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Uspekhi Mat. Nauk, 1972, Volume 27, Issue 5(167), Pages 119–184 (Mi umn5111)  

This article is cited in 99 scientific papers (total in 100 papers)

Works cyrcle on the theory of singularities of smooth mappings
Lectures on bifurcations in versal families

V. I. Arnol'd

Abstract: In these lectures we consider the ways in which the disposition of the phase curves of a vector field can alter in a neighbourhood of a singularity as the parameters on which the vector field depends vary. A technical convenience in the study of such changes are certain deformations having a special universality property – the so-called versal families. Our results are presented mainly in the form of explicit formulae for versal families and an analysis of the corresponding bifurcation diagrams. As an application of the general theory we give a classification of the singularities of the decrement of general two-parameter families of linear autonomous systems and a classification of the singularities of the neutral surface (stability boundary) of general three-parameter families of linear systems; we also treat the topologically versal deformations of singular points of non-linear systems of ordinary differential equations for all cases of degeneracy of codimension 1 and for some of codimension 2; we indicate applications to the theory of hydrodynamical stability.

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Russian Mathematical Surveys, 1972, 27:5, 54–123

Bibliographic databases:

UDC: 519.9+513.83
MSC: 37G10, 58K60

Citation: V. I. Arnol'd, “Lectures on bifurcations in versal families”, Uspekhi Mat. Nauk, 27:5(167) (1972), 119–184; Russian Math. Surveys, 27:5 (1972), 54–123

Citation in format AMSBIB
\by V.~I.~Arnol'd
\paper Lectures on bifurcations in versal families
\jour Uspekhi Mat. Nauk
\yr 1972
\vol 27
\issue 5(167)
\pages 119--184
\jour Russian Math. Surveys
\yr 1972
\vol 27
\issue 5
\pages 54--123

