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Volumn 121, Issue 5-6, 2005, Pages 697-748

Multiparameter critical situations, universality and scaling in two-dimensional period-doubling maps

Author keywords

Coupled maps; Coupled oscillators; Multi parameter analysis; Onset of chaos; Period doubling; Renormalization group; Scaling; Universality

Indexed keywords


EID: 29044435646     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10955-005-6973-6     Document Type: Article
Times cited : (20)

References (133)
  • 1
    • 0001816458 scopus 로고
    • Quantitative universality for a class of nonlinear transformations
    • M. J. Feigenbaum, Quantitative universality for a class of nonlinear transformations, J. Stat. Phys. 19:25-52 (1978).
    • (1978) J. Stat. Phys. , vol.19 , pp. 25-52
    • Feigenbaum, M.J.1
  • 2
    • 0347447535 scopus 로고
    • The universal metric properties of nonlinear transformations
    • M. J. Feigenbaum, The universal metric properties of nonlinear transformations, J. Stat. Phys. 21:669-706 (1979).
    • (1979) J. Stat. Phys. , vol.21 , pp. 669-706
    • Feigenbaum, M.J.1
  • 5
    • 84927989803 scopus 로고
    • Feigenbaum universality and thermodynamic formalism
    • E. B. Vul, Ya. G. Sinai, and K. M. Khanin, Feigenbaum universality and thermodynamic formalism, Russ. Math. Sun. 39:1-40 (1984).
    • (1984) Russ. Math. Sun. , vol.39 , pp. 1-40
    • Vul, E.B.1    Sinai, Ya.G.2    Khanin, K.M.3
  • 6
    • 0001623717 scopus 로고
    • Fluctuations and the onset of chaos
    • J. P. Crutchfleld and B. A. Huberman, Fluctuations and the onset of chaos, Phys. Lett. A 77:407-410 (1980).
    • (1980) Phys. Lett. A , vol.77 , pp. 407-410
    • Crutchfleld, J.P.1    Huberman, B.A.2
  • 8
    • 0001404303 scopus 로고
    • Structure of the parameter space of the Hénon map
    • J. A. C. Gallas, Structure of the parameter space of the Hénon map, Phys. Rev. Lett. 70:2714-2717 (1993).
    • (1993) Phys. Rev. Lett. , vol.70 , pp. 2714-2717
    • Gallas, J.A.C.1
  • 9
    • 0542360754 scopus 로고
    • Oscillatory instabilities leading to "optical turbulence" in a bistable ring cavity
    • H. J. Carmichael, R. R. Snapp, and W. C. Schieve, Oscillatory instabilities leading to "optical turbulence" in a bistable ring cavity, Phys. Rev. A 26:3408-3422 (1982).
    • (1982) Phys. Rev. A , vol.26 , pp. 3408-3422
    • Carmichael, H.J.1    Snapp, R.R.2    Schieve, W.C.3
  • 10
    • 0001634226 scopus 로고
    • Fine structure of phase locking
    • L. Glass and R. Perez, Fine Structure of Phase Locking, Phys. Rev. Lett. 48:1772-1775 (1982).
    • (1982) Phys. Rev. Lett. , vol.48 , pp. 1772-1775
    • Glass, L.1    Perez, R.2
  • 11
    • 0003180475 scopus 로고
    • Frehlich Transition to chaos in the Duffing oscillator
    • S. R. G. Novak, Frehlich Transition to chaos in the Duffing oscillator, Phys. Rev. A 26:3660-3663 (1982).
    • (1982) Phys. Rev. A , vol.26 , pp. 3660-3663
    • Novak, S.R.G.1
  • 12
    • 0018494738 scopus 로고
    • A magnetoelastic strange attractor
    • W. Moon and P. J. Holmes, A magnetoelastic strange attractor, J. Sound Vib. 65:285-296 (1979).
    • (1979) J. Sound Vib. , vol.65 , pp. 285-296
    • Moon, W.1    Holmes, P.J.2
  • 13
    • 0001282865 scopus 로고
    • Evidence for universal chaotic behavior of a driven nonlinear oscillator
    • J. Testa, J. Perez, and C. Jeffries, Evidence for universal chaotic behavior of a driven nonlinear oscillator, Phys. Rev. Lett. 48:714-717 (1982).
    • (1982) Phys. Rev. Lett. , vol.48 , pp. 714-717
    • Testa, J.1    Perez, J.2    Jeffries, C.3
  • 14
    • 0040763059 scopus 로고
    • Topology of the invariant manifolds of a period-doubling attractors for some forced nonlinear oscillators
    • P. Beiersdorfer, J. M. Wersinger, Topology of the Invariant Manifolds of a Period-doubling Attractors for Some Forced Nonlinear Oscillators, Phys. Lett. A 96:269-272 (1983).
    • (1983) Phys. Lett. A , vol.96 , pp. 269-272
    • Beiersdorfer, P.1    Wersinger, J.M.2
  • 15
    • 0042517510 scopus 로고    scopus 로고
    • On the period-doubling scenario in dynamical systems with time delay
    • Wissenschaft and Technik
    • 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).
    • (1997) Proceedings of the 15th IMACS World Congress 1997 , vol.1 , pp. 215-220
    • Schanz, M.1    Pelster, A.2
  • 19
    • 0032330473 scopus 로고    scopus 로고
    • Transition to the chaotic regime in a system composed of an electron beam and an inverse electromagnetic wave
    • N. M. Ryskin, V. N. Titov, and D. I. Trubetskov, Transition to the Chaotic Regime in a System Composed of an Electron Beam and an Inverse Electromagnetic Wave, Doklady Physics 43:90-93 (1998).
    • (1998) Doklady Physics , vol.43 , pp. 90-93
    • Ryskin, N.M.1    Titov, V.N.2    Trubetskov, D.I.3
  • 20
    • 0018505940 scopus 로고
    • Rayleigh-Bénard experiment in liquid helium: Frequency locking and the onset of turbulence
    • J. Maurer and A. Libchaber, Rayleigh-Bénard experiment in liquid helium: frequency locking and the onset of turbulence, J. de Physique Lettres 40:419-423 (1979).
