Gevrey normal forms for nilpotent contact points of order two

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February 2014, 34(2): 677-688. doi: 10.3934/dcds.2014.34.677

Gevrey normal forms for nilpotent contact points of order two

P. De Maesschalck ^1,

1.	Hasselt University, Campus Diepenbeek, Agoralaan-Gebouw D, B-3590 Diepenbeek, Belgium

Received January 2013 Revised March 2013 Published August 2013

Abstract
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This paper deals with normal forms about contact points (`turning points') of nilpotent type that one frequently encounters in the study of planar slow-fast systems. In case the contact point of an analytic slow-fast vector field is of order two, we prove that the slow-fast vector field can locally be written as a slow-fast Liénard equation up to exponentially small error. The proof is based on the use of Gevrey asymptotics. Furthermore, for slow-fast jump points, we eliminate the exponentially small remainder.

Keywords: slow-fast vector field, Liénard system., Gevrey asymptotics, Singular perturbations, normal forms.

Mathematics Subject Classification: Primary: 34D14, 34A26; Secondary: 34M30, 34M6.

Citation: P. De Maesschalck. Gevrey normal forms for nilpotent contact points of order two. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 677-688. doi: 10.3934/dcds.2014.34.677

References:

[1]	P. Bonckaert and P. De Maesschalck, Gevrey and analytic local models for families of vector fields, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 377-400. doi: 10.3934/dcdsb.2008.10.377.
[2]	Éric Benoît, Perturbation singulière en dimension trois: Canards en un point pseudo-singulier nœud, Bull. Soc. Math. France, 129 (2001), 91-113.
[3]	Bernard Candelpergher, Francine Diener and Marc Diener, Retard à la bifurcation: Du local au global, In "Bifurcations of Planar Vector Fields" (Luminy, 1989), Lecture Notes in Math., 1455, Springer, Berlin, (1990), 1-19. doi: 10.1007/BFb0085388.
[4]	M. Canalis-Durand, J. P. Ramis, R. Schäfke and Y. Sibuya, Gevrey solutions of singularly perturbed differential equations, J. Reine Angew. Math., 518 (2000), 95-129. doi: 10.1515/crll.2000.008.
[5]	Mireille Canalis-Durand and Reinhard Schäfke, Divergence and summability of normal forms of systems of differential equations with nilpotent linear part, Ann. Fac. Sci. Toulouse Math. (6), 13 (2004), 493-513. doi: 10.5802/afst.1079.
[6]	P. De Maesschalck, F. Dumortier and R. Roussarie, Cyclicity of common slow-fast cycles, Indag. Math. (N.S.), 22 (2011), 165-206. doi: 10.1016/j.indag.2011.09.008.
[7]	Peter De Maesschalck and Nikola Popović, Gevrey properties of the asymptotic critical wave speed in a family of scalar reaction-diffusion equations, J. Math. Anal. Appl., 386 (2012), 542-558. doi: 10.1016/j.jmaa.2011.08.016.
[8]	Freddy Dumortier, Compactification and desingularization of spaces of polynomial Liénard equations, J. Differential Equations, 224 (2006), 296-313. doi: 10.1016/j.jde.2005.08.011.
[9]	A. Fruchard and R. Schäfke, Overstability and resonance, Ann. Inst. Fourier (Grenoble), 53 (2003), 227-264. doi: 10.5802/aif.1943.
[10]	Masaki Hibino, Borel summability of divergent solutions for singularly perturbed first-order ordinary differential equations, Tohoku Math. J. (2), 58 (2006), 237-258.
[11]	Gérard Iooss and Eric Lombardi, Normal forms with exponentially small remainder: application to homoclinic connections for the reversible $0^{2+}i\omega$ resonance, C. R. Math. Acad. Sci. Paris, 339 (2004), 831-838. doi: 10.1016/j.crma.2004.10.002.
[12]	Frank Loray, Réduction formelle des singularités cuspidales de champs de vecteurs analytiques, J. Differential Equations, 158 (1999), 152-173. doi: 10.1016/S0022-0396(99)80021-7.
[13]	Eric Lombardi and Laurent Stolovitch, Normal forms of analytic perturbations of quasihomogeneous vector fields: rigidity, invariant analytic sets and exponentially small approximation, Ann. Sci. Éc. Norm. Supér. (4), 43 (2010), 659-718.
[14]	Robert Roussarie, Putting a boundary to the space of Liénard equations, Discrete Contin. Dyn. Syst., 17 (2007), 441-448. doi: 10.3934/dcds.2007.17.441.
[15]	Reinhard Schäfke, Gevrey asymptotics in singular perturbations of ODE, in "International Conference on Differential Equations, Vol. 1, 2" (Berlin, 1999), World Sci. Publ., River Edge, NJ, (2000), 118-123.
[16]	Yasutaka Sibuya, The Gevrey asymptotics in the case of singular perturbations, J. Differential Equations, 165 (2000), 255-314. doi: 10.1006/jdeq.2000.3787.
[17]	Ewa Stróżyna and Henryk Żoładek, The analytic and formal normal form for the nilpotent singularity, J. Differential Equations, 179 (2002), 479-537. doi: 10.1006/jdeq.2001.4043.
[18]	Ewa Stróżyna and Henryk Żoładek, Orbital formal normal forms for general Bogdanov-Takens singularity, J. Differential Equations, 193 (2003), 239-259. doi: 10.1016/S0022-0396(03)00137-2.