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    This publication is cited in the following articles:
    1. V. I. Arnol'd, “Normal forms for functions near degenerate critical points, the Weyl groups of $A_k$, $D_k$, $E_k$ and Lagrangian singularities”, Funct. Anal. Appl., 6:4 (1972), 254–272  mathnet  crossref  mathscinet  zmath
    2. R. Tom, “Topologiya i lingvistika”, UMN, 30:1(181) (1975), 199–221  mathnet  mathscinet  zmath
    3. V. I. Arnol'd, “Critical points of smooth functions and their normal forms”, Russian Math. Surveys, 30:5 (1975), 1–75  mathnet  crossref  mathscinet  zmath
    4. R. I. Bogdanov, “Versal deformations of a singular point of a vector field on the plane in the case of zero eigenvalues”, Funct. Anal. Appl., 9:2 (1975), 144–145  mathnet  crossref  mathscinet  zmath
    5. R. I. Bogdanov, “Reduction to orbital normal form of a vector field on the plane”, Funct. Anal. Appl., 10:1 (1976), 61–62  mathnet  crossref  mathscinet  zmath
    6. P.J. Holmes, D.A. Rand, “The bifurcations of duffing's equation: An application of catastrophe theory”, Journal of Sound and Vibration, 44:2 (1976), 237  crossref
    7. R. Gilmore, “Structural stability of the phase transition in Dicke-like models”, J Math Phys (N Y ), 18:1 (1977), 17  crossref  adsnasa
    8. V. V. Bykov, “O rozhdenii periodicheskikh dvizhenii iz separatrisnogo kontura trekhmernoi sistemy”, UMN, 32:6(198) (1977), 213–214  mathnet  mathscinet  zmath
    9. V. D. Sedykh, “Singularities of the convex hull of a curve in $\mathbb{R}^3$”, Funct. Anal. Appl., 11:1 (1977), 72–73  mathnet  crossref  mathscinet  zmath
    10. V. I. Arnol'd, “Loss of stability of self-oscillations close to resonance and versal deformations of equivariant vector fields”, Funct. Anal. Appl., 11:2 (1977), 85–92  mathnet  crossref  mathscinet  zmath
    11. R. I. Bogdanov, “Singularities of vector fields on a plane”, Funct. Anal. Appl., 11:4 (1977), 303–304  mathnet  crossref  mathscinet  zmath
    12. P.J. Holmes, “Bifurcations to divergence and flutter in flow-induced oscillations: A finite dimensional analysis”, Journal of Sound and Vibration, 53:4 (1977), 471  crossref
    13. Philip Holmes, “‘Strange’ phenomena in dynamical systems and their physical implications”, Applied Mathematical Modelling, 1:7 (1977), 362  crossref
    14. J. Palis, F. Takens, “Topological equivalence of normally hyperbolic dynamical systems”, Topology, 16:4 (1977), 335  crossref
    15. R. Gilmore, “Relation between the equilibrium and nonequilibrium critical properties of the Dicke model”, Phys Rev A, 17:5 (1978), 1747  crossref  adsnasa
    16. Martin Golubitsky, “An Introduction to Catastrophe Theory and Its Applications”, SIAM Rev, 20:2 (1978), 352  crossref  mathscinet  zmath
    17. G. R. Belitskii, “Equivalence and normal forms of germs of smooth mappings”, Russian Math. Surveys, 33:1 (1978), 107–177  mathnet  crossref  mathscinet  zmath
    18. N. V. Nikolenko, “On perturbed Korteweg–de Vries equations”, Russian Math. Surveys, 33:3 (1978), 167–168  mathnet  crossref  mathscinet  zmath
    19. Philip Holmes, Jerrold Marsden, “Bifurcation to divergence and flutter in flow-induced oscillations: an infinite dimensional analysis”, Automatica, 14:4 (1978), 367  crossref
    20. R. Gilmore, “Catastrophe time scales and conventions”, Phys Rev A, 20:6 (1979), 2510  crossref  mathscinet  adsnasa  isi
    21. Philip Holmes, Jerrold E. Marsden, “QUALITATIVE TECHNIQUES FOR BIFURCATION ANALYSIS OF COMPLEX SYSTEMS”, Ann N Y Acad Sci, 316:1 (1979), 608  crossref  mathscinet  zmath
    22. R. I. Bogdanov, “Versal deformations of singular points of vector fields on a plane”, Funct. Anal. Appl., 13:1 (1979), 50–51  mathnet  crossref  mathscinet  zmath
    23. Yu. S. Ilyashenko, “Divergence of series reducing an analytic differential equation to linear normal form at a singular point”, Funct. Anal. Appl., 13:3 (1979), 227–229  mathnet  crossref  mathscinet  zmath
    24. Marc Mangel, “Relaxation at critical points: Deterministic and stochastic theory”, Physica A: Statistical Mechanics and its Applications, 97:3 (1979), 616  crossref