    • (1979) J. de Physique Lettres , vol.40 , pp. 419-423
    • Maurer, J.1    Libchaber, A.2
  • 21
    • 0020114162 scopus 로고
    • Period doubling in mercury, a quantative measurement
    • A. Libchaber, C. Laroche, and S. Fauve, Period doubling in mercury, a quantative measurement, J. de Phys. Lett. 43:211-216 (1982).
    • (1982) J. de Phys. Lett. , vol.43 , pp. 211-216
    • Libchaber, A.1    Laroche, C.2    Fauve, S.3
  • 23
    • 29144481818 scopus 로고
    • Experimental evidence of sub-harmonic bifurcations, multistability, and turbulence in a q-switched gas laser
    • F. T. Arecci, R. Meucci, G. Puccioni, and J. Tredicce, Experimental Evidence of Sub-harmonic Bifurcations, Multistability, and Turbulence in a Q-Switched Gas Laser, Phys. Rev. Lett. 49:1217-1220 (1982).
    • (1982) Phys. Rev. Lett. , vol.49 , pp. 1217-1220
    • Arecci, F.T.1    Meucci, R.2    Puccioni, G.3    Tredicce, J.4
  • 24
    • 0010766147 scopus 로고
    • Route to chaos in an acousto-optic bistable device
    • R. Vallée and C. Delisle, Route to chaos in an acousto-optic bistable device, Phys. Rev. A 31:2390-2396 (1985).
    • (1985) Phys. Rev. A , vol.31 , pp. 2390-2396
    • Vallée, R.1    Delisle, C.2
  • 29
    • 0001476091 scopus 로고
    • Existence of a fixed point of the doubling transformation for area-preserving maps of the plane
    • J.-P. Eckmann, H. Koch, and P. Wittwer, Existence of a fixed point of the doubling transformation for area-preserving maps of the plane, Phys. Rev. A 26:720-722 (1982).
    • (1982) Phys. Rev. A , vol.26 , pp. 720-722
    • Eckmann, J.-P.1    Koch, H.2    Wittwer, P.3
  • 30
    • 29044439748 scopus 로고
    • Universal functional equation for period doubling in constant-Jacobian maps
    • G. R. W Quispel, Universal functional equation for period doubling in constant-Jacobian maps, Phys. Lett. A 118:457-462 (1986).
    • (1986) Phys. Lett. A , vol.118 , pp. 457-462
    • Quispel, G.R.W.1
  • 31
    • 0002500575 scopus 로고    scopus 로고
    • A variety of period-doubling universality classes in multi-parameter analysis of transition to chaos
    • A. P. Kuznetsov, S. P. Kuznetsov and I. R. Sataev, A variety of period-doubling universality classes in multi-parameter analysis of transition to chaos, Physica D 109:91-112 (1997). See also: http://sgtnd.narod.ru/science/alphabet/eng/index.htm.
    • (1997) Physica D , vol.109 , pp. 91-112
    • Kuznetsov, A.P.1    Kuznetsov, S.P.2    Sataev, I.R.3
  • 32
    • 0346932128 scopus 로고
    • Period doubling as a universal route to stochasticity
    • eds. W. Horton, L. E. Reichl, and V. Szebehely (J. Wiley & Sons, New York)
    • R. S. MacKay, Period doubling as a universal route to stochasticity. In book: Long time prediction in dynamics, eds. W. Horton, L. E. Reichl, and V. Szebehely (J. Wiley & Sons, New York, 1983).
    • (1983) Book: Long Time Prediction in Dynamics
    • MacKay, R.S.1
  • 33
    • 4243410350 scopus 로고
    • Universal transition between Hamiltonian and dissipative chaos
    • C. Chen, G. Gyorgyi, and G. Schmidt, Universal transition between Hamiltonian and dissipative chaos, Phys. Rev. A 34:2568-2570 (1986).
    • (1986) Phys. Rev. A , vol.34 , pp. 2568-2570
    • Chen, C.1    Gyorgyi, G.2    Schmidt, G.3
  • 34
    • 0000237730 scopus 로고
    • Universal behavior in families of area preserving maps
    • J. M. Greene, R. S. MacKay, F. Vivaldi, and M. J. Feigenbaum, Universal behavior in families of area preserving maps, Physica D 3:468-486 (1981).
    • (1981) Physica D , vol.3 , pp. 468-486
    • Greene, J.M.1    MacKay, R.S.2    Vivaldi, F.3    Feigenbaum, M.J.4
  • 35
    • 22244435161 scopus 로고
    • Renormalization group analysis of bifurcations in area-preserving maps
    • M. Widom and L. P. Kadanoff, Renormalization group analysis of bifurcations in area-preserving maps, Physica D 5:287-292 (1982).
    • (1982) Physica D , vol.5 , pp. 287-292
    • Widom, M.1    Kadanoff, L.P.2
  • 36
    • 0007267636 scopus 로고
    • A physical interpretation of the Henon map
    • J. F. Heagy, A physical interpretation of the Henon map, Physica D 57:436-446 (1992).
    • (1992) Physica D , vol.57 , pp. 436-446
    • Heagy, J.F.1
  • 37
    • 23544458663 scopus 로고
    • Universal scaling in dissipative systems
    • C. Chen, G. Gyorgyi, and G. Schmidt, Universal scaling in dissipative systems, Phys. Rev. A 35:2660-2668 (1987).
    • (1987) Phys. Rev. A , vol.35 , pp. 2660-2668
    • Chen, C.1    Gyorgyi, G.2    Schmidt, G.3
  • 38
    • 4243484844 scopus 로고
    • Universal corrections to parameter scaling in period-doubling systems: Multiple scaling and crossover
    • C. Reick, Universal corrections to parameter scaling in period-doubling systems: Multiple scaling and crossover, Phys. Rev. A 45:777-792 (1992).
    • (1992) Phys. Rev. A , vol.45 , pp. 777-792
    • Reick, C.1
  • 39
    • 0001479650 scopus 로고
    • New type of critical behavior of coupled systems at the transition to chaos
    • B. P. Bezruchko, Yu. V. Gulyaev, S. P. Kuznetsov, and E. P. Seleznev, New type of critical behavior of coupled systems at the transition to chaos, Sov. Phys. Dokl. 31:258-260 (1986).
