메뉴 건너뛰기




Volumn 14, Issue 7, 2004, Pages 2143-2160

On some mathematical topics in classical synchronization. A tutorial

Author keywords

Homoclinic bifurcation; Saddle node; Synchronization

Indexed keywords

BIFURCATION (MATHEMATICS); DYNAMICS; FREQUENCIES; MATHEMATICAL MODELS; PROBLEM SOLVING; RESONANCE; THEOREM PROVING;

EID: 13844299122     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127404010539     Document Type: Article
Times cited : (51)

References (39)
  • 1
    • 0038883553 scopus 로고
    • On some global bifurcations connected with the disappearance of saddle-node fixed point
    • Afraimovich, V. S. & Shilnikov, L. P. [1974a] "On some global bifurcations connected with the disappearance of saddle-node fixed point," Dokl. Acad. Nauk SSSR 219, 1981-1985.
    • (1974) Dokl. Acad. Nauk SSSR , vol.219 , pp. 1981-1985
    • Afraimovich, V.S.1    Shilnikov, L.P.2
  • 2
    • 13844252148 scopus 로고
    • On small periodic perturbations of autonomous systems
    • Afraimovich, V. S. & Shilnikov, L. P. [1974b] "On small periodic perturbations of autonomous systems," Dokl. Acad. Nauk SSSR 5, 734-742.
    • (1974) Dokl. Acad. Nauk SSSR , vol.5 , pp. 734-742
    • Afraimovich, V.S.1    Shilnikov, L.P.2
  • 3
    • 0040539385 scopus 로고
    • The annulus principle and problems on interaction of two self-oscillation systems
    • Afraimovich, V. S. & Shilnikov, L. P. [1977] "The annulus principle and problems on interaction of two self-oscillation systems," Prikladnaja Matematika i Mehanika 41, 618-627.
    • (1977) Prikladnaja Matematika I Mehanika , vol.41 , pp. 618-627
    • Afraimovich, V.S.1    Shilnikov, L.P.2
  • 4
    • 0000697407 scopus 로고
    • Invariant tori, their breakdown and stochasticity
    • Afraimovich, V. S. & Shilnikov, L. P. [1991] "Invariant tori, their breakdown and stochasticity," Am. Math. Soc. Transl. 149, 201-211.
    • (1991) Am. Math. Soc. Transl. , vol.149 , pp. 201-211
    • Afraimovich, V.S.1    Shilnikov, L.P.2
  • 5
    • 33646141213 scopus 로고
    • Zur theorie des mitmehmens von van der Pol
    • Andronov, A. A. & Vitt, A. A. [1930] "Zur theorie des mitmehmens von van der Pol," Archiv für Elektrotechnik XXIV, 99-110.
    • (1930) Archiv für Elektrotechnik , vol.24 , pp. 99-110
    • Andronov, A.A.1    Vitt, A.A.2
  • 6
    • 13844262991 scopus 로고
    • Generation of limit cycles from a separatrix forming a loop and from the separatrix of an equilibrium state of saddle-node type
    • Andronov, A. A. & Leontovich, E. A. [1963] "Generation of limit cycles from a separatrix forming a loop and from the separatrix of an equilibrium state of saddle-node type," Am. Math. Soc. Transl. II. Ser. 33, 189-231; translation from Mat. Sbornik 48, 1959.
    • (1963) Am. Math. Soc. Transl. II. Ser. , vol.33 , pp. 189-231
    • Andronov, A.A.1    Leontovich, E.A.2
  • 7
    • 13844269392 scopus 로고
    • translation from
    • Andronov, A. A. & Leontovich, E. A. [1963] "Generation of limit cycles from a separatrix forming a loop and from the separatrix of an equilibrium state of saddle-node type," Am. Math. Soc. Transl. II. Ser. 33, 189-231; translation from Mat. Sbornik 48, 1959.
    • (1959) Mat. Sbornik , vol.48
  • 10
    • 79955577320 scopus 로고
    • On three-dimensional dynamical systems close to systems with a structurally unstable homoclinic curve I
    • Gavrilov, N. K. & Shilnikov, L. P. [1972] "On three-dimensional dynamical systems close to systems with a structurally unstable homoclinic curve I," Math. USSR Sbornik 88, 467-485.