    25. Philip Holmes, “UNFOLDING A DEGENERATE NONLINEAR OSCILLATOR: A CODIMENSION TWO BIFURCATION”, Ann N Y Acad Sci, 357:1 (1980), 473  crossref  mathscinet  zmath
    26. Marc Mangel, “Small Fluctuations in Systems with Multiple Limit Cycles”, SIAM J Appl Math, 38:1 (1980), 120  crossref  mathscinet  isi
    27. N. V. Nikolenko, “Invariant asymptotically stable tori of the perturbed Korteweg–de Vries equation”, Russian Math. Surveys, 35:5 (1980), 139–207  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    28. John Guckenheimer, “Symbolic dynamics and relaxation oscillations”, Physica D: Nonlinear Phenomena, 1:2 (1980), 227  crossref
    29. David Chillingworth, “A global genericity theorem for bifurcations in variational problems”, Journal of Functional Analysis, 35:2 (1980), 251  crossref
    30. Philip J. Holmes, “A strange family of three-dimensional vector fields near a degenerate singularity”, Journal of Differential Equations, 37:3 (1980), 382  crossref
    31. H. TROGER, K. ZEMAN, “Application of Bifurcation Theory to Tractor-Semitrailer Dynamics”, Vehicle System Dynamics, 10:2-3 (1981), 156  crossref
    32. A.D. Bazykin, F.S. Berezovskaya, G.A. Denisov, Yu.A. Kuznetzov, “The influence of predator saturation effect and competition among predators on predator-prey system dynamics”, Ecological Modelling, 14:1-2 (1981), 39  crossref  elib
    33. Philip J. Holmes, “Center manifolds, normal forms and bifurcations of vector fields with application to coupling between periodic and steady motions”, Physica D: Nonlinear Phenomena, 2:3 (1981), 449  crossref
    34. Ian Stewart, “Applications of catastrophe theory to the physical sciences”, Physica D: Nonlinear Phenomena, 2:2 (1981), 245  crossref
    35. P. H. Coullet, E. A. Spiegel, “Amplitude Equations for Systems with Competing Instabilities”, SIAM J Appl Math, 43:4 (1983), 776  crossref  mathscinet  zmath  isi
    36. John Guckenheimer, “Multiple Bifurcation Problems of Codimension Two”, SIAM J Math Anal, 15:1 (1984), 1  crossref  mathscinet  zmath  isi
    37. Hiroshi Kokubu, “Normal forms for parametrized vector fields and its application to bifurcations of some reaction diffusion equations”, Japan J Appl Math, 1:2 (1984), 273  crossref  mathscinet  zmath
    38. Shigehiro Ushiki, “Normal forms for singularities of vector fields”, Japan J Appl Math, 1:1 (1984), 1  crossref  mathscinet  zmath
    39. P.H. Coullet, “Chaotic behaviours in the unfolding of singular vector fields”, Physics Reports, 103:1-4 (1984), 95  crossref
    40. G. Lyberatos, B. Kuszta, J.E. Bailey, “Discrimination and identification of dynamic catalytic reaction models via introduction of feedback”, Chemical Engineering Science, 39:4 (1984), 739  crossref
    41. R. Scheidl, H. Troger, K. Zeman, “Coupled flutter and divergence bifurcation of a double pendulum”, International Journal of Non-Linear Mechanics, 19:2 (1984), 163  crossref
    42. Hüseyi̇in Koçak, “Normal forms and versal deformations of linear Hamiltonian systems”, Journal of Differential Equations, 51:3 (1984), 359  crossref
    43. Riccardo Benedetti, Paolo Cragnolini, “Versal families of matrices with respect to unitary conjugation”, Advances in Mathematics, 54:3 (1984), 314  crossref
    44. Hiroe Oka, Hiroshi Kokubu, “Constrained Lorenz-like attractors”, Japan J Appl Math, 2:2 (1985), 495  crossref  mathscinet  zmath
    45. A. Arneodo, P.H. Coullet, E.A. Spiegel, C. Tresser, “Asymptotic chaos”, Physica D: Nonlinear Phenomena, 14:3 (1985), 327  crossref
    46. Jack Carr, Shui-Nee Chow, Jack K Hale, “Abelian integrals and bifurcation theory”, Journal of Differential Equations, 59:3 (1985), 413  crossref
    47. R Cushman, J.A Sanders, “A codimension two bifurcation with a third order Picard-Fuchs equation”, Journal of Differential Equations, 59:2 (1985), 243  crossref
    48. Jan A. Sanders, Richard Cushman, “Limit Cycles in the Josephson Equation”, SIAM J Math Anal, 17:3 (1986), 495  crossref  mathscinet  zmath  isi
    49. E. Knobloch, “Normal forms for bifurcations at a double zero eigenvalue”, Physics Letters A, 115:5 (1986), 199  crossref
    50. Jack Carr, Jan A. Sanders, Stephan A. van Gils, “Nonresonant Bifurcations with Symmetry”, SIAM J Math Anal, 18:3 (1987), 579  crossref  mathscinet  zmath  isi