    • (1986) Sov. Phys. Dokl. , vol.31 , pp. 258-260
    • Bezruchko, B.P.1    Gulyaev, Yu.V.2    Kuznetsov, S.P.3    Seleznev, E.P.4
  • 40
    • 0000129456 scopus 로고
    • Bicritical dynamics of period-doubling systems with unidirectional coupling
    • A. P. Kuznetsov, S. P. Kuznetsov, and I. R. Sataev, Bicritical dynamics of period-doubling systems with unidirectional coupling, Int. J. Bifurcation Chaos 1:839-848 (1991).
    • (1991) Int. J. Bifurcation Chaos , vol.1 , pp. 839-848
    • Kuznetsov, A.P.1    Kuznetsov, S.P.2    Sataev, I.R.3
  • 41
    • 0001555345 scopus 로고
    • Spatial period-doubling in open flow
    • K. Kaneko, Spatial period-doubling in open flow, Phys. Lett. A 111:321-325 (1985).
    • (1985) Phys. Lett. A , vol.111 , pp. 321-325
    • Kaneko, K.1
  • 42
    • 0024088719 scopus 로고
    • The onset and spatial development of turbulence in flow systems
    • I. S. Aranson, A. V. Gaponov-Grekhov, and M. I. Rabinovich, The onset and spatial development of turbulence in flow systems, Physica D 33:1-20 (1988).
    • (1988) Physica D , vol.33 , pp. 1-20
    • Aranson, I.S.1    Gaponov-Grekhov, A.V.2    Rabinovich, M.I.3
  • 43
    • 0034817519 scopus 로고    scopus 로고
    • Characterization of chaotic attractors at bifurcations in Murali-Lakshmanan-Chua's circuit and one-way coupled map lattice system
    • P. Philominathan and P. Neelamegam, Characterization of chaotic attractors at bifurcations in Murali-Lakshmanan-Chua's circuit and one-way coupled map lattice system, Chaos, Soliton and Fractals 12:1005-1017 (2001).
    • (2001) Chaos, Soliton and Fractals , vol.12 , pp. 1005-1017
    • Philominathan, P.1    Neelamegam, P.2
  • 44
    • 0036732432 scopus 로고    scopus 로고
    • ON/OFF phase shift keying for chaos-encrypted communication using external-cavity semiconductor lasers
    • T. Heil, J. Mulet, I. Fischer, C. R. Mirasso, M. Peil, P. Colet, and W. Elsäber, ON/OFF Phase Shift Keying for Chaos-Encrypted Communication Using External-Cavity Semiconductor Lasers, IEEE J. Quant. Electronics 38:1162-1170 (2002).
    • (2002) IEEE J. Quant. Electronics , vol.38 , pp. 1162-1170
    • Heil, T.1    Mulet, J.2    Fischer, I.3    Mirasso, C.R.4    Peil, M.5    Colet, P.6    Elsäber, W.7
  • 45
    • 37649026967 scopus 로고    scopus 로고
    • Chaos-based secure communications in a large community
    • S. Wang, J. Kuang, J. Li, Y. Luo, H. Lu and G. Hu, Chaos-based secure communications in a large community, Phys. Rev. E 66:065202 (2002).
    • (2002) Phys. Rev. E , vol.66 , pp. 065202
    • Wang, S.1    Kuang, J.2    Li, J.3    Luo, Y.4    Lu, H.5    Hu, G.6
  • 46
    • 4243221295 scopus 로고
    • Variety of types of critical behavior and multistability in period doubling systems with unidirectional coupling near the onset of chaos
    • A. P. Kuznetsov, S. P. Kuznetsov, and I. R. Sataev, Variety of types of critical behavior and multistability in period doubling systems with unidirectional coupling near the onset of chaos, Int. J. Bifurcation Chaos 3:139-152 (1993).
    • (1993) Int. J. Bifurcation Chaos , vol.3 , pp. 139-152
    • Kuznetsov, A.P.1    Kuznetsov, S.P.2    Sataev, I.R.3
  • 47
    • 4243901540 scopus 로고    scopus 로고
    • Period-doubling for two- Dimensional non-invertible maps: Renormalization group analysis and quantitative universality
    • S. P. Kuznetsov and I. R. Sataev, Period-doubling for two- dimensional non-invertible maps: Renormalization group analysis and quantitative universality, Physica D 101:249-269 (1997).
    • (1997) Physica D , vol.101 , pp. 249-269
    • Kuznetsov, S.P.1    Sataev, I.R.2
  • 48
    • 0000770380 scopus 로고
    • On singularities of mappings of euclidean spaces. I. Mappings of the plane into the plane
    • H. Whitney, On Singularities of Mappings of Euclidean Spaces. I. Mappings of the Plane into the Plane, Ann. Math. 62:374-410 (1955).
    • (1955) Ann. Math. , vol.62 , pp. 374-410
    • Whitney, H.1
  • 49
    • 30644456858 scopus 로고
    • Scaling relations for forced oscillators at the transition from a dissipative to a Hamiltonian system
    • K. Flensberg and H. Svensmark, Scaling relations for forced oscillators at the transition from a dissipative to a Hamiltonian system, Phys. Rev. E 47:289-305 (1993).
    • (1993) Phys. Rev. E , vol.47 , pp. 289-305
    • Flensberg, K.1    Svensmark, H.2
  • 50
    • 0001341410 scopus 로고
    • Universality and scaling of period-doubling bifurcations in a dissipative distributed medium
    • S. P. Kuznetsov and A.S.Pikovsky, Universality and scaling of period-doubling bifurcations in a dissipative distributed medium, Physica D 19:384-396 (1986).
    • (1986) Physica D , vol.19 , pp. 384-396
    • Kuznetsov, S.P.1    Pikovsky, A.S.2
  • 51
    • 0029682875 scopus 로고    scopus 로고
    • Multi-parameter criticality in Chua's circuit at period-doubling transition to chaos
    • A. P. Kuznetsov, S. P. Kuznetsov, I. R. Sataev, and L. O. Chua, Multi-parameter criticality in Chua's circuit at period-doubling transition to chaos, Int. J. Bifurcation Chaos 6:119-148 (1996).
    • (1996) Int. J. Bifurcation Chaos , vol.6 , pp. 119-148
    • Kuznetsov, A.P.1    Kuznetsov, S.P.2    Sataev, I.R.3    Chua, L.O.4
  • 52
    • 0035275671 scopus 로고    scopus 로고
    • Bicritical scaling behavior in unidirectionally coupled oscillators
    • S.-Y. Kim and W. Lim, Bicritical scaling behavior in unidirectionally coupled oscillators, Phys. Rev. E 63:036223 (2001).
