Parametric normal forms for Bogdanov--Takens singularity; the generalized saddle-node case

Majid Gazor; Mojtaba Moazeni

doi:10.3934/dcds.2015.35.205

Article Contents

2015, Volume 35, Issue 1: 205-224. Doi: 10.3934/dcds.2015.35.205

This issue Previous Article Small-divisor equation with large-variable coefficient and periodic solutions of DNLS equations Next Article Self-trapping and Josephson tunneling solutions to the nonlinear Schrödinger / Gross-Pitaevskii equation

Parametric normal forms for Bogdanov--Takens singularity; the generalized saddle-node case

Majid Gazor^1, and
Mojtaba Moazeni^1,

1.
Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111

Received: April 30, 2013

Revised: May 31, 2014

Published: December 2014

Abstract Related Papers Cited by

Abstract

We obtain a parametric (and an orbital) normal form for any non-degenerate perturbation of the generalized saddle-node case of Bogdanov--Takens singularity. Explicit formulas are derived and greatly simplified for an efficient implementation in any computer algebra system. A Maple program is prepared for an automatic parametric normal form computation. A section is devoted to present some practical formulas which avoid technical details of the paper.

Keywords:

Mathematics Subject Classification: Primary: 34C20; Secondary: 34A34, 16W50.

Citation:

Related Papers

Cited by

References

[1]	A. Algaba, E. Freire, E. Gamero and C. Garcia, Quasi-homogeneous normal forms, J. Comp. and Appl. Math., 150 (2003), 193-216.doi: 10.1016/S0377-0427(02)00660-X.
[2]	A. Baider and R. C. Churchill, Unique normal forms for planar vector fields, Math. Z., 199 (1988), 303-310.doi: 10.1007/BF01159780.
[3]	A. Baider and J. A. Sanders, Unique normal forms: The nilpotent Hamiltonian case, J. Differential Equations, 92 (1991), 282-304.doi: 10.1016/0022-0396(91)90050-J.
[4]	A. Baider and J. A. Sanders, Further reductions of the Takens-Bogdanov normal form, J. Differential Equations, 99 (1992), 205-244.doi: 10.1016/0022-0396(92)90022-F.
[5]	G. Chen and J. D. Dora, Further reductions of normal forms for dynamical systems, J. Differential Equations, 166 (2000), 79-106.doi: 10.1006/jdeq.2000.3783.
[6]	G. Chen, D. Wang and X. Wang, Unique normal forms for nilpotent planar vector fields, Internat. J. Bifur. Chaos, 12 (2002), 2159-2174.doi: 10.1142/S0218127402005741.
[7]	M. Gazor and F. Mokhtari, Normal forms of Hopf-Zero singularity, preprint, arXiv:1210.4467v4.
[8]	M. Gazor and F. Mokhtari, Volume-preserving normal forms of Hopf-Zero singularity, Nonlinearity, 26 (2013), 2809-2832.doi: 10.1088/0951-7715/26/10/2809.
[9]	M. Gazor, F. Mokhtari and J. A. Sanders, Normal forms for Hopf-Zero singularities with nonconservative nonlinear part, J. Differential Equations, 254 (2013), 1571-1581.doi: 10.1016/j.jde.2012.11.004.
[10]	M. Gazor and P. Yu, Spectral sequences and parametric normal forms, J. Differential Equations, 252 (2012), 1003-1031.doi: 10.1016/j.jde.2011.09.043.
[11]	M. Gazor and P. Yu, Formal decomposition method and parametric normal forms, Internat. J. Bifur. Chaos, 20 (2010), 3487-3515.doi: 10.1142/S0218127410027830.
[12]	M. Gazor and P. Yu, Infinite order parametric normal form of Hopf singularity, Internat. J. Bifur. Chaos, 18 (2008), 3393-3408.doi: 10.1142/S0218127408022445.
[13]	H. Kokubu, H. Oka and D. Wang, Linear grading function and further reduction of normal forms, J. Differential Equations, 132 (1996), 293-318.doi: 10.1006/jdeq.1996.0181.
[14]	M. Moazeni, Asymptotic Unfoldings and Normal Forms of the Generalized Saddle-Node case of Bogdanov-Takens Singularity, Master Thesis (in persian), Isfahan University of Technology, Isfahan, Iran, 2011.
[15]	J. Murdock, Asymptotic unfoldings of dynamical systems by normalizing beyond the normal form, J. Differential Equations, 143 (1998), 151-190.doi: 10.1006/jdeq.1997.3368.
[16]	J. Murdock, Normal Forms and Unfoldings for Local Dynamical Systems, Springer, New York, 2003.doi: 10.1007/b97515.
[17]	J. Murdock, Hypernormal form theory: Foundations and algorithms, J. Differential Equations, 205 (2004), 424-465.doi: 10.1016/j.jde.2004.02.015.
[18]	J. Murdock and D. Malonza, An improved theory of asymtotic unfoldings, J. Differential Equations, 247 (2009), 685-709.doi: 10.1016/j.jde.2009.04.014.
[19]	J. Peng and D. Wang, A suffiecient condition for the uniqueness of normal forms and unique normal forms of generalized Hopf singularities, Internat. J. Bifur. Chaos, 14 (2004), 3337-3345.doi: 10.1142/S0218127404011247.
[20]	E. Stróżyna, The analytic and formal normal form for the nilpotent singularity. The case of generalized saddle-node, Bull. Sci. Math., 126 (2002), 555-579.doi: 10.1016/S0007-4497(02)01127-2.
[21]	E. Stróżyna and H. Żoladek, The complete formal normal form for the Bogdanov-Takens singularity, preprint, (2013), private communication.
[22]	E. Stróżyna and H. Żoladek, Divergence of the reduction to the multidimensional nilpotent Takens normal form, Nonlinearity, 24 (2011), 3129-3141.doi: 10.1088/0951-7715/24/11/007.
[23]	E. Stróżyna and H. Żoladek, The analytic and formal normal form for the nilpotent singularity, J. Differential Equations, 179 (2002), 479-537.doi: 10.1006/jdeq.2001.4043.
[24]	E. Stróżyna and H. Żoladek, Orbital formal normal forms for general Bogdanov-Takens singularity, J. Differential Equations, 193 (2003), 239-259.doi: 10.1016/S0022-0396(03)00137-2.
[25]	D. Wang, J. Li, M. Huang and Y. Jiang, Unique normal form of Bogdanos-Takens singularities, J. Differential Equations, 163 (2000), 223-238.doi: 10.1006/jdeq.1999.3739.
[26]	J. Li, L. Zhang and D. Wang, Unique normal form of a class of 3 dimensional vector fields with symmetries, J. Differential Equations, 257 (2014), 2341-2359.doi: 10.1016/j.jde.2014.05.039.
[27]	P. Yu, Computation of the simplest normal forms with perturbation parameters based on Lie transform and rescaling, J. Comp. and App. Math., 144 (2002), 359-373.doi: 10.1016/S0377-0427(01)00573-8.
[28]	P. Yu and Y. Yuan, A matching pursuit technique for computing the simplest normal forms of vector fields, J. Symbolic Computation, 35 (2003), 591-615.doi: 10.1016/S0747-7171(03)00021-X.
[29]	Y. Yuan and P. Yu, Computation of simplest normal forms of differential equations associated with a double-zero eigenvalues, Internat. J. Bifur. Chaos, 11 (2001), 1307-1330.doi: 10.1142/S0218127401002742.
[30]	P. Yu and A. Y. T. Leung, The simplest normal form of Hopf bifurcation, Nonlinearity, 16 (2003), 277-300.doi: 10.1088/0951-7715/16/1/317.