    51. Shigehiro Ushiki, René Lozi, “Confinor and anti-confinor in constrained “Lorenz” system”, Japan J Appl Math, 4:3 (1987), 433  crossref  mathscinet  zmath
    52. C. Elphick, E. Tirapegui, M.E. Brachet, P. Coullet, G. Iooss, “A simple global characterization for normal forms of singular vector fields”, Physica D: Nonlinear Phenomena, 29:1-2 (1987), 95  crossref
    53. Paul Bryant, Carson Jeffries, “The dynamics of phase locking and points of resonance in a forced magnetic oscillator”, Physica D: Nonlinear Phenomena, 25:1-3 (1987), 196  crossref
    54. D.G. Aronson, E.J. Doedel, H.G. Othmer, “An analytical and numerical study of the bifurcations in a system of linearly-coupled oscillators”, Physica D: Nonlinear Phenomena, 25:1-3 (1987), 20  crossref
    55. G Iooss, “Global characterization of the normal form for a vector field near a closed orbit”, Journal of Differential Equations, 76:1 (1988), 47  crossref
    56. J.-D. Jin, Y. Matsuzaki, “Bifurcations in a two-degree-of-freedom elastic system with follower forces”, Journal of Sound and Vibration, 126:2 (1988), 265  crossref
    57. H. Sakaguchi, S. Shinomoto, Y. Kuramoto, “Phase Transitions and Their Bifurcation Analysis in a Large Population of Active Rotators with Mean-Field Coupling”, Progress of Theoretical Physics, 79:3 (1988), 600  crossref
    58. Hu Gang, H. Haken, “Multimode instability criterion in optical bistable systems”, Phys Rev A, 40:4 (1989), 1899  crossref  mathscinet  adsnasa  isi
    59. Hu Gang, “Codimension-three bifurcation point in a laser system with an injected signal”, Phys Rev A, 40:2 (1989), 834  crossref  mathscinet  adsnasa  isi
    60. Hu Gang, Cun-Zheng Ning, H. Haken, “Codimension-two bifurcations in single-mode optical bistable systems”, Phys Rev A, 41:5 (1990), 2702  crossref  adsnasa  isi
    61. Dwight Barkley, “Theory and predictions for finite-amplitude waves in two-dimensional plane Poiseuille flow”, Phys Fluids A, 2:6 (1990), 955  crossref  mathscinet  zmath  isi
    62. L. F. Burlaga, “Heliospheric shocks and catastrophe theory”, Geophys Res Lett, 17:10 (1990), 1633  crossref  adsnasa  isi
    63. John David Crawford, “Introduction to bifurcation theory”, Rev Mod Phys, 63:4 (1991), 991  crossref  mathscinet  isi
    64. A Edalat, E C Zeeman, Nonlinearity, 5:4 (1992), 921  crossref  mathscinet  zmath  adsnasa  isi
    65. D. Afolabi, “Flutter analysis using transversality theory”, Acta Mech, 103:1-4 (1994), 1  crossref  mathscinet  zmath  isi
    66. Guanrong Chen, Jorge L. Moiola, “An overview of bifurcation, chaos and nonlinear dynamics in control systems”, Journal of the Franklin Institute, 331:6 (1994), 819  crossref
    67. Lijun Yang, Bodo Werner, “Analysis of Takens-Bogdanov bifurcation by characteristic functions”, Nonlinear Analysis: Theory, Methods & Applications, 26:2 (1996), 363  crossref
    68. John Guckenheimer, Mark Myers, Bernd Sturmfels, “Computing Hopf Bifurcations I”, SIAM J Numer Anal, 34:1 (1997), 1  crossref  mathscinet  isi
    69. D. V. Anosov, A. A. Bolibrukh, V. A. Vassiliev, A. M. Vershik, A. A. Gonchar, M. L. Gromov, S. M. Gusein-Zade, V. M. Zakalyukin, Yu. S. Ilyashenko, V. V. Kozlov, M. L. Kontsevich, Yu. I. Manin, A. I. Neishtadt, S. P. Novikov, Yu. S. Osipov, M. B. Sevryuk, Ya. G. Sinai, A. N. Tyurin, L. D. Faddeev, B. A. Khesin, A. G. Khovanskii, “Vladimir Igorevich Arnol'd (on his 60th birthday)”, Russian Math. Surveys, 52:5 (1997), 1117–1139  mathnet  crossref  crossref  mathscinet  adsnasa  isi
    70. Xu Jian-Xue, Norio Hasebe, “The problem of an elastic-plastic beam dynamics and an incomplete co-dimension two bifurcation”, International Journal of Non-Linear Mechanics, 32:1 (1997), 127  crossref
    71. Christof Schütte, Folkmar A. Bornemann, “On the Singular Limit of the Quantum-Classical Molecular Dynamics Model”, SIAM J Appl Math, 59:4 (1999), 1208  crossref  mathscinet  zmath  isi
    72. Osita D. I. Nwokah, E. Borzova, Gemunu S. Happawana, Daré Afolabi, “Catastrophes in Optimal Control”, J Dyn Sys Meas Control, 121:4 (1999), 577  crossref  isi