    • (2001) Phys. Rev. E , vol.63 , pp. 036223
    • Kim, S.-Y.1    Lim, W.2
  • 54
    • 0347208502 scopus 로고
    • Application of ID Chua's map from Chua's circuit: A pictorial guide
    • M. Genot, Application of ID Chua's map From Chua's circuit: A pictorial guide, J. Circuits, Systems and Computers 3:431-440 (1993).
    • (1993) J. Circuits, Systems and Computers , vol.3 , pp. 431-440
    • Genot, M.1
  • 55
    • 0346578272 scopus 로고
    • Two-parameter study of transition to chaos in Chua's circuit: Renormalization group, universality and scaling
    • A. P. Kuznetsov, S. P. Kuznetsov, I. R. Sataev, and L. O. Chua, Two-parameter study of transition to chaos in Chua's circuit: renormalization group, universality and scaling, Int. J. Bifurcation Chaos 3:943-962 (1993).
    • (1993) Int. J. Bifurcation Chaos , vol.3 , pp. 943-962
    • Kuznetsov, A.P.1    Kuznetsov, S.P.2    Sataev, I.R.3    Chua, L.O.4
  • 56
    • 0000370984 scopus 로고
    • New types of critical dynamics for two-dimensional maps
    • S. P. Kuznetsov and I. R. Sataev, New types of critical dynamics for two-dimensional maps, Phys. Lett. A 162:236-242 (1992).
    • (1992) Phys. Lett. A , vol.162 , pp. 236-242
    • Kuznetsov, S.P.1    Sataev, I.R.2
  • 57
    • 29044434529 scopus 로고    scopus 로고
    • On the criticality of the FQ-type in the system of coupled maps with period-doubling
    • A. P. Kuznetsov, A. V. Savin, and S.-Y. Kim, On the Criticality of the FQ-Type in the System of Coupled Maps with Period-Doubling, Nonlinear Phenomena in Complex Systems 7:69-77 (2004).
    • (2004) Nonlinear Phenomena in Complex Systems , vol.7 , pp. 69-77
    • Kuznetsov, A.P.1    Savin, A.V.2    Kim, S.-Y.3
  • 58
    • 17144450765 scopus 로고    scopus 로고
    • Universality and scaling for the breakup of phase synchronization at the onset of chaos in a periodically driven Rössler oscillator
    • S. P. Kuznetsov and I. R. Sataev, Universality and scaling for the breakup of phase synchronization at the onset of chaos in a periodically driven Rössler oscillator, Phys. Rev. E 64:046214 (2001).
    • (2001) Phys. Rev. E , vol.64 , pp. 046214
    • Kuznetsov, S.P.1    Sataev, I.R.2
  • 59
    • 0002144874 scopus 로고
    • On the problem of turbulence
    • In Russian
    • L. D. Landau, On the problem of turbulence, Doklady Akademii Nauk SSSR 44:311 (1944). (In Russian.)
    • (1944) Doklady Akademii Nauk SSSR , vol.44 , pp. 311
    • Landau, L.D.1
  • 60
    • 84980080650 scopus 로고
    • A mathematical example displaying features of turbulence
    • E. Hopf, A mathematical example displaying features of turbulence, Commun. Appl. Math. 1:303-322 (1948).
    • (1948) Commun. Appl. Math. , vol.1 , pp. 303-322
    • Hopf, E.1
  • 61
    • 34250451946 scopus 로고
    • On the nature of turbulence
    • D. Ruelle and F. Takens, On the nature of turbulence, Commun. Math. Phys. 20:167-192 (1971).
    • (1971) Commun. Math. Phys. , vol.20 , pp. 167-192
    • Ruelle, D.1    Takens, F.2
  • 63
  • 64
    • 29044449982 scopus 로고    scopus 로고
    • On critical behavior of a map with Neimark-Sacker bifurcation at destruction of the phase syncronization at the limit point of the Feigenbaum cascade
    • In Russian
    • A. P. Kuznetsov, A. Yu. Kuznetsova, and I. R. Sataev, On critical behavior of a map with Neimark-Sacker bifurcation at destruction of the phase syncronization at the limit point of the Feigenbaum cascade, Izvestija VUZov - Prikladnaja Nelineinaja Dinamika (Saratov) 11(1):12-18 (2003). (In Russian.)
    • (2003) Izvestija VUZov - Prikladnaja Nelineinaja Dinamika (Saratov) , vol.11 , Issue.1 , pp. 12-18
    • Kuznetsov, A.P.1    Kuznetsova, A.Yu.2    Sataev, I.R.3
  • 66
    • 4243954599 scopus 로고
    • Degenerate routes to chaos
    • J. A. C. Gallas, Degenerate routes to chaos, Phys. Rev. E 48:4156-4159 (1993).
    • (1993) Phys. Rev. E , vol.48 , pp. 4156-4159
    • Gallas, J.A.C.1
  • 67
    • 18344389610 scopus 로고
    • Period doubling: Universality and critical-point order
    • B. Hu and J. M. Mao, Period doubling: Universality and critical-point order, Phys. Rev. A 25:3259-3261 (1982).
    • (1982) Phys. Rev. A , vol.25 , pp. 3259-3261
    • Hu, B.1    Mao, J.M.2
  • 68
    • 0011273794 scopus 로고
    • A spectrum of universality classes in period-doubling and period tripling
    • B. Hu and I. I. Satija, A spectrum of universality classes in period-doubling and period tripling, Phys. Lett. A 98:143-146 (1983).
    • (1983) Phys. Lett. A , vol.98 , pp. 143-146
    • Hu, B.1    Satija, I.I.2
  • 70
    • 85204082666 scopus 로고
    • Universalities and scaling at the transition to chaos through homoclinic bifurcations
    • eds. D. V. Shirkov, D. I. Kazakov, and A. A. Vladimirov (World Scientific, Singapore, New Jersey, Hong Kong)
    • D. V. Lubimov, A. S. Pikovsky, and M. A. Zaks, Universalities and scaling at the transition to chaos through homoclinic bifurcations, Renormalization Group, eds. D. V. Shirkov, D. I. Kazakov, and A. A. Vladimirov (World Scientific, Singapore, New Jersey, Hong Kong, 1988), pp. 278-289.
