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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 & Continuous Dynamical Systems - A, 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. 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. doi: 10.1007/BF01159780.

[3]

A. Baider and J. A. Sanders, Unique normal forms: The nilpotent Hamiltonian case,, J. Differential Equations, 92 (1991), 282. 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. 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. 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. doi: 10.1142/S0218127402005741.

[7]

M. Gazor and F. Mokhtari, Normal forms of Hopf-Zero singularity,, preprint, ().

[8]

M. Gazor and F. Mokhtari, Volume-preserving normal forms of Hopf-Zero singularity,, Nonlinearity, 26 (2013), 2809. 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. 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. 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. 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. 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. 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), (2011).

[15]

J. Murdock, Asymptotic unfoldings of dynamical systems by normalizing beyond the normal form,, J. Differential Equations, 143 (1998), 151. doi: 10.1006/jdeq.1997.3368.

[16]

J. Murdock, Normal Forms and Unfoldings for Local Dynamical Systems,, Springer, (2003). doi: 10.1007/b97515.

[17]

J. Murdock, Hypernormal form theory: Foundations and algorithms,, J. Differential Equations, 205 (2004), 424. 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. 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. 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. 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).

[22]

E. Stróżyna and H. Żoladek, Divergence of the reduction to the multidimensional nilpotent Takens normal form,, Nonlinearity, 24 (2011), 3129. 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. 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. 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. 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. 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. 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. 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. doi: 10.1142/S0218127401002742.

[30]

P. Yu and A. Y. T. Leung, The simplest normal form of Hopf bifurcation,, Nonlinearity, 16 (2003), 277. 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. 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. doi: 10.1007/BF01159780.

[3]

A. Baider and J. A. Sanders, Unique normal forms: The nilpotent Hamiltonian case,, J. Differential Equations, 92 (1991), 282. 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. 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. 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. doi: 10.1142/S0218127402005741.

[7]

M. Gazor and F. Mokhtari, Normal forms of Hopf-Zero singularity,, preprint, ().

[8]

M. Gazor and F. Mokhtari, Volume-preserving normal forms of Hopf-Zero singularity,, Nonlinearity, 26 (2013), 2809. 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. 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. 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. 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. 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. 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), (2011).

[15]

J. Murdock, Asymptotic unfoldings of dynamical systems by normalizing beyond the normal form,, J. Differential Equations, 143 (1998), 151. doi: 10.1006/jdeq.1997.3368.

[16]

J. Murdock, Normal Forms and Unfoldings for Local Dynamical Systems,, Springer, (2003). doi: 10.1007/b97515.

[17]

J. Murdock, Hypernormal form theory: Foundations and algorithms,, J. Differential Equations, 205 (2004), 424. 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. 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. 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. 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).

[22]

E. Stróżyna and H. Żoladek, Divergence of the reduction to the multidimensional nilpotent Takens normal form,, Nonlinearity, 24 (2011), 3129. 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. 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. 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. 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. 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. 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. 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. doi: 10.1142/S0218127401002742.

[30]

P. Yu and A. Y. T. Leung, The simplest normal form of Hopf bifurcation,, Nonlinearity, 16 (2003), 277. doi: 10.1088/0951-7715/16/1/317.

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