    • (1972) Math. USSR Sbornik , vol.88 , pp. 467-485
    • Gavrilov, N.K.1    Shilnikov, L.P.2
  • 11
    • 0012144124 scopus 로고
    • On some bifurcations of an equilibrium state with one zero and a pair of purely imaginary roots
    • Gorky
    • Gavrilov, N. K. [1978] "On some bifurcations of an equilibrium state with one zero and a pair of purely imaginary roots," in Methods of Qualitative Theory of Differential Equations (Gorky), pp. 41-49.
    • (1978) Methods of Qualitative Theory of Differential Equations , pp. 41-49
    • Gavrilov, N.K.1
  • 13
    • 0008340999 scopus 로고
    • On a codimension two bifurcation
    • Coventry 1980, Lecture Notes in Mathematics
    • Guckenheimer, J. [1981] "On a codimension two bifurcation," Dynamical Systems and Turbulence, Proc. Symp., Coventry 1980, Lecture Notes in Mathematics, Vol. 898, pp. 99-142.
    • (1981) Dynamical Systems and Turbulence, Proc. Symp. , vol.898 , pp. 99-142
    • Guckenheimer, J.1
  • 14
    • 51649161951 scopus 로고
    • Lyapunov exponents, entropy and periodic orbits for diffeomorphisms
    • Katok, A. [1980] "Lyapunov exponents, entropy and periodic orbits for diffeomorphisms," Inst. Hautes Etudes Sci. Publ. Math. 51, 137-173.
    • (1980) Inst. Hautes Etudes Sci. Publ. Math. , vol.51 , pp. 137-173
    • Katok, A.1
  • 15
    • 0036888068 scopus 로고    scopus 로고
    • Accumulating regions of winding periodic orbits in optically driven lasers
    • Krauskopf, B. & Wieczorek, S. [2002] "Accumulating regions of winding periodic orbits in optically driven lasers," Physica D173(1-2), 114-129.
    • (2002) Physica , vol.D173 , Issue.1-2 , pp. 114-129
    • Krauskopf, B.1    Wieczorek, S.2
  • 17
    • 0002887375 scopus 로고
    • Qualitative analysis of the periodically forced relaxation oscillations
    • Levi, M. [1981] "Qualitative analysis of the periodically forced relaxation oscillations," Mem. Am. Math. Soc. 32, 1-146.
    • (1981) Mem. Am. Math. Soc. , vol.32 , pp. 1-146
    • Levi, M.1
  • 18
    • 0000057066 scopus 로고
    • A second order differential equation with singular solutions
    • Levinson, N. [1949] "A second order differential equation with singular solutions," Ann. Math. 2, 127-153.
    • (1949) Ann. Math. , vol.2 , pp. 127-153
    • Levinson, N.1
  • 19
    • 34250577695 scopus 로고
    • 2)ẏ + y = bk cos(t + α) for large k, and its generalizations
    • 2)ẏ + y = bk cos(t + α) for large k, and its generalizations," Acta Math. 97, 267-308.
    • (1957) Acta Math. , vol.97 , pp. 267-308
    • Littlewood, J.E.1
  • 20
    • 34250601358 scopus 로고
    • On non-linear differential equations of the second order. IV. The general equation ÿ + kf(y)ẏ + g(y) = bkp(φ), φ = t + α
    • Littlewood, J. E. [1957b] "On non-linear differential equations of the second order. IV. The general equation ÿ + kf(y)ẏ + g(y) = bkp(φ), φ = t + α," Acta Math. 98, 1-110.
    • (1957) Acta Math. , vol.98 , pp. 1-110
    • Littlewood, J.E.1
  • 21
    • 0242313111 scopus 로고
    • Some bifurcations of dynamical system with homoclinic structures
    • Lukyanov, V. & Shilnikov, L. P. [1978] "Some bifurcations of dynamical system with homoclinic structures," Soviet Math. Dokl. 19, 1314-1318.
    • (1978) Soviet Math. Dokl. , vol.19 , pp. 1314-1318
    • Lukyanov, V.1    Shilnikov, L.P.2
  • 22
    • 0010926346 scopus 로고
    • On a new type of bifurcations on manifolds
    • Medvedev, V. S. [1980] "On a new type of bifurcations on manifolds," Mat. Sbornik 113, 487-492.
    • (1980) Mat. Sbornik , vol.113 , pp. 487-492