    • (1988) Renormalization Group , pp. 278-289
    • Lubimov, D.V.1    Pikovsky, A.S.2    Zaks, M.A.3
  • 71
    • 0000793023 scopus 로고
    • New universal scenarios for the onset of chaos in Lorenz-type flows
    • J.-M. Gambaudo, I. Procaccia, S. Thomae, and C. Tresser, New universal scenarios for the onset of chaos in Lorenz-type flows, Phys. Rev. Lett. 57:925-928 (1986).
    • (1986) Phys. Rev. Lett. , vol.57 , pp. 925-928
    • Gambaudo, J.-M.1    Procaccia, I.2    Thomae, S.3    Tresser, C.4
  • 72
    • 29044432561 scopus 로고
    • Bifurcation sequences in the dissipative systems with saddle equilibria
    • Banach Center Publications, Warszawa
    • M. A. Zaks and D. V. Lubimov, Bifurcation sequences in the dissipative systems with saddle equilibria, Dynamical Systems and Ergodic Theory 23:367-380. (Banach Center Publications, Warszawa, 1989).
    • (1989) Dynamical Systems and Ergodic Theory , vol.23 , pp. 367-380
    • Zaks, M.A.1    Lubimov, D.V.2
  • 73
    • 0001570377 scopus 로고
    • Bounds on the unstable eigenvalue for period doubling
    • J.-P. Eckmann and H. Epstein, Bounds on the unstable eigenvalue for period doubling, Commun. Math. Phys. 128:427-435 (1990).
    • (1990) Commun. Math. Phys. , vol.128 , pp. 427-435
    • Eckmann, J.-P.1    Epstein, H.2
  • 74
    • 0032008442 scopus 로고    scopus 로고
    • Analytic solution of the Cvitanovié-Feigenbaum and Feigenbaum-Kadanoff-Shenker equations
    • K. M. Briggs, T. W. Dixon, and G. Szekeres, Analytic solution of the Cvitanovié-Feigenbaum and Feigenbaum-Kadanoff-Shenker equations, Int. J. Bifurcation Chaos 8:347-357 (1998).
    • (1998) Int. J. Bifurcation Chaos , vol.8 , pp. 347-357
    • Briggs, K.M.1    Dixon, T.W.2    Szekeres, G.3
  • 75
    • 0010208772 scopus 로고
    • A renormalization group with periodic behaviour
    • A. Arneodo, P. Coullet, and V. Tresser, A renormalization group with periodic behaviour, Phys. Lett. A 70:74-76 (1979).
    • (1979) Phys. Lett. A , vol.70 , pp. 74-76
    • Arneodo, A.1    Coullet, P.2    Tresser, V.3
  • 76
    • 0032332116 scopus 로고    scopus 로고
    • Feigenbaum theory for unimodal maps with asymmetric critical point: Rigorous results
    • B. D. Mestel and A. H. Osbaldestin, Feigenbaum theory for unimodal maps with asymmetric critical point: rigorous results, Commun. Math. Phys. 197:211-228 (1998).
    • (1998) Commun. Math. Phys. , vol.197 , pp. 211-228
    • Mestel, B.D.1    Osbaldestin, A.H.2
  • 77
    • 5644289775 scopus 로고    scopus 로고
    • Bounds on the unstable eigen-value for the asymmetric renormalization operator for period doubling
    • B. D. Mestel, A. H. Osbaldestin, and A. V. Tsygvintsev, Bounds on the Unstable Eigen-value for the Asymmetric Renormalization Operator for Period Doubling, Commun. Math. Phys. 250:241-257 (2004).
    • (2004) Commun. Math. Phys. , vol.250 , pp. 241-257
    • Mestel, B.D.1    Osbaldestin, A.H.2    Tsygvintsev, A.V.3
  • 78
    • 0001384723 scopus 로고
    • Nonuniversal behavior of asymmetric unimodal maps
    • R. V. Jensen and L. K. H. Ma, Nonuniversal behavior of asymmetric unimodal maps, Phys. Rev. A 31:3993-3995 (1985).
    • (1985) Phys. Rev. A , vol.31 , pp. 3993-3995
    • Jensen, R.V.1    Ma, L.K.H.2
  • 79
    • 0039149506 scopus 로고    scopus 로고
    • Death of period-doublings: Locating the homoclinic-doubling cascade
    • B. E. Oldeman, B. Krauskopf, and A. R. Champneys, Death of period-doublings: locating the homoclinic-doubling cascade, Physica D 146:100-120 (2000).
    • (2000) Physica D , vol.146 , pp. 100-120
    • Oldeman, B.E.1    Krauskopf, B.2    Champneys, A.R.3
  • 80
    • 0000277257 scopus 로고
    • Iterative properties of a one-dimensional quartic map: Critical lines and tricritical behavior
    • S.-J. Chang, M. Wortis, and J. A. Wright, Iterative properties of a one-dimensional quartic map: Critical lines and tricritical behavior, Phys. Rev. A 24:2669-2684 (1981).
    • (1981) Phys. Rev. A , vol.24 , pp. 2669-2684
    • Chang, S.-J.1    Wortis, M.2    Wright, J.A.3
  • 81
    • 0013554362 scopus 로고
    • Universal vector scaling in one-dimensional maps
    • S. Fraser and R. Kapral, Universal vector scaling in one-dimensional maps, Phys. Rev. A 30:1017-1025 (1984).
    • (1984) Phys. Rev. A , vol.30 , pp. 1017-1025
    • Fraser, S.1    Kapral, R.2
  • 82
    • 0000640007 scopus 로고
    • Tricriticality in two-dimensional maps
    • S. P. Kuznetsov, Tricriticality in two-dimensional maps, Phys. Lett. A 169:438-444 (1992).
    • (1992) Phys. Lett. A , vol.169 , pp. 438-444
    • Kuznetsov, S.P.1
  • 83
    • 0035889622 scopus 로고    scopus 로고
    • Two-parameter analysis of the scaling behavior at the onset of chaos: Tricritical and pseudo-tricritical points
    • A. P. Kuznetsov, S. P. Kuznetsov, L. V. Turukina, and E. Mosekilde, Two-parameter analysis of the scaling behavior at the onset of chaos: Tricritical and pseudo-tricritical points, Physica A 300:367-385 (2001).
