An unfolding of the Takens-Bogdanov singularity
Authors:
P. Hirschberg and E. Knobloch
Journal:
Quart. Appl. Math. 49 (1991), 281-287
MSC:
Primary 58F14; Secondary 34C23
DOI:
https://doi.org/10.1090/qam/1106393
MathSciNet review:
MR1106393
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Abstract: A complete analysis for $\left | \mu \right |, \left | \nu \right | \ll 1$ of the equation \[ \dot x = y, \dot y = \mu x + \nu y + M{x^2} + \Gamma xy\] describing a particular unfolding of the Takens-Bogdanov singularity is presented.
- R. I. Bogdanov, Versal deformation of a singular point of a vector field on the plane in the case of zero eigenvalues, Funkcional Anal. i Priložen. 9 (1975), no. 2, 63 (Russian). MR 0380878
- Jack Carr, Applications of centre manifold theory, Applied Mathematical Sciences, vol. 35, Springer-Verlag, New York-Berlin, 1981. MR 635782
- John Guckenheimer and Philip Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Applied Mathematical Sciences, vol. 42, Springer-Verlag, New York, 1990. Revised and corrected reprint of the 1983 original. MR 1139515
- Barbara Lee Keyfitz, Admissibility conditions for shocks in conservation laws that change type, SIAM J. Math. Anal. 22 (1991), no. 5, 1284–1292. MR 1112508, DOI https://doi.org/10.1137/0522082
- Barbara Lee Keyfitz and Michael Shearer (eds.), Nonlinear evolution equations that change type, The IMA Volumes in Mathematics and its Applications, vol. 27, Springer-Verlag, New York, 1990. MR 1074180
- E. Knobloch and M. R. E. Proctor, Nonlinear periodic convection in double-diffusive systems, J. Fluid Mech. 108 (1981), 291–316. MR 627230, DOI https://doi.org/10.1017/S0022112081002139
- N. Kopell and L. N. Howard, Bifurcations and trajectories joining critical points, Advances in Math. 18 (1975), no. 3, 306–358. MR 397078, DOI https://doi.org/10.1016/0001-8708%2875%2990048-1
- William J. Merryfield, Juri Toomre, and Douglas Gough, Nonlinear behavior of solar gravity modes driven by $^3{\rm He}$ in the core. I. Bifurcation analysis, Astrophys. J. 353 (1990), no. 2, 678–697. MR 1049653, DOI https://doi.org/10.1086/168658
- Floris Takens, Singularities of vector fields, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 47–100. MR 339292
R. I. Bogdanov, Versal deformations of a singular point on the plane in the case of zero eigenvalues, Functional Anal. Appl. (2) 9, 144–145 (1975)
J. Carr, Applications of Center Manifold Theory, Springer-Verlag, New York-Heidelberg-Berlin, 1981
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York-Heidelberg-Berlin, 1986
B. Keyfitz, Admissibility conditions for shocks in conservation laws that change type, SIAM J. Math. Anal., in press (1991)
B. Keyfitz, Shocks near the sonic line: A comparison between steady and unsteady models for change of type, Nonlinear Evolution Equations that Change Type, B. L. Keyfitz and M. Shearer, eds., Springer-Verlag, New York-Heidelberg-Berlin, 1990
E. Knobloch and M. R. E. Proctor, Nonlinear periodic convection in double-diffusive systems, J. Fluid Mech. 108, 291–316 (1981)
N. Kopell and L. N. Howard, Bifurcations and trajectories connecting critical points, Adv. Math. (3) 18, 306–358 (1975)
W. Merryfield, J. Toomre, and D. Gough, Nonlinear behavior of solar gravity modes driven by $^{3}$He in the core I. Bifurcation analysis, Astrophys. J. 353, 678–697 (1990)
F. Takens, Singularities of vector fields, Inst. Hautes Études Sci. Publ. Math. 43, 47–100 (1974)
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Article copyright:
© Copyright 1991
American Mathematical Society