    • Medvedev, V.S.1
  • 23
    • 51249181043 scopus 로고
    • The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms
    • Newhouse, S. E. [1979] "The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms," Inst. Hautes Etudes Sci. Publ. Math. 50, 101-151.
    • (1979) Inst. Hautes Etudes Sci. Publ. Math. , vol.50 , pp. 101-151
    • Newhouse, S.E.1
  • 24
    • 0003377111 scopus 로고
    • Cycles and bifurcation theory
    • Newhouse, S. & Palis, J. [1976] "Cycles and bifurcation theory," Astérisque 31, 44-140.
    • (1976) Astérisque , vol.31 , pp. 44-140
    • Newhouse, S.1    Palis, J.2
  • 26
    • 13844261470 scopus 로고
    • The hyperbolicity of a basic family for Levinson's equation
    • in Russian
    • Osipov, A. V. [1975] "The hyperbolicity of a basic family for Levinson's equation," Differencialnye Uravnenija 11, 1795-1800 (in Russian).
    • (1975) Differencialnye Uravnenija , vol.11 , pp. 1795-1800
    • Osipov, A.V.1
  • 30
  • 31
    • 0000063614 scopus 로고
    • On a Poincaré-Birkhoff problem
    • Shilnikov, L. P. [1967] "On a Poincaré-Birkhoff problem," Math. USSR Sbornik 3, 415-443.
    • (1967) Math. USSR Sbornik , vol.3 , pp. 415-443
    • Shilnikov, L.P.1
  • 32
    • 0031189220 scopus 로고    scopus 로고
    • On simple bifurcations leading to hyperbolic attractors
    • Shilnikov, L. P. & Turaev, D. V. [1997] "On simple bifurcations leading to hyperbolic attractors," Comput. Math. Appl. 34, 441-457.
    • (1997) Comput. Math. Appl. , vol.34 , pp. 441-457
    • Shilnikov, L.P.1    Turaev, D.V.2
  • 33
    • 3242698870 scopus 로고    scopus 로고
    • A new simple bifurcation of a periodic orbit of blue sky catastrophe type
    • Amer. Math. Soc. Transl., II Ser. (AMS, Providence, RI)
    • Shilnikov, L. P. & Turaev, D. V. [2000] "A new simple bifurcation of a periodic orbit of blue sky catastrophe type," in Methods of Qualitative Theory of Differential Equations and Related Topics, Amer. Math. Soc. Transl., II Ser. 200 (AMS, Providence, RI), pp. 165-188.
    • (2000) Methods of Qualitative Theory of Differential Equations and Related Topics , vol.200 , pp. 165-188
    • Shilnikov, L.P.1    Turaev, D.V.2
  • 34
    • 0032380870 scopus 로고    scopus 로고
    • Homoclinic Ω-explosion and domains of hyperbolicity
    • Sten'kin, O. V. & Shilnikov, L. P. [1998] "Homoclinic Ω-explosion and domains of hyperbolicity," Sb. Math. 189, 603-622.
    • (1998) Sb. Math. , vol.189 , pp. 603-622
    • Sten'kin, O.V.1    Shilnikov, L.P.2
  • 35
    • 51649178043 scopus 로고
    • Singularities of vector fields
    • Takens, F. [1974] "Singularities of vector fields," Publ. Math. IHES 43, 47-100.
    • (1974) Publ. Math. IHES , vol.43 , pp. 47-100
    • Takens, F.1
  • 37
  • 38
    • 0001874281 scopus 로고
    • Forced oscillations in a circuit with nonlinear resistance (receptance with reactive triode)
    • Reprinted [1964] (Bellman & Kakaba)
    • Van der Pol, B. [1927] "Forced oscillations in a circuit with nonlinear resistance (receptance with reactive triode)," London, Edinburgh and Dublin Phil. Mag. 3, 65-80; Reprinted [1964] (Bellman & Kakaba).
    • (1927) London, Edinburgh and Dublin Phil. Mag. , vol.3 , pp. 65-80
    • Van Der Pol, B.1
  • 39
    • 30244449696 scopus 로고
    • Frequency demultiplication
    • Van der Pol, B. & Van der Mark, J. [1927] "Frequency demultiplication," Nature 120, 363-364.
    • (1927) Nature , vol.120 , pp. 363-364
    • Van Der Pol, B.1    Van Der Mark, J.2


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.