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January  2015, 35(1): 205-224. doi: 10.3934/dcds.2015.35.205

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

1. 

Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, Iran, Iran

Received  May 2013 Revised  June 2014 Published  August 2014

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.
Citation: Majid Gazor, Mojtaba Moazeni. Parametric normal forms for Bogdanov--Takens singularity; the generalized saddle-node case. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 205-224. doi: 10.3934/dcds.2015.35.205
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.

show all references

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.

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