    73. V. I. Yudovich, L. G. Kurakin, “Bifurcation of a limit cycle from the equilibrium submanifold in a system with multiple cosymmetries”, Math. Notes, 66:2 (1999), 254–258  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    74. P. YU, “SIMPLEST NORMAL FORMS OF Hopf AND GENERALIZED Hopf BIFURCATIONS”, Int. J. Bifurcation Chaos, 09:10 (1999), 1917  crossref
    75. Matías Navarro, Federico Sánchez-Bringas, “Bifurcations of simple umbilical points defined by vector fields normal to a surface immersed in ℝ4 ”, Qual Th Dyn Syst, 2:2 (2001), 359  crossref  zmath
    76. Alexei A. Mailybaev, “Transformation to versal deformations of matrices”, Linear Algebra and its Applications, 337:1-3 (2001), 87  crossref
    77. P. Yu, A.Y.T. Leung, “A perturbation method for computing the simplest normal forms of dynamical systems”, Journal of Sound and Vibration, 261:1 (2003), 123  crossref
    78. L. G. Kurakin, V. I. Yudovich, “On equilibrium bifurcations in the cosymmetry collapse of a dynamical system”, Siberian Math. J., 45:2 (2004), 294–310  mathnet  crossref  mathscinet  zmath  isi  elib
    79. Pei Yu, Songhui Zhu, “Computation of the normal forms for general M-DOF systems using multiple time scales. Part I: autonomous systems”, Communications in Nonlinear Science and Numerical Simulation, 10:8 (2005), 869  crossref
    80. J. Palis, “A global perspective for non-conservative dynamics”, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 22:4 (2005), 485  crossref
    82. Palis J., “A Global Perspective for Non-Conservative Dynamics”, Ann. Inst. Henri Poincare-Anal. Non Lineaire, 22:4 (2005), 485–507  crossref  isi
    83. Stefan C. Mancas, S. Roy Choudhury, “Traveling wavetrains in the complex cubic–quintic Ginzburg–Landau equation”, Chaos, Solitons & Fractals, 28:3 (2006), 834  crossref
    84. H.W. Broer, J. Hoo, V. Naudot, “Normal linear stability of quasi-periodic tori”, Journal of Differential Equations, 232:2 (2007), 355  crossref
    85. Stefan C. Mancas, S.Roy Choudhury, “The complex cubic–quintic Ginzburg–Landau equation: Hopf bifurcations yielding traveling waves”, Mathematics and Computers in Simulation, 74:4-5 (2007), 281  crossref
    86. I. M. Peshkov, “Vetvlenie reshenii matematicheskikh modelei gipoteticheskikh gennykh setei”, Vestn. NGU. Ser. matem., mekh., inform., 7:3 (2007), 59–72  mathnet
    87. Mikhail B Sevryuk, “KAM tori: persistence and smoothness”, Nonlinearity, 21:10 (2008), T177  crossref
    88. Robert Sacker, “Introduction to the 2009 re-publication of the 'Neimark-Sacker bifurcation theorem'”, J. of Difference Equations & Applications, 15:8 (2009), 753  crossref
    89. D.A. Kulikov, “Self-similar cycles and their local bifurcations in the problem of two weakly coupled oscillators☆”, Journal of Applied Mathematics and Mechanics, 74:4 (2010), 389  crossref
    90. Guojun Peng, Yaolin Jiang, “Practical computation of normal forms of the Bogdanov–Takens bifurcation”, Nonlinear Dyn, 2011  mathnet  crossref
    91. Andrii R. Dmytryshyn, Vyacheslav Futorny, Vladimir V. Sergeichuk, “Miniversal deformations of matrices of bilinear forms”, Linear Algebra and its Applications, 2012  crossref
    92. R. I. Bogdanov, M. R. Bogdanov, P. S. Kuzin, “Continuous billiards and the weakly dissipative Kolmogorov – Arnold – Moser theory”, J Math Sci, 2012  crossref
    93. Kaschenko I.S., Kaschenko S.A., “Kvazinormalnye formy dvukhkomponentnykh singulyarno vozmuschennykh sistem”, Doklady akademii nauk, 447:4 (2012), 376–376  elib
    94. Baoying Chen, “An Approach for the Construction of Systems That Self-Generate Chaotic Solitons”, AM, 03:07 (2012), 755  crossref
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    97. Andrii Dmytryshyn, Vyacheslav Futorny, V.V.. Sergeichuk, “Miniversal deformations of matrices under *congruence and reducing transformations”, Linear Algebra and its Applications, 446 (2014), 388  crossref
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    100. I. S. Kashchenko, S. A. Kashchenko, “Local dynamics of two-component singularly perturbed parabolic systems”, Trans. Moscow Math. Soc., 77 (2016), 55–68  mathnet  crossref  elib
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