46L143L147Google Scholar
  66. 66.
    Gallas, J.A.C. 1993Degenerate routes to chaosPhys. Rev. E4841564159CrossRefADSMathSciNetGoogle Scholar
  67. 67.
    Hu, B., Mao, J.M. 1982Period doubling: Universality and critical-point orderPhys. Rev. A2532593261ADSMathSciNetGoogle Scholar
  68. 68.
    Hu, B., Satija, I.I. 1983A spectrum of universality classes in period-doubling and period triplingPhys. Lett. A98143146ADSMathSciNetGoogle Scholar
  69. 69.
    Hauser, P.R., Tsallis, C., Curado, E.M.F. 1984Criticality of rouds to chaos of the 1−a|x| z Phys. Rev. A3020742079CrossRefADSMathSciNetGoogle Scholar
  70. 70.
    Lubimov, D.V., Pikovsky, A.S., Zaks, M.A. 1988Universalities and scaling at the transition to chaos through homoclinic bifurcationsShirkov, D.V.Kazakov, D.I.Vladimirov, A.A eds. Renormalization Group.World ScientificSingapore, New Jersey, Hong Kong278289Google Scholar
  71. 71.
    Gambaudo, J.-M., Procaccia, I., Thomae, S., Tresser, C. 1986New universal scenarios for the onset of chaos in Lorenz-type flowsPhys. Rev. Lett.57925928CrossRefADSMathSciNetGoogle Scholar
  72. 72.
    Zaks, M.A., Lubimov, D.V. 1989Bifurcation sequences in the dissipative systems with saddle equilibriaDynamical Systems and Ergodic Theory23367380(Banach Center Publications, Warszawa)Google Scholar
  73. 73.
    Eckmann, J.-P., Epstein, H. 1990Bounds on the unstable eigenvalue for period doublingCommun. Math. Phys.128427435CrossRefMathSciNetzbMATHADSGoogle Scholar
  74. 74.
    Briggs, K.M., Dixon, T.W., Szekeres, G. 1998Analytic solution of the Cvitanović-Feigenbaum and Feigenbaum-Kadanoff-Shenker equationsInt. J. Bifurcation Chaos8347357MathSciNetzbMATHGoogle Scholar
  75. 75.
    Arneodo, A., Coullet, P., Tresser, V. 1979A renormalization group with periodic behaviourPhys. Lett. A707476CrossRefADSMathSciNetGoogle Scholar
  76. 76.
    Mestel, B.D., Osbaldestin, A.H. 1998Feigenbaum theory for unimodal maps with asymmetric critical point: rigorous resultsCommun. Math. Phys.197211228CrossRefADSMathSciNetzbMATHGoogle Scholar
  77. 77.
    Mestel, B.D., Osbaldestin, A.H., Tsygvintsev, A.V. 2004Bounds on the Unstable Eigenvalue for the Asymmetric Renormalization Operator for Period DoublingCommun. Math. Phys.250241257CrossRefADSMathSciNetzbMATHGoogle Scholar
  78. 78.
    Jensen, R.V., Ma, L.K.H. 1985Nonuniversal behavior of asymmetric unimodal mapsPhys. Rev. A3139933995CrossRefADSMathSciNetGoogle Scholar
  79. 79.
    Oldeman, B.E., Krauskopf, B., Champneys, A.R. 2000Death of period-doublings: locating the homoclinic-doubling cascadePhysica D146100120CrossRefADSMathSciNetzbMATHGoogle Scholar
  80. 80.
    Chang, S.-J., Wortis, M., Wright, J.A. 1981Iterative properties of a one-dimensional quartic map: Critical lines and tricritical behaviorPhys. Rev. A2426692684ADSMathSciNetGoogle Scholar
  81. 81.
    Fraser, S., Kapral, R. 1984Universal vector scaling in one-dimensional mapsPhys. Rev. A3010171025CrossRefADSMathSciNetGoogle Scholar
  82. 82.