    • (2001) Physica A , vol.300 , pp. 367-385
    • Kuznetsov, A.P.1    Kuznetsov, S.P.2    Turukina, L.V.3    Mosekilde, E.4
  • 84
    • 84962991750 scopus 로고
    • Boundary of topological chaos for bimodal maps of the interval
    • R. S. MacKay and C. Tresser, Boundary of topological chaos for bimodal maps of the interval, J. London Math. Soc. 37:164-181 (1988).
    • (1988) J. London Math. Soc. , vol.37 , pp. 164-181
    • MacKay, R.S.1    Tresser, C.2
  • 85
    • 84956235619 scopus 로고
    • Period doubling for bimodal maps: A horseshoe for a renormalisation operator
    • R.S. MacKay and J. B. J. van Zeijts, Period doubling for bimodal maps: a horseshoe for a renormalisation operator, Nonlinearity 1:253-277 (1988).
    • (1988) Nonlinearity , vol.1 , pp. 253-277
    • MacKay, R.S.1    Van Zeijts, J.B.J.2
  • 86
    • 0000282722 scopus 로고
    • From bimodal one-dimensional maps to Hénon-like two-dimensional maps: Does quantitative universality survive?
    • A. P. Kuznetsov, S. P. Kuznetsov, and I. R. Sataev, From bimodal one-dimensional maps to Hénon-like two-dimensional maps: does quantitative universality survive? Phys. Lett. A 164:413-421 (1994).
    • (1994) Phys. Lett. A , vol.164 , pp. 413-421
    • Kuznetsov, A.P.1    Kuznetsov, S.P.2    Sataev, I.R.3
  • 88
    • 0003083332 scopus 로고
    • Remarks on iterated cubic maps
    • J. Milnor, Remarks on iterated cubic maps, Experimental Mathematics 1:5-24 (1992).
    • (1992) Experimental Mathematics , vol.1 , pp. 5-24
    • Milnor, J.1
  • 90
    • 0000554193 scopus 로고
    • Three-parameter scaling for one-dimensional maps
    • A. P. Kuznetsov, S. P. Kuznetsov, and I. R. Sataev, Three-parameter scaling for one-dimensional maps, Phys. Lett. A 189:367-373 (1994).
    • (1994) Phys. Lett. A , vol.189 , pp. 367-373
    • Kuznetsov, A.P.1    Kuznetsov, S.P.2    Sataev, I.R.3
  • 94
    • 0037982301 scopus 로고
    • Universal properties for the period-tripling bifurcations
    • A. I. Golberg, Ya. G. Sinai, and K. M. Khanin, Universal properties for the period-tripling bifurcations, Uspekhi Matem. Nauk 38:159-160 (1983).
    • (1983) Uspekhi Matem. Nauk , vol.38 , pp. 159-160
    • Golberg, A.I.1    Sinai, Ya.G.2    Khanin, K.M.3
  • 95
    • 30244535403 scopus 로고
    • Universality for period n-tupling in complex mappings
    • P. Cvitanović, J. Myrheim, Universality for period n-tupling in complex mappings Phys. Lett. A 94:329-333 (1983).
    • (1983) Phys. Lett. A , vol.94 , pp. 329-333
    • Cvitanović, P.1    Myrheim, J.2
  • 96
    • 0009582324 scopus 로고
    • 2-μ for complex μ and z: The fractal structure of its M set, and scaling
    • 2-μ for complex μ and z: The fractal structure of its M set, and scaling, Physica D 7:224-239 (1983).
    • (1983) Physica D , vol.7 , pp. 224-239
    • Mandelbrot, B.B.1
  • 97
    • 2542441916 scopus 로고    scopus 로고
    • On scaling properties of two-dimensional maps near the accumulation point of the period-tripling cascade
    • O. B. Isaeva and S. P. Kuznetsov, On scaling properties of two-dimensional maps near the accumulation point of the period-tripling cascade, Regular and Chaotic Dynamics 5:459-476 (2000).
    • (2000) Regular and Chaotic Dynamics , vol.5 , pp. 459-476
    • Isaeva, O.B.1    Kuznetsov, S.P.2
  • 98
    • 4243232651 scopus 로고
    • Period doubling in four-dimensional symplectic maps
    • J. M. Mao, I. I. Satija, and B. Hu, Period doubling in four-dimensional symplectic maps, Phys. Rev. A 34:4325-4332 (1986).
    • (1986) Phys. Rev. A , vol.34 , pp. 4325-4332
    • Mao, J.M.1    Satija, I.I.2    Hu, B.3
  • 99
    • 29044435345 scopus 로고
    • Inverted period-doubling sequences in four-dimensional reversible maps and solutions to the renormalization equations
    • A. Lahiri, Inverted period-doubling sequences in four-dimensional reversible maps and solutions to the renormalization equations, Phys. Rev. A 45:757-762 (1992).
    • (1992) Phys. Rev. A , vol.45 , pp. 757-762
    • Lahiri, A.1
  • 100
    • 0001405884 scopus 로고
    • Scaling behavior in a map of a circle onto itself: Empirical results
    • S. J. Shenker, Scaling behavior in a map of a circle onto itself: Empirical results, Physica D 5:405-411 (1982).
    • (1982) Physica D , vol.5 , pp. 405-411
    • Shenker, S.J.1
  • 101
    • 0001540965 scopus 로고
    • Quasiperiodicity in dissipative systems: A renormalization group analysis
    • M. J. Feigenbaum, L. P. Kadanoff, and S. J. Shenker, Quasiperiodicity in dissipative systems: A renormalization group analysis, Physica D 5:370-386 (1982).
    • (1982) Physica D , vol.5 , pp. 370-386
    • Feigenbaum, M.J.1    Kadanoff, L.P.2    Shenker, S.J.3
  • 102
    • 0020810173 scopus 로고
    • Universal properties of the transition from quasi-periodicity to chaos in dissipative systems
    • S. Ostlund, D. Rand, J. Sethna, and E. Siggia, Universal properties of the transition from quasi-periodicity to chaos in dissipative systems, Physica D 8:303-342 (1983).