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References:

[1]	P. Bonckaert and P. De Maesschalck, Gevrey and analytic local models for families of vector fields, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 377-400. doi: 10.3934/dcdsb.2008.10.377.
[2]	Éric Benoît, Perturbation singulière en dimension trois: Canards en un point pseudo-singulier nœud, Bull. Soc. Math. France, 129 (2001), 91-113.
[3]	Bernard Candelpergher, Francine Diener and Marc Diener, Retard à la bifurcation: Du local au global, In "Bifurcations of Planar Vector Fields" (Luminy, 1989), Lecture Notes in Math., 1455, Springer, Berlin, (1990), 1-19. doi: 10.1007/BFb0085388.
[4]	M. Canalis-Durand, J. P. Ramis, R. Schäfke and Y. Sibuya, Gevrey solutions of singularly perturbed differential equations, J. Reine Angew. Math., 518 (2000), 95-129. doi: 10.1515/crll.2000.008.
[5]	Mireille Canalis-Durand and Reinhard Schäfke, Divergence and summability of normal forms of systems of differential equations with nilpotent linear part, Ann. Fac. Sci. Toulouse Math. (6), 13 (2004), 493-513. doi: 10.5802/afst.1079.
[6]	P. De Maesschalck, F. Dumortier and R. Roussarie, Cyclicity of common slow-fast cycles, Indag. Math. (N.S.), 22 (2011), 165-206. doi: 10.1016/j.indag.2011.09.008.
[7]	Peter De Maesschalck and Nikola Popović, Gevrey properties of the asymptotic critical wave speed in a family of scalar reaction-diffusion equations, J. Math. Anal. Appl., 386 (2012), 542-558. doi: 10.1016/j.jmaa.2011.08.016.
[8]	Freddy Dumortier, Compactification and desingularization of spaces of polynomial Liénard equations, J. Differential Equations, 224 (2006), 296-313. doi: 10.1016/j.jde.2005.08.011.
[9]	A. Fruchard and R. Schäfke, Overstability and resonance, Ann. Inst. Fourier (Grenoble), 53 (2003), 227-264. doi: 10.5802/aif.1943.
[10]	Masaki Hibino, Borel summability of divergent solutions for singularly perturbed first-order ordinary differential equations, Tohoku Math. J. (2), 58 (2006), 237-258.
[11]	Gérard Iooss and Eric Lombardi, Normal forms with exponentially small remainder: application to homoclinic connections for the reversible $0^{2+}i\omega$ resonance, C. R. Math. Acad. Sci. Paris, 339 (2004), 831-838. doi: 10.1016/j.crma.2004.10.002.
[12]	Frank Loray, Réduction formelle des singularités cuspidales de champs de vecteurs analytiques, J. Differential Equations, 158 (1999), 152-173. doi: 10.1016/S0022-0396(99)80021-7.
[13]	Eric Lombardi and Laurent Stolovitch, Normal forms of analytic perturbations of quasihomogeneous vector fields: rigidity, invariant analytic sets and exponentially small approximation, Ann. Sci. Éc. Norm. Supér. (4), 43 (2010), 659-718.
[14]	Robert Roussarie, Putting a boundary to the space of Liénard equations, Discrete Contin. Dyn. Syst., 17 (2007), 441-448. doi: 10.3934/dcds.2007.17.441.
[15]	Reinhard Schäfke, Gevrey asymptotics in singular perturbations of ODE, in "International Conference on Differential Equations, Vol. 1, 2" (Berlin, 1999), World Sci. Publ., River Edge, NJ, (2000), 118-123.
[16]	Yasutaka Sibuya, The Gevrey asymptotics in the case of singular perturbations, J. Differential Equations, 165 (2000), 255-314. doi: 10.1006/jdeq.2000.3787.
[17]	Ewa Stróżyna and Henryk Żoładek, The analytic and formal normal form for the nilpotent singularity, J. Differential Equations, 179 (2002), 479-537. doi: 10.1006/jdeq.2001.4043.
[18]	Ewa Stróżyna and Henryk Żoładek, Orbital formal normal forms for general Bogdanov-Takens singularity, J. Differential Equations, 193 (2003), 239-259. doi: 10.1016/S0022-0396(03)00137-2.

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