    Kuznetsov, S.P. 1992Tricriticality in two-dimensional mapsPhys. Lett. A169438444CrossRefADSMathSciNetGoogle Scholar
  83. 83.
    Kuznetsov, A.P., Kuznetsov, S.P., Turukina, L.V., Mosekilde, E. 2001Two-parameter analysis of the scaling behavior at the onset of chaos: Tricritical and pseudo-tricritical pointsPhysica A300367385CrossRefADSzbMATHGoogle Scholar
  84. 84.
    MacKay, R.S., Tresser, C. 1988Boundary of topological chaos for bimodal maps of the intervalJ. London Math. Soc.37164181MathSciNetzbMATHGoogle Scholar
  85. 85.
    MacKay, R.S., Zeijts, J.B.J. 1988Period doubling for bimodal maps: a horseshoe for a renormalisation operatorNonlinearity1253277CrossRefADSMathSciNetzbMATHGoogle Scholar
  86. 86.
    Kuznetsov, A.P., Kuznetsov, S.P., Sataev, I.R. 1994From bimodal one-dimensional maps to Hénon-like two-dimensional maps: does quantitative universality survive?Phys. Lett. A164413421ADSMathSciNetGoogle Scholar
  87. 87.
    Carcasses, J., Mira, C., Bosch, M., Simo, C., Tatjer, J.C. 1991Crossroad area–spring area transition (I) Parameter plane representationInt. J. Bifurcation Chaos1183196MathSciNetzbMATHGoogle Scholar
  88. 88.
    Milnor, J. 1992Remarks on iterated cubic mapsExperimental Mathematics1524ADSzbMATHMathSciNetGoogle Scholar
  89. 89.
    Hunt, B.R., Gallas, J.A.C., Grebogi, C., Yorke, J.A., Koçak, H. 1999Bifurcation RigidityPhysica D1293556CrossRefADSMathSciNetzbMATHGoogle Scholar
  90. 90.
    Kuznetsov, A.P., Kuznetsov, S.P., Sataev, I.R. 1994Three-parameter scaling for one-dimensional mapsPhys. Lett. A189367373CrossRefADSMathSciNetzbMATHGoogle Scholar
  91. 91.
    Briggs, K.M., Quispel, G.R.W., Thompson, C.J. 1991Feigenvalues for MandelsetsJ. Physics A: Math. General2433633368ADSMathSciNetzbMATHGoogle Scholar
  92. 92.
    K. M. Briggs, Feigenbaum scaling in discrete dynamical systems. PhD Dissertation (University of Melbourne, 1997).Google Scholar
  93. 93.
    Kuznetsov, S.P. 1996Cascade of period doubling in complex cubic mapIzvestija VUZov – Prikladnaja Nelineinaja Dinamika (Saratov)4312(In Russian.)zbMATHGoogle Scholar
  94. 94.
    Golberg, A.I., Sinai, Ya.G., Khanin, K.M. 1983Universal properties for the period-tripling bifurcationsUspekhi Matem. Nauk38159160MathSciNetGoogle Scholar
  95. 95.
    Cvitanović, P., Myrheim, J. 1983Universality for period n-tupling in complex mappingsPhys. Lett A94329333ADSMathSciNetGoogle Scholar
  96. 96.
    Mandelbrot, B.B. 1983On the quadratic mapping zz 2−μ for complex μ and z: The fractal structure of its M set, and scalingPhysica D7224239CrossRefADSMathSciNetGoogle Scholar
  97. 97.
    Isaeva, O.B., Kuznetsov, S.P. 2000On scaling properties of two-dimensional maps near the accumulation point of the period-tripling cascadeRegular and Chaotic Dynamics5459476CrossRefMathSciNetzbMATHGoogle Scholar
  98. 98.
    Mao, J.M., Satija, I.I., Hu, B. 1986Period doubling in four-dimensional symplectic mapsPhys. Rev. A3443254332CrossRefADSMathSciNetGoogle Scholar
  99. 99.
    Lahiri, A. 1992Inverted period-doubling sequences in four-dimensional reversible maps and solutions to the renormalization equationsPhys. Rev. A45757762CrossRefADSMathSciNetGoogle Scholar
  100. 100.