    • (1983) Physica D , vol.8 , pp. 303-342
    • Ostlund, S.1    Rand, D.2    Sethna, J.3    Siggia, E.4
  • 103
    • 0343728839 scopus 로고
    • Circle-map scaling in a two-dimensional setting
    • X. Wang, R. Mainieri, and J. H. Lowenstein, Circle-map scaling in a two-dimensional setting, Phys. Rev. A 40:5382-5389 (1989).
    • (1989) Phys. Rev. A , vol.40 , pp. 5382-5389
    • Wang, X.1    Mainieri, R.2    Lowenstein, J.H.3
  • 104
    • 0008978763 scopus 로고
    • Fixed winding number and the quasiperiodic route to chaos in a convective fluid
    • J. Stavans, F. Heslot, and A. Libchaber, Fixed Winding Number and the Quasiperiodic Route to Chaos in a Convective Fluid, Phys. Rev. Lett. 55:596-599 (1985).
    • (1985) Phys. Rev. Lett. , vol.55 , pp. 596-599
    • Stavans, J.1    Heslot, F.2    Libchaber, A.3
  • 105
    • 84956065976 scopus 로고
    • Experimental studies of mode-locking and circle maps in inductively shunted Josephson-junctions
    • M.Bauer, U. Krueger, and W. Martienssen, Experimental studies of mode-locking and circle maps in inductively shunted Josephson-junctions, Europhysics Letters 9:191-196 (1989).
    • (1989) Europhysics Letters , vol.9 , pp. 191-196
    • Bauer, M.1    Krueger, U.2    Martienssen, W.3
  • 106
    • 0024036863 scopus 로고
    • Quasi-periodicity and dynamical systems - An experimentalists view
    • J. A. Glazier and A. Libchaber, Quasi-periodicity and dynamical systems - An experimentalists view, IEEE Trans. Circuits Systems 35:790-809 (1988).
    • (1988) IEEE Trans. Circuits Systems , vol.35 , pp. 790-809
    • Glazier, J.A.1    Libchaber, A.2
  • 107
    • 0009097213 scopus 로고
    • Renormalization of the quasiperiodic transition to chaos for arbitrary winding numbers
    • J. D. Farmer and I. I. Satija, Renormalization of the quasiperiodic transition to chaos for arbitrary winding numbers, Phys. Rev. A 31:3520-3522 (1985).
    • (1985) Phys. Rev. A , vol.31 , pp. 3520-3522
    • Farmer, J.D.1    Satija, I.I.2
  • 108
    • 29044435536 scopus 로고
    • Farey level separation in mode-locking structure of circle mappings
    • M. A. Zaks and A. S. Pikovsky, Farey level separation in mode-locking structure of circle mappings, Physica D 59:255-269 (1992).
    • (1992) Physica D , vol.59 , pp. 255-269
    • Zaks, M.A.1    Pikovsky, A.S.2
  • 109
    • 29044443293 scopus 로고
    • Renormalization in a circle map with two inflection points
    • J. A. Ketoja, Renormalization in a circle map with two inflection points, Physica D 55:45-68 (1992).
    • (1992) Physica D , vol.55 , pp. 45-68
    • Ketoja, J.A.1
  • 111
    • 0039933205 scopus 로고
    • Fractal dimension associated with a critical circle map with an arbitrary-order inflection point
    • R. Delbourgo and B. G. Kenny, Fractal dimension associated with a critical circle map with an arbitrary-order inflection point, Phys. Rev. A 42:6230-6233 (1990).
    • (1990) Phys. Rev. A , vol.42 , pp. 6230-6233
    • Delbourgo, R.1    Kenny, B.G.2
  • 112
    • 0000296339 scopus 로고
    • Critical behavior of a KAM surface. I. Empirical results
    • S. J. Shenker and L. P. Kadanoff, Critical Behavior of a KAM Surface. I. Empirical Results, J. Stat. Phys. 27:631-656 (1982).
    • (1982) J. Stat. Phys. , vol.27 , pp. 631-656
    • Shenker, S.J.1    Kadanoff, L.P.2
  • 113
    • 0000272187 scopus 로고
    • Scaling for a critical Kolmogorov-Arnold-Moser trajectory
    • L. P. Kadanoff, Scaling for a critical Kolmogorov-Arnold-Moser trajectory, Phys. Rev. Lett. A 47:1641-1634 (1981).
    • (1981) Phys. Rev. Lett. A , vol.47 , pp. 1641-11634
    • Kadanoff, L.P.1
  • 114
    • 0000427928 scopus 로고
    • A renormalization approach to invariant circles in area-preserving maps
    • R. S. MacKay, A renormalization approach to invariant circles in area-preserving maps, Phystca D 7:283-300 (1983).
    • (1983) Phystca D , vol.7 , pp. 283-300
    • MacKay, R.S.1
  • 115
    • 22244435161 scopus 로고
    • Renormalization group analysis of bifurcations in area-preserving maps
    • M. Widom and L. P. Kadanoff, Renormalization group analysis of bifurcations in area-preserving maps, Physica D 5:287-292 (1982).
    • (1982) Physica D , vol.5 , pp. 287-292
    • Widom, M.1    Kadanoff, L.P.2
  • 116
    • 85204084303 scopus 로고    scopus 로고
    • New fixed point of the renormalisation operator associated with the recurrence of invariant circles in generic Hamiltonian maps
    • J. Wilbrink, New fixed point of the renormalisation operator associated with the recurrence of invariant circles in generic Hamiltonian maps, Nonlinearity 16:1565-1571 (2003).
    • (2003) Nonlinearity , vol.16 , pp. 1565-1571
    • Wilbrink, J.1
  • 117
    • 0010294123 scopus 로고    scopus 로고
    • Renormalization and transition to chaos in area preserving nontwist maps
    • D. Del-Castillo-Negrete, J. M. Greene, and P. J. Morrison, Renormalization and transition to chaos in area preserving nontwist maps, Physica D 100:311-329 (1997).
    • (1997) Physica D , vol.100 , pp. 311-329
    • Del-Castillo-Negrete, D.1    Greene, J.M.2    Morrison, P.J.3
  • 118
    • 0002707370 scopus 로고
    • Higher-order fixed points of the renormalisation operator for invariant circles
    • J. M. Greene and J. Mao, Higher-order fixed points of the renormalisation operator for invariant circles, Nonlinearity 3:69-78 (1990).