    Shenker, S.J. 1982Scaling behavior in a map of a circle onto itself: Empirical resultsPhysica D5405411CrossRefADSMathSciNetGoogle Scholar
  101. 101.
    Feigenbaum, M.J., Kadanoff, L.P., Shenker, S.J. 1982Quasiperiodicity in dissipative systems: A renormalization group analysisPhysica D5370386CrossRefADSMathSciNetGoogle Scholar
  102. 102.
    Ostlund, S., Rand, D., Sethna, J., Siggia, E. 1983Universal properties of the transition from quasi-periodicity to chaos in dissipative systemsPhysica D8303342CrossRefADSMathSciNetzbMATHGoogle Scholar
  103. 103.
    Wang, X., Mainieri, R., Lowenstein, J.H. 1989Circle-map scaling in a two-dimensional settingPhys. Rev. A4053825389ADSMathSciNetGoogle Scholar
  104. 104.
    Stavans, J., Heslot, F., Libchaber, A. 1985Fixed Winding Number and the Quasiperiodic Route to Chaos in a Convective FluidPhys. Rev. Lett.55596599ADSGoogle Scholar
  105. 105.
    Bauer, M., Krueger, U., Martienssen, W. 1989Experimental studies of mode-locking and circle maps in inductively shunted Josephson-junctionsEurophysics Letters9191196ADSGoogle Scholar
  106. 106.
    Glazier, J.A., Libchaber, A. 1988Quasi-periodicity and dynamical systems – An experimentalists viewIEEE Trans. Circuits Systems35790809CrossRefMathSciNetGoogle Scholar
  107. 107.
    Farmer, J.D., Satija, I.I. 1985Renormalization of the quasiperiodic transition to chaos for arbitrary winding numbersPhys. Rev. A3135203522CrossRefADSMathSciNetGoogle Scholar
  108. 108.
    Zaks, M.A., Pikovsky, A.S. 1992Farey level separation in mode-locking structure of circle mappingsPhysica D59255269ADSMathSciNetzbMATHGoogle Scholar
  109. 109.
    Ketoja, J.A. 1992Renormalization in a circle map with two inflection pointsPhysica D554568CrossRefADSzbMATHMathSciNetGoogle Scholar
  110. 110.
    Tseng, H.-C., Tai, M.-F., Chen, H.-J., Li, P.-C., Chou, C.-H., Hu, C.-K. 1999Some scaling behaviors in a circle map with two inflection pointsInt. J. Modern Phys. B1331493158ADSGoogle Scholar
  111. 111.
    Delbourgo, R., Kenny, B.G. 1990Fractal dimension associated with a critical circle map with an arbitrary-order inflection pointPhys. Rev. A4262306233CrossRefADSMathSciNetGoogle Scholar
  112. 112.
    Shenker, S.J., Kadanoff, L.P. 1982Critical Behavior of a KAM Surface. I. Empirical ResultsJ. Stat. Phys.27631656CrossRefMathSciNetADSGoogle Scholar
  113. 113.
    Kadanoff, L.P. 1981Scaling for a critical Kolmogorov-Arnold-Moser trajectoryPhys. Rev. Lett. A4716411634ADSzbMATHMathSciNetCrossRefGoogle Scholar
  114. 114.
    MacKay, R.S. 1983A renormalization approach to invariant circles in area-preserving mapsPhysica D7283300CrossRefADSMathSciNetGoogle Scholar
  115. 115.
    Widom, M., Kadanoff, L.P. 1982Renormalization group analysis of bifurcations in area-preserving mapsPhysica D5287292CrossRefADSMathSciNetGoogle Scholar
  116. 116.
    Wilbrink, J. 2003New fixed point of the renormalisation operator associated with the recurrence of invariant circles in generic Hamiltonian mapsNonlinearity1615651571MathSciNetGoogle Scholar
  117. 117.
    Del-Castillo-Negrete, D., Greene, J.M., Morrison, P.J. 1997Renormalization and transition to chaos in area preserving nontwist mapsPhysica D100311329ADSMathSciNetzbMATHGoogle Scholar
  118. 118.
    Greene, J.M., Mao, J. 1990Higher-order fixed points of the renormalisation operator for invariant circlesNonlinearity36978CrossRefADSMathSciNetzbMATHGoogle Scholar
  119. 119.
    Dixon, T.W., Gherghetta, T., Kenny, B.G. 1996Universality in the quasiperiodic route to chaosCHAOS63242CrossRefADSMathSciNetzbMATHGoogle Scholar
  120. 120.
    Chirikov, B.V., Shepelyansky, D.L. 1988Chaos border and statistical anomaliesShirkov, D.V.Kazakov, D.I.Vladimirov, A.A eds. Renormalization Group.World ScientificSingapore, New Jersey, Hong Kong221250Google Scholar
  121. 121.
    Grebogi, C., Ott, E., Pelikan, S., Yorke, J.A. 1984Strange attractors that are not chaoticPhysica D13261268CrossRefADSMathSciNetzbMATHGoogle Scholar
  122. 122.
    Kuznetsov, S.P., Pikovsky, A.S., Feudel, U. 1995Birth of a strange nonchaotic attractor: A renormalization group analysisPhys. Rev. E5116291632CrossRefADSMathSciNetGoogle Scholar
  123. 123.
    Kuznetsov, S., Feudel, U., Pikovsky, A. 1998Renormalization group for scaling at the torus-doubling terminal pointPhys. Rev. E5715851590CrossRefADSMathSciNetGoogle Scholar
  124. 124.
    Kuznetsov, S.P., Neumann, E., Pikovsky, A., Sataev, I.R. 2000Critical point of tori collision in quasiperiodically forced systemsPhys. Rev. E6219952007CrossRefADSMathSciNetGoogle Scholar
  125. 125.
    Kuznetsov, S.P. 2002Torus fractalization and intermittencyPhys. Rev. E65066209CrossRefADSMathSciNetGoogle Scholar
  126. 126.
    Pomeau, Y., Manneville, P. 1980Intermittent transition to turbulence in dissipative dynamical systemsCommun. Math. Phys.74189197CrossRefMathSciNetADSGoogle Scholar
  127. 127.
    Hu, B., Rudnik, J. 1982Exact solution of the Feigenbaum renormalization group equations for intermittencyPhys. Rev. Lett. A2630353036ADSGoogle Scholar
  128. 128.
    Hirsch, J.E., Huberman, B.A., Scalapino, D.J. 1982Theory of intermittencyPhys. Rev. A25519532CrossRefADSGoogle Scholar
  129. 129.
    Hu, B., Rudnik, J. 1986Differential-equation approach to functional equations: exact solutions intermittencyPhys. Rev. A3424532457ADSGoogle Scholar
  130. 130.
    Procaccia, I., Schuster, H. 1983Functional renormalization-group theory of universal 1/f noise in dynamical systemsPhys. Rev. A2812101212CrossRefADSMathSciNetGoogle Scholar
  131. 131.
    Zisook, A.B. 1982Intermittency in area-preserving mappingsPhys. Rev. A2522892292ADSMathSciNetGoogle Scholar
  132. 132.
    Zisook, A.B., Shenker, S.J. 1982Renormalization group for intermittency in area-preserving mapsPhys. Rev. A2528242826ADSMathSciNetGoogle Scholar
  133. 133.
    Zisook, A.B. 1984The Complete Set of Hamiltonian Intermittency Scaling BehaviorsCommun. Math. Phys.96361371CrossRefzbMATHMathSciNetADSGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • S. P. Kuznetsov
    • 1
    • 2
    Email author
  • A. P. Kuznetsov
    • 1
    • 2
  • I. R. Sataev
    • 1
  1. 1.Institute of Radio-Engineering and Electronics of RASSaratovRussian Federation
  2. 2.Faculty of Nonlinear ProcessesSaratov State UniversitySaratovRussian Federation

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