    • (1990) Nonlinearity , vol.3 , pp. 69-78
    • Greene, J.M.1    Mao, J.2
  • 119
    • 0040955222 scopus 로고    scopus 로고
    • Universality in the quasiperiodic route to chaos
    • T. W. Dixon, T. Gherghetta, and B. G Kenny, Universality in the quasiperiodic route to chaos, CHAOS 6:32-42 (1996).
    • (1996) CHAOS , vol.6 , pp. 32-42
    • Dixon, T.W.1    Gherghetta, T.2    Kenny, B.G.3
  • 120
    • 0038199809 scopus 로고
    • Chaos border and statistical anomalies
    • eds. D. V. Shirkov, D. I. Kazakov, and A. A. Vladimirov (World Scientific, Singapore, New Jersey, Hong Kong)
    • B. V. Chirikov and D. L. Shepelyansky, Chaos border and statistical anomalies, Renormalization Group, eds. D. V. Shirkov, D. I. Kazakov, and A. A. Vladimirov (World Scientific, Singapore, New Jersey, Hong Kong, 1988), pp. 221-250.
    • (1988) Renormalization Group , pp. 221-250
    • Chirikov, B.V.1    Shepelyansky, D.L.2
  • 121
    • 48549114635 scopus 로고
    • Strange attractors that are not chaotic
    • C. Grebogi, E. Ott, S. Pelikan, and J. A. Yorke, Strange attractors that are not chaotic, Physica D 13:261-268 (1984).
    • (1984) Physica D , vol.13 , pp. 261-268
    • Grebogi, C.1    Ott, E.2    Pelikan, S.3    Yorke, J.A.4
  • 122
    • 0001341663 scopus 로고
    • Birth of a strange nonchaotic attractor: A renormalization group analysis
    • S. P. Kuznetsov, A. S. Pikovsky, and U. Feudel, Birth of a strange nonchaotic attractor: A renormalization group analysis, Phys. Rev. E 51:1629-1632 (1995).
    • (1995) Phys. Rev. E , vol.51 , pp. 1629-1632
    • Kuznetsov, S.P.1    Pikovsky, A.S.2    Feudel, U.3
  • 123
    • 0000376963 scopus 로고    scopus 로고
    • Renormalization group for scaling at the torus-doubling terminal point
    • S. Kuznetsov, U. Feudel, and A. Pikovsky, Renormalization group for scaling at the torus-doubling terminal point, Phys. Rev. E 57:1585-1590 (1998).
    • (1998) Phys. Rev. E , vol.57 , pp. 1585-1590
    • Kuznetsov, S.1    Feudel, U.2    Pikovsky, A.3
  • 124
    • 0034239256 scopus 로고    scopus 로고
    • Critical point of tori collision in quasiperiodically forced systems
    • S. P. Kuznetsov, E. Neumann, A. Pikovsky, and I. R. Sataev, Critical point of tori collision in quasiperiodically forced systems, Phys. Rev. E 62:1995-2007 (2000).
    • (2000) Phys. Rev. E , vol.62 , pp. 1995-2007
    • Kuznetsov, S.P.1    Neumann, E.2    Pikovsky, A.3    Sataev, I.R.4
  • 125
    • 42749108729 scopus 로고    scopus 로고
    • Torus fractalization and intermittency
    • S. P. Kuznetsov, Torus fractalization and intermittency, Phys. Rev. E 65:066209 (2002).
    • (2002) Phys. Rev. E , vol.65 , pp. 066209
    • Kuznetsov, S.P.1
  • 126
    • 34250254669 scopus 로고
    • Intermittent transition to turbulence in dissipative dynamical systems
    • Y. Pomeau and P. Manneville, Intermittent transition to turbulence in dissipative dynamical systems, Commun. Math. Phys. 74:189-197 (1980).
    • (1980) Commun. Math. Phys. , vol.74 , pp. 189-197
    • Pomeau, Y.1    Manneville, P.2
  • 127
    • 0141803196 scopus 로고
    • Exact solution of the Feigenbaum renormalization group equations for intermittency
    • B. Hu and J. Rudnik, Exact solution of the Feigenbaum renormalization group equations for intermittency, Phys. Rev. Lett. A 26:3035-3036 (1982).
    • (1982) Phys. Rev. Lett. A , vol.26 , pp. 3035-3036
    • Hu, B.1    Rudnik, J.2
  • 129
    • 29044440383 scopus 로고
    • Differential-equation approach to functional equations: Exact solutions intermittency
    • B. Hu and J. Rudnik, Differential-equation approach to functional equations: exact solutions intermittency, Phys. Rev. A 34:2453-2457 (1986).
    • (1986) Phys. Rev. A , vol.34 , pp. 2453-2457
    • Hu, B.1    Rudnik, J.2
  • 130
    • 0000467194 scopus 로고
    • Functional renormalization-group theory of universal 1/f noise in dynamical systems
    • I. Procaccia and H. Schuster, Functional renormalization-group theory of universal 1/f noise in dynamical systems, Phys. Rev. A 28:1210-1212 (1983).
    • (1983) Phys. Rev. A , vol.28 , pp. 1210-1212
    • Procaccia, I.1    Schuster, H.2
  • 131
    • 0001286418 scopus 로고
    • Intermittency in area-preserving mappings
    • A. B. Zisook, Intermittency in area-preserving mappings, Phys. Rev. A 25:2289-2292 (1982).
    • (1982) Phys. Rev. A , vol.25 , pp. 2289-2292
    • Zisook, A.B.1
  • 132
    • 4244069922 scopus 로고
    • Renormalization group for intermittency in area-preserving maps
    • A. B. Zisook and S. J. Shenker, Renormalization group for intermittency in area-preserving maps, Phys. Rev. A 25:2824-2826 (1982).
    • (1982) Phys. Rev. A , vol.25 , pp. 2824-2826
    • Zisook, A.B.1    Shenker, S.J.2
  • 133
    • 22244450030 scopus 로고
    • The complete set of hamiltonian intermittency scaling behaviors
    • A. B. Zisook, The Complete Set of Hamiltonian Intermittency Scaling Behaviors, Commun. Math. Phys. 96:361-371 (1984).
    • (1984) Commun. Math. Phys. , vol.96 , pp. 361-371
    • Zisook, A